Handbook of special functions: derivatives, integrals, series and other formulas

H A N D B O O K O F Special Functions Derivatives, Integrals, Series and Other Formulas H A N D B O O K O F Speci...

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H A N D B O O K

O F

Special Functions Derivatives, Integrals, Series and Other Formulas

H A N D B O O K

O F

Special Functions Derivatives, Integrals, Series and Other Formulas

Yury A. Brychkov Computing Center of the Russian Academy of Sciences Moscow, Russia

Chapman & Hall/CRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2008 by Taylor & Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-13: 978-1-58488-956-4 (Hardcover) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http:// www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

Contents

The Derivatives # " # $!%" & ' & !! ( ( )* ' !!"!! '' **, + + * *, " " & +**,!"! & ( # +**,!"! ( ' +**,!"! . & +**,!"! . . ! ! ( +* *, " . !!"!! **, -- + " " , * * + ! ! / , " * * / + ! ! 0 +* *, " # 1

1.1.

Elementary Functions

1.2.

The Hurwitz Zeta Function ζ (ν, z )

1.3.

The Exponential Integral Ei (z )

1.4.

The Sine si (z ) and Cosine ci (z ) Integrals

1.5.

The Error Functions erf (z ) and erfc (z )

1.6.

The Fresnel Integrals S (z ) and C (z )

1.7.

The Generalized Fresnel Integrals S (z, ν ) and C (z, ν )

1.8.

The Incomplete Gamma Functions γ (ν, z ) and Γ(ν, z )

1.9.

The Parabolic Cylinder Function Dν (z )

!!"!! ****, .. + + , " 0 ! ! , " * * + +**,!"!0 !!"!! ****, + + , " 0 !!"!! **, + 0 , " * * + !!"!! ****, ## + + , " 0 !!"!! ****, '' + + , " 0 !!"!! ****, && + + , " 0 !!"!! ****, (( + 0 + , " - +**,!"! / +**,!"! . +**,!"! !!"!! ****, + + , " " !!"!! ****, + + , " " !!"! ****, + + , "" 1

1.10.

The Bessel Function Jν (z )

1.11.

The Bessel Function Yν (z )

1.12.

The Hankel Functions Hν(1) (z ) and Hν(2) (z )

1.13.

The Modified Bessel Function Iν (z )

1.14.

The Macdonald Function Kν (z )

1.15.

The Struve Functions Hν (z ) and Lν (z )

1.16.

The Anger Jν (z ) and Weber Eν (z ) Functions

1.17.

The Kelvin Functions berν (z ), beiν (z ), kerν (z ) and keiν (z )

1.18.

The Legendre Polynomials Pn (z )

1.19.

The Chebyshev Polynomials Tn (z ) and Un (z )

1.20.

The Hermite Polynomials Hn (z )

1.21.

The Laguerre Polynomials Lλn (z )

1.22.

The Gegenbauer Polynomials Cnλ (z )

1.23.

The Jacobi Polynomials Pn(ρ,σ ) (z )

' ' / / ' & &. # # # #' #' ##( #-#' '& '& '/ '/ & & & &&# &# &# && && && &/

/ & ! ! # +* *, " &./ (. ! ! ' , " * * ' + +**,!"" (( ( ! ! ' **, " && + ( ! + * *, " " ( (& ! ! , " * * (( + (& +**,!"" -. (( - +**,!"! -. -. ! ! / -. **, " / + ! + * *, " " --' !!"! --&& **, .. + + * *, "" ( /' /' /' !!0 0

! ** 0 /' 0"% /' # !! % 0 //&& ' ! !% !*" & $"% /( ( !"% /( - !"%! //. !"% / $%" // " .. !! . 0

# . !## # . 1

1.24.

The Complete Elliptic Integrals K (z ), E (z ) and D (z )

1.25.

The Legendre Function Pνµ (z )

1.26.

The Kummer Confluent Hypergeometric Function

1 F 1 (a ; b ; z )

1.27.

The Tricomi Confluent Hypergeometric Function Ψ(a; b; z )

1.28.

The Whittaker Functions Mµ,ν (z ) and W µ,ν (z )

1.29.

The Gauss Hypergeometric Function 2 F1 (a, b; c; z )

1.30.

The Generalized Hypergeometric Function p Fq ((ap ); (bq ); z )

Limits

2.1.

Special Functions

H

Indefinite Integrals

3.1.

Elementary Functions

3.2.

Special Functions

L

!!*00 .' # % .' ./ " ## ! & ### $%" ( ! ! ' '' ##& )* 0 ( ## ))0 ( / ( 0 ! ! -. ### ))0* -. -- 0 ## ))0 - 00!! - ### )) 0 * 0 - ' # )" --# 0 0! --'## #### )) 0 " ! -& ! ##### )) 0 0* ( ##' ) ---( 0 0 " ##& ) -/

00 -/ ##'' )) 0! ! /. ' / ##'# )) 0 / ' ' 0 / ##'& )) 0!! / / 00! / ##&& )) 0" / / #& ) 1 H

L

Definite Integrals

4.1.

Elementary Functions

4.2.

The Dilogarithm Li2 (z )

4.3.

The Sine Si (z ) and Cosine ci (z ) Integrals

4.4.

The Error Functions erf (z ), erfi (z ) and erfc (z )

4.5.

The Fresnel Integrals S (z ) and C (z )

4.6.

The Incomplete Gamma Function γ (ν, z )

0!! /' #&#' ) 0 0 /' #& ) #&& ) " /' /& 0 /& #( ) 00!" //( #( ) ! ! //( #( ) 0 // #(#' ) 0 0 .. * #( ) #(& )"0 . #(( ) . 0 .## #-- ) .# # ) 0 .' 0 .' #// ) 0!" .& #/ ) 00! ! .#/ ) 0 . #/#' ) 0* #/ ) " 0 #/& )" # ( ) & 0

. 0 & #. ) # ) " & ( 0 ( # ) 0! / %" # ) 0 / 0! ! . # ) 0 ##' ) * # )

0 # ) 0 0 # )" 0 ## ) 00! ! ## ) & ## )

4.7.

The Bessel Function Jν (z )

4.8.

The Bessel Function Yν (z )

4.9.

The Modified Bessel Function Iν (z )

4.10.

The Macdonald Function Kν (z )

4.11.

The Struve Functions Hν (z ) and Lν (z ) H L H H L H L H L

4.12.

The Kelvin Functions berν (z ), beiν (z ), kerν (z ) and keiν (z )

4.13.

The Airy Functions Ai (z ) and Bi (z )

4.14.

The Legendre Polynomials Pn (z )

0 0 #####' )) " # 0 ' 0 ##' )) # #' )0" #& 00 # ##&& )) # # 0

##&&# ))"0 #& #& 00! #( ##(( )) -/ " #( # 0 0 '# ##((# )) 0

' ##((& ))"0 ' ' '# 0 '# #-- ) 0 '' #- ) 0 '& #- ) ## ) "0 '& '( 0 '( #// ) 0 '( #/ ) # ) "0 & & ! !0 & #.. ) !0 & #. ) 0! & #. ) # # ) "0 &#' !0! 0 &&' ## )) " %" &( 0 ! ! ### )) 0* &( #' ) 0 (#

4.15.

The Chebyshev Polynomials Tn (z )

4.16.

The Chebyshev Polynomials Un (z )

4.17.

The Hermite Polynomials Hn (z )

4.18.

The Laguerre Polynomials Lλn (z )

4.19.

The Gegenbauer Polynomials Cnλ (z )

4.20.

The Jacobi Polynomials Pn(ρ,σ ) (z )

4.21.

The Complete Elliptic Integral K (z ) K K K K K

0 (# #& ) 0 (' #-( ) 0 (' #/ ) 0 (' # . ) 0 (& #) 0 (( #) 0 0 (#) # ) " (-/ 0 ((/ # ) !"!%"0 # ) 0 ! ! - & # ) 0* ##' ) 0 -/ # ) 0 / #& ) 0 / #-( ) 0 / #/ ) 0 / # . ) 0 / # # #) 0 / & #) 0 / / #)"00 /& //& #)"0 # #) .. 0% .. # ) . # ) "0 0 . 0 . ## ) 0 .# ## ) 0!! .& ## ) 0 .& ###' ) # # ) "0 .& ./ ./ ' ./ ' " 0 ' K K K K K K

H

L

K

K

4.22.

The Complete Elliptic Integral E (z ) E E E E E E E E E E E

H

L

E

E E

4.23.

4.24.

K

The Complete Elliptic Integral D (z ) D D K

The Generalized Hypergeometric Function p Fq ((ap ); (bq ); z )

K

Finite Sums

5.1.

E

The Psi Function ψ (z )

E

'' 0 " ' # ' 0 # ' " # ' ''# 0 0 '' "0 '## " ( ''''

- 0

" '' "0 / / '& 0 / . ''( 0 . ' 0 " '(( " ' '(# & ''(' 0 (& " / ''-- 0 / "

''-- 0 '-#' ' " '-& ' ( ''// 0 0 " #.( '/ # ''//#' " ## '/& "0 ## ## '. ##

5.2.

The Incomplete Gamma Functions γ (ν, z ) and Γ(ν, z )

5.3.

The Bessel Function Jν (z )

5.4.

The Modified Bessel Function Iν (z )

5.5.

The Macdonald Function Kν (z )

5.6.

The Struve Functions Hν (z ) and Lν (z ) H L

5.7.

The Legendre Polynomials Pn (z )

5.8.

The Chebyshev Polynomials Tn (z ) and Un (z )

5.9.

The Hermite Polynomials Hn (z )

5.10.

The Laguerre Polynomials Lλn (z )

#& '. #'. 0" '' '.# '( '.' "0 '/ '.& 0" & '.( & '.- "0 & & ' ' &' ' &' 0" ( '# '' "0 ( - '& '( 0" -'- "0 /. / ' !! / ' / ' !0" #. '# "0 ! #.' '' ! #.' '& ! 0" #.'( "0 ! #. ' # # # ''# # 00" '## # " # # '''' # 0 " ## # '& # ##

5.11.

The Gegenbauer Polynomials Cnλ (z )

5.12.

The Jacobi Polynomials Pn(ρ,σ ) (z )

5.13.

The Legendre Function Pνµ (z )

5.14.

The Kummer Confluent Hypergeometric Function

1 F 1 (a ; b ; z )

5.15.

The Tricomi Confluent Hypergeometric Function Ψ(a; b; z )

5.16.

The Gauss Hypergeometric Function 2 F1 (a, b; c; z )

'& 00" ' '& " # #('(

0 # # # '( '(

" # 0 '(# " ## '(' "0 ' '(& ##' #& ''-- # & ! " % #( #/ / # / && # ! / " # ! . # &&# %" # # & 0 # 0 && ##( ##( " &# ##'/ # & #''

# ##'' &# #0 #' &# #' &&### "0# # #'

#

#00 ##' &&'' #'' # #

0

' # # # #'&' 0 & # "

#0 # #'/ #'/

&& "0 #'/. && #& 1

5.17.

The Generalized Hypergeometric Function p Fq ((ap ); (bq ); z )

5.18.

Multiple Sums

Infinite Series

6.1.

Elementary Functions

6.2.

The Psi Function ψ (z )

6.3.

The Hurwitz Zeta Function ζ (s, z )

6.4.

The Sine Si (z ) and Cosine ci (z ) Integrals

6.5.

The Fresnel Integrals S (x) and C (x)

6.6.

The Incomplete Gamma Function γ (ν, z )

. #& 0 &( % #&. #& &-- #& , &- ! #&' &- #& &-#' 0 #&& &- # #&( &-& #0 #&( &--( "0 # #(.. &-/ "0 #0 #( &- . #0 # #( &- # #0 # #( &- #( &- #( #(# &- & # #('' #(' / &/ 0 #( &/ "0 #(& &/ #(( -. 0 &/#' " # & #-. - # 0 #- &.. # #- &.

0 #- &. ## &.#' # 0 # #0 # #--# &. #0 # #-'# &.& "0 # #-' # &( -& # 0 #--& & #& & 0 " & #- #-&# -/ # 0 #-/ & #-/ & 1

6.7.

The Parabolic Cylinder Function Dν (z )

6.8.

The Bessel Functions Jν (z ) and Yν (z )

6.9.

The Modified Bessel Function Iν (z )

6.10.

The Struve Functions Hν (z ) and Lν (z ) H L H H H H H H

6.11.

The Legendre Polynomials Pn (z )

6.12.

The Chebyshev Polynomials Tk (z ) and Uk (z )

0 #/. & #/. &# / # 0 / && # 0 " #/

0 #/ & #//# " &#' 0

" ## & /' # #/' &# #/& 0" &# "0 #/( &# "0 # #/( &##' #/( &# 0" #/ &#& 0 &#( " #/// # #// &'' '.. 0" &' 0 " '. &' '. &'#' & 0" '.# '.' ! && ! 0 '.' '.& " && 0 && "! ! '. ( '. &&#' ! 0 '.( && " ( '.

'.&&(( ' 0

0 ' " &&((# "0 '#( # '( &&((&' '

0" ' &(-( " 0 '& &( 1

6.13.

Hermite Polynomials Hn (z )

6.14.

The Laguerre Polynomials Lλn (z )

6.15.

The Gegenbauer Polynomials Cnλ (z )

6.16.

The Jacobi Polynomials Pn(ρ,σ ) (z )

6.17.

The Generalized Hypergeometric Function p Fq ((ap ); (bq ); z )

' ( ' '( ( ' ( '( ! " (( !" 0 '-( %0! ! ' ( !!" 0 '/ 0 ((#' ! '# * ! 0 ''#' ((&( ! % 0 # !

* ((/ !0"% 0 ''#& / # ! ! ! 0 %" *" '''#/ % ((. !$ % % '' " ( !! "%! '' ( ! ''' "% ((#' !"% ! % '' ! " " ''/ ((&( ! " " '&. ! 0 (- '& ''& '& -- !!!!%" & '-/ - !!%" &. --#' !!!!%" & %" &-/ --& !!!!%" %" & & --( !!%" & %"

!%" & --/. !! ! %" & -- !!!!%" &' &/ - !!%" &/ %" --#' !!!!%"

& & %" -& !!%" & 1 The Connection Formulas

7.1.

Elementary Functions

7.2.

Special Functions

H

J

L

E

K E

Representations of Hypergeometric Functions and of the Meijer G Function

8.1.

The Hypergeometric Functions

---( !!!!%" & & %" --/. !!!!%" & &'# -- !!!!%" & %" && %" - ! 0! %" &( & & -- &/ ' -- % &&' " &'' &&/ 0 &(/ ))00 % &( (a p ) (b q )

8.2.

The Meijer Function Gm,n z p,q

8.3.

Representation in Terms of Hypergeometric Functions

1

Preface

$!% &'(!#%)!"* +" ,&")-.!%&!"-)/" * !"#! ,& )!-$,.))!%<)-= &"45%/ $6(!# $%(#)!"%" "4%#!&' 710;23 ) + 0 8

7

3 9 ; 78 : 1023 > .!%&"

@

A 7 ! $

5" ?

8

8

B

>

9 C

7

"

7

9

7

EMENNN B

F K L

G H I J DE E E E E E

Chapter 1

The Derivatives 1.1. Elementary Functions 1.1.1. General formulas 1.

2.

3. 4. 5.

6.

! '#" " . ')(* $ ! % ,+ &-

¤ & $ %

'65 $ (87! % . & -21 0/ 0 43 ¢ $9% ')( $ (:7! % £ / -;&- <+ &- / 7 "" ' . 7 & -21 ' ¢ ¢ I G H ) ' ( : ( 7 ! $ % ,+ &- F G ED EF 3 3 KJ $L 7 7 EDM &-21 EF GIH 3 ¢ -;&-21ON F G < PQ , & N & ¤ 1 & R 0& ¤¤4¤ ST0& U X[<\]W ')( $ Y &- -< Z Z Z [<\#U ^ 7 EDM SEa F G S D S -21Ab

2. 3. 4.

= '8>[email protected]

U4. & - U ')( $ XW4. ' + & + 0& ¤¤4¤ $ $ L V V L

U4 J 1 XW4 J (_BB B<( $ ( " [<\#U . U X[<\#U4. ¤ $( " & ` S + 2- 1 & R S + &- -
1.1.2. Algebraic functions 1.

= '8>[email protected]

&c# 43 ¢ 43Pd2 & c -; ¤ . &e< fg3h& i f & e ;- 0fg3h& 0i -;j +ke -;9l i -; F . &e< fg3h& i &e0fg3h& i -; j + - e - i -21Rl i -; F ¢ . m &e< fg3h& i &e -; 0fg3h& i j +ke -;9lC- e - i -21 F m

¢ 3 O G ¤ m m ¤ 3 :G 5n ¤ (* G

5.

6. 7.

8.

( &! ' m c f & c F G c -; 0f 1 ( L' J

3 ¢ 0 fI& c -; - c F O 5 m 5 m -21 0 f & -21 m 5 £ 3 ¢ a 1 f -21 f & -;&-21 ¢ 3 F G - c -21 ] fg3h& c £ -

9.

c

10.

11.

12.

13.

14.

15.

16.

17.

18.

,

& c -; c -; a 1 F ¢ m G ¤ ¤ G

. 5nO + a 1 a F m 5 G! ¤ m 1

'! % m ¤ a (1 L ' - c -21 0fg3h& c -; c -; 1 F#" G J

5 m fg3h& &- c -21 4 3 ¢ f

c -; 0fg3h& - c -21$ - c F m G ¤ &- c -21 0f 7 & c (9! ' m ¤ F3 G ¢ a c ; 8G f & ( L ' &- c -21 c -; 1 F 1 % J

- c -!&4 0fg3h& c &L ' m ¤ 3 ' % a ( 6J ( 1 L '%& f -21 O - c -21 fg3h& c -; c -;a 1 1 F#" G ' J

1L'

( f -21 0fg3h& c -; c -; a 1 F " m G ¤ &-21 0fg3h& c J 1 J L' a 1 0f & c '65_7! % 3 £ - &-21 ( 5_ 7X! '%& $ fI c a 1 - c a -;&-21 5 m-, ¤ 1 )+* '! % &-21 0f & c £ - ( 5 7! ' -21 f- & c. - c -; )+* m 5 , ¤ m 5 . 0f & -21 f 0f & - + a 1 j F m 5 m G ¤ f & &-21 f f & -21 j F " ¢ m G ¤ 0 f & a 1 f a 1 j a 1 F " ¢ m G ¤ 0 m f & -;&-21 0 f & -;&-21 j ) * m 5 , ¤ 0 ,

19. 20. 21. 22. 23.

24.

25. 26. 27. 28. 29. 30. 31. 32. 33.

34. 35.

0f & -;&-!&4 f -21 ] f- & -;&-21Oj 0 f & f j F ¢ m G ¤ -;&-21 0 f & 3 ¢ -;&-21 j F ¢

m 5 , ¤ a m

1 )+* m ¤ G

5m : , ¤ &-21 fh& -;&-21 43 ¢ -21 0f h & -;&-21 j )+* &-21 0 f & 3PfI -21 j F " m G ¤ &L ' a 1 0 f & a 1 'gJ 5 7 f 1 0 f & 1 F ¢ m G ¤ L' #J 'Q& 5_ 7! f a 1 1 a 1 F " ¢ m G ¤ 7 &-21 0f & a 1 F G f a 1 -21 a 1 F#" ¢ m G ¤ 7 a 1 0f & &-21 F G f 1 0f & -21 F " ¢ m G ¤ 7 ¢ a 1 0fg3h& &-21 43 F G f a 1 fg3h& -21 a 1 F#" m G ¤ 7 &-21 0 f & &-21 F G f -21 f & -21 #F " ¢ m G ¤ 7 m &-21 0 f & -; F G -21 f & -; )+* m 5 , ¤ 7 m a 1 0 f & -;&-21 F G 1 f & -;&-21 )+* m 5n , ¤ 43 ¢ F 7 G f & -;&-21 a 1 ) * m 5 , ¤ &-21 0 f & -;&-21 m F 7 G fI& -21 0 f & -;&-21 a 1 ) * m 5 , ¤ 43 ¢ F 7 G -21 0 f h -;&-21 )+* m 5: , ¤ a 1 0 f & -;&- ( m ¤ 43 ¢ £ - O'6'65 5_77X! ! % % 1

; & f & F 5 m G 3

36. 37. 38.

39. 40. 41.

42. 43. 44. 45. 46. 47. 48.

49.

L ' 1

m #J 'Q& 5_ 7! F m G f & -;&-!&4 a 1 5n , ¤ m +) * ( ' 9 ! ' 0f9 3h

c 3 £ fI ' ( &! ' 0f9 3h c -; c -; a 1 F m G ¤ 0f9 3h

-21 ( m a . f -21 fg3h& -;&-21 ¢ 3 F E 5 m G + 1 a 1 F ( m G ¤ m

0f9 3h

-21 £ f -21 f9 3h O -;&-21 a 1 F m (8 G ¤ m

a 1 0f9 3h

-21 £ ¢ f -21 ; f9 3h O -;&-!&4 a 1 F m (8 G ¤ -21 0f & -21 43 ¢ - + a 1 . 0f & - + a 1 . j F 5 m m 5 . 0f9

-21 R 43 ¢ f9 - + a 1 j F m 5 G ¤ 0f9 3h

43 £ fI j F m G ¤ 7 0f9 3h

&-21 R 43 £ fI F G f9 3h

-21 F m G ¤ 7 0fIQ3h

&-21 R 43fI F G 0fIQ3h

-21 F" m G ¤ ( m ! ' a 0f 3h 1 '65_7 F G 0f 3h 1 F m G ¤ m £ -;&-21 0f9 3h

43 fI -;&-21 j F G ¤ &- c -21 0 3hf9

c ('9! ' £ fI ')( &! &- c -21 0 3hf c -; c -; a 1 F ' 7 5 m F $ f G 1

3 ¢ F 7 G 0 f9

- + a 1 . F O'6 5_7 m

1.1.3. The exponential function 1. 2.

3

k c - c -; - c -; 0fI& ¤ m c -R 43 ¢ c -; -R - c -21 F G ¤ 5

G ¤

m ¤ G ¤ m G

W W 3A f R/ f& ¤ 3. ED H W W m f

-;&-21 R F G ¤ 4. EDM &-21 R H W W ¢ H 5. ED 3 f - / f& ¤ - W W m 6. EDM &-21 #-R H f -;&-21 #-R F G ¤ m a . 7. / 8F G 1 + 1A- &-21 f;/ 1 -; 0f;/ ¤ 7 m a . 8. D H / *F G 1 - + 1 a a 1 f;/ -;&-21 f;/ 0 ¤ 9. k &-21 R m m ¤ 3 ¢ / *F mG a 1 - + a . D G F

F &-21

1 -; GIH 10. k &-21 R m 43 ¢ / F m G a 1 - + & a . D a F m G ¤ F 1

-;&-21 GIH m']" ' & . ¤ ¢ 3 11. - I+ 1A- &-21 f / 7 m'#" '% & ¤ ¢ a . 43 12. D - H - + 1 a 1 0f / m ¤ ']m " '% & a . 13. &-21 -R - + &-21 F G m ¤ ']m " '% & a . 14. k -;&-21 #-R - + & a 1 F G m a . U . . 15. D + -21 3 ¢ a 1 R/ *F G 1 + 1A- + f;/ H 1 -; = 7@CB m a 1

. . ¢ G +

16. 43 / 8F 1A- + 0f;/ = 7][email protected] & 2 1 7 . U 17. D + -21 H 3 ¢ a a 1 R/ *F m G a 1 - + a 1 . + . 0f;/ = 7@CB a 1

/ F m G a 1 - + a 1 . + . 0f / = 7][email protected] 18. ; -;&-21

5

1.1.4. Hyperbolic functions 1.

&c

2.

c

' % fI& 0 &c -; ' % fI& 0 c -;

c ;- 4 3fI& <3 #- c -; 4 3fI& -

¤ c ;- fI& ¤ c -; fI&

fI& 0 ")" $ 5 7! " £ 3fI ¢ S $ L ¤ " 5_7! 3 "'& 8 4 3 J S W W ( ! ' ( 7! ' f R/ f & 3 f - 0/ f& ¤ 4. 0fI

W W ( ! ' ( 7X! ' f R/ f& f V- 0/ f& ¤ 5. fI 0 / 8F mG a 1 I+ 1A- . ¤ 6. 0f;/ 0 1 -; f;/ / 8F m G a 1 I+ 1A- . ¤ 7. f;/ 0 &-21 f;/ ! m m a . H / 8F G 1 - + 1 a a 1 f;/ ¤ 8. D ! m m a . H / *F G 1 - + 1 a -;&-21 0f / ¤ 9. ED a ¢ 10. / 3 ?F 5 G 1 7 ( &B B B ( 5 &B BB 5 #$ & 1 1 1 1 ¤ Y a N a

& 1 & ( &B BB & ( & 5 &B BB & 5 "! ( 7

'

% ' ] ( X 7 ! ' % % ¢ H 43 a 1 '%& '%& 11. ED #$ 7V( &B B B V( &B BB ¤ Y a N a

& 7 ( 9BB B #7( #7,5 &B B B #7<5 ! ( 7 & a H £ 43 ¢ F 5 G 1 12. D 7 ( &B BB ( 5 9BB B 5 #$ 1 1 1 1 ¤ Y a N a

& & ( 9BB B & ( & 5 &B B B & 5 "! 7

3.

:

' % ED H 4 3 ¢ a 1 '%& #$ 7 ( & B B B ( &B B B # ( 7! ' ' % ' & a & N a 7 ( 9 BB B #7( # 7,5 &B B B #7<5 7 ¤ ! m m ¤ 3 ¢ / F m G a 1 - + a . 14. D &-21 F GIH F 1 -; G m m ¤ 3 ¢ / *F m G a 1 - + a . 15. ED &-21 F GIH F &-21 G m 3 ¢ / F m G a 1 - + & a . a F m G ¤ 16. D &-21 F GIH 1 m m ¤ 3 ¢ / F m G a 1 - + & a . 17. ED &-21 F GIH F -;&-21 G 18. 0f / '%& f a 1

I+ 1A- . & -21 $&% '6')5 ( $ $ ! ! 43fI - - a 0f / -; 1

= '*>[email protected] 13.

0 f / ' & f a 1

I+ 1A- . & -21 $9% '6 ')5 ( $ $ ! ! 43fI - - & - -21 f / = '*>[email protected] m ! 20. ' & f a 1 - + a 1 . & -21 $9% '6')5 ( $ $ ! ! 3PfI - - a f / &- 1

= '*>[email protected] m ! 21. '%& f a 1 - + a 1 . & -21 $&% '6' 5 ( $ $ ! ! 43fI - - -;&-21 0f / = '*>[email protected] 19.

1.1.5. Trigonometric functions 1.

0fI& f

F fI

' ¤ G <

2. 3.

4.

5.

6.

7.

8.

9.

0 fI& f

F fI

;

' ¤ G

. 0fI& 4 3 ¢ + a 1 f $ £ Y S J L 3 " 3 ¢ + a 1 . f ( 7X" ! 1 (* . S $ +

¢ Y

5 L 3 J

' 5_7 ( 7 ! " 6 " J $ 5 7 L 1 7 ( D 3 £ fIQ3

a

1 fI& ( 7! ' H

= = @ C! @CB

'65_7 $ £ 5_7 L SEa 1 fI& S J L 3 J$ ' a

! = @O !#B

fI&

. 7 " + - ¢ S fI& S 43 J Y SEa 0fI& 43 ¢ $ L £L ¢ J " $ 5_7X! £ AfI ¢ S 5 7! 3 "'& 8 S 3

$

3 ¢ + a 1 . f

5 L

'

= @ ! #B !

$ J L

= = @ [email protected]

' 3 £ a 1 m 5 ' % ! %' & m' m m . 5n . 5n m ! '%& D + F G 3 + F 9G H ( m 5 m ¤ m ' '%& . * . n ¢ 3 43 ! D + F G 3 + F 3 IG H

fI& 0 " 5_7 3 ¢ f ( 7" ! '6 5_7 L J $ S . S 1 $ (* Y + - - 3 ¢ 5 L a J 3 ¢ f fI& 7 " 43 S 1 . $ +

a ¢ Y P5 L

3 J L

J

$

0fI&

L 3 £

SEa 1 fI&

'

!

= @ C7X O ! B

¢ S $ L £ ¢ J ' 0fI& ! = @ C7 O ! B

m ' %' & . 5 m . m 10. D + F G 3 + F G9H ! m '' & . ( m . m ¢ G 3 + F3 G9H ¤ 3 43 ! D + F $"'& " m ]! ¢ £ a 43 1 fI 11. 0fI& 7( ( 7! 1 ' $ ¢ Y D 3 4fI$ H 3 ¢ S J L = = @ C7 [email protected] S 1 $ 7" ¢ £ a 43 4 1 fI fI& 3 ¢ S J L 12. 1 S 1 a . 8 ( 7 $ ' @ ! B Y + V- 3 ¢ Q( Q5_7 L - a 1 fI& = ! J 43 ¢ a 1 ' ' % & m

m ' '&

m ' '&

. 7 m . 7 m D + F G 3 3 ¢ + F 3 IG H ¤ $ 7" . £ ¢ 14. fI& 0 43 + &-21 C fI fI& 3 ¢ S J L S 1 a . ' (:7 Y + V- 43 ¢ Q$ ( 5_7 L - a 1 fI& ! = @ ! B J

13.

15.

16.

c

17.

c

. m . D 3 ¢ + F]3 G 3 + ' % fI& 0 c -; c -; 43 ' % 0fI& c -; c -; 43

' fI

0 ( 7X! f + &- . D W & 19. 0fI'

( 7X! f D W & m 20. ED &-21 H 3PfI -;&-21 F

18.

21.

D &-21

m H 3PfI -;&-21

P

/ fE3

m

m F

m F ¢ G9H ¤ AfI& 3 - AfI& -

/ f ' ¤ G ' ¤ G

c -; AfI& ¤

c -; AfI& ¤

-

W

W

/ f H ¤

/ f H ¤

;

m a . f / 0 / *F G 1 I + 1A- 1 -; f / ¤ 3 ¢ / *F m G a 1 I+ 1A- . ¤ 23. 0f;/ &-21 0f;/ S 24. 0f;/ < ' & S - + 1 a + -21 . [ .

m 43 ¢ S ( &(']" '#" ¢ $ L 3 3 8 J 7 ( ( 7X! ( £) a ) £ Y ( 3 1 1 -; 4 3 4f;/ , 7 h 5 ( 7! £! a > [email protected] ¢ 43 3 1 &-21 3 £! 4f;/ = ? S 25. 0f;/ ' & S £) a ¤ £) 43 ¢ ( & '% & m '#" 3 8 J $ L 3 1 &-21 4 3 4 f / 22.

26. 27. 28.

29.

30.

31.

m 2! m a . H 3 ¢ / *F G 1 - + 1 a a 1 0f / ¤ ! m m a . ED H / *F G 1 - + 1 a -;&-21 f;/ ¤ a / ?F ( G 1 7 ( &B BB ( 5 9BB B 5 #$ & 1 1 1 1 Y a N a

& 1 & ( &B BB & ( & 5 &B B B & 5 ! ( 7

' %

% ] ( 7X ! ' ' %

¢ a D H 3 1 ' & '%& 7 V( &B BB V( &B B B ( 7 #$ ! Y a N a

& 7 ( &B BB #7( #7<5 &B B B #7<5 a £ ED H #F ( G 1 7 ( 9BB B ( 5 &B B B 5 #$ 1 1 1 1 Y a N a

( &

B

B B ( 5 &

B B B 5 7

& & & & & !

£ ¢ D H 43 -;&-21 7 V( 9BB B V( &B B B 7 #$ ! ¢ '%' & % Y a N a 3 4 3

& 7 ( &B BB #7( #7<5 &B B B V7<5 ED

N

¤

¤

¤

¤

32.

ED &-21

33.

ED &-21

34.

D &-21

35.

ED &-21

m ¢ m a m ¤ . H 4 3 / 8F G 1 - + a 1 -; F G

m m a m ¤ . H / 8F G 1 - + a & a 1 F G

m m a m ¤ a . H / F G 1 - + &-21 F G

m ¢ m a m ¤ . H 4 3 / 8F G 1 - + a & -;&-21 F G

. f / 0 £ - &-21 O/ f a 1

I+ 1A- '65 ! Y & -21 43fI - $9% ')( $ $ ! - a -; 1

. 0f / 3 ¢ £ - &-21 / f a 1

I+ 1A- '65 ! Y & -21 f - $9% ')( $ $ ! - &- -21

m ! 43 ¢ £ - &-21 / f a 1 O - + 1 a . '65 ! Y & -21 f - $9% ')( $ $ ! - a &- 1

m ! £ - &-21 / f a 1 - + 1 a . '65 ! Y & -21 43 ¢ f - $&% ')( $ $ ! - -;&-21

36.

f / = '8>[email protected] B

37.

f / = '8>[email protected]

38.

f / = '8>[email protected]

39.

0f / = '8>[email protected] B

H a 1;D H a D H a & D H D H a 1 D H

41.

42.

43.

44.

45.

RED

40.

R R R R R

3#

3#

643 ¢ £ a 1

3V

3 ¢ a 1 £ a 1

3V

¤

¤

¤

& ¤

& ¤

& ¤

R a & D

47.

R a D

46.

48.

49.

50.

51.

H 643 ¢ £ a 1

:

H 3 ¢ a 1 £ a 1

¤

a 1

& ¤

m . 0f;/ ¢ / 8 G F I +

A 1 f;/ Q ' 5 ! £ £ <1 f / ' <1 f / H ¢ a 1 / *F m G a 1 I+ 1A- . 0f;/ f;/ O'65 ! f;/ £ O'65 ! £ D f / K3 f / H 2 1 2 1 7 ¢ / *F m G a 1 - + a 1 . 0f;/ f;/ f;/ O'65 ! Q ' 5 ! £ ' / D

f f / £ H 2 <1

2<1

7 f;/ 0f / f;/ ¢ a 1 / *F m G a 1 - + a 1 . O'65 ! 'Q5 ! £ £ D f / E3 f / H 1 1

f;/ O'65 ! Y D Y Y Y

¤

¤

¤

¤

1.1.6. The logarithmic function 1.

2.

/ / f 7X! % 5 m 43 ¢ & -21 ')(: -; f9 -; j & -21 5 m , ) k &-21 / f / f 5n m ')O(8 7! % 3 ')O(8 7! % 5 m j F m m 5 m G &-21

3.

D F $

4.

D &-21

5.

f G9H 43 ¢ & -21 3 ¢ 0 9f O ;- j F &-21 5 m G

= '*>[email protected] = '*>[email protected]

F f f GIH ) ' (87! % ) ' (:7X! % m f f9

-; j & -21 F 5 m G 3 m m 5

ED m (8
-;&-21 &2- 1 F m * ( G

= '*>[email protected]

!,

= '*>[email protected] = '8>[email protected]

m 5 m (

13. ED 1 PED -21 0f 3h& &-21 m ( HOH 43 ¢ F 7 G f f 3h& -;&-21 m 5 14. ED &-21 PED f 3h& &-21 m ( HOH 43 ¢ F 7 G f -21 f 3h& -;&-21 ¢ ¢ ¡ 15. &-21 3hfI& 3hfI& 0 ¢ ¢ ¡ 16. 3hfI& &-21 3hfI& 0 ¢ ¢ a ¢ 17. 1 3hfI -21 -21 3hfI 3hfI& 43 ¢ f ¢ 3hfI& -;&-21 ¢ 4

f ¢ 3hfI& ¤ a ¢ 18. 1 3hfI -21 3hfI& 0 6.

a 1;D m 5 D m (

1.1.7. Inverse trigonometric functions 1.

= '*>[email protected] = '*>[email protected] = '8>[email protected] = '8>[email protected] m 5 ¤ m (

m 5 ¤ m ( m 5 ¤ m ( = '8>[email protected] = '8>[email protected] ¢ 3hfI& ¤

m fI& 0 3A &-21 3 ¢ f ¢ h 3 f9

-; j &-21 F 7( m G ' >[email protected] =* 3

2.

3.

4.

5.

6.

7.

0 fI& m 3 ¢ &-21 3 ¢ f ¢ h 3 f9

-; j & -21 F 7 ( m

fI& 0 7 43 ¢ £ 3 ¢ f9 a 1 ; ¢ f9 O

-;&-21

&-21 F 7<5 m

a 1 fI& 0 43 ¢ £ f9 a 1 ¢ f9 O

-;&-21 a F

1

fI& 0 7 43 ¢ a 1 £ 3 ¢ f9 a 1 ; ¢ f9 O

-;&-21 F

&-21 7<5 m

a 1 fI& 0 43 ¢ a 1 £ f9 a 1 ¢ f9 O

-;&-21 a F

1 ( f;! / ']" 0 7( m ¢ 3 f 0Q3hf -; j &-21 m m (8 )

<

G

= '*>[email protected] G

= '*>[email protected] 7

¤ 7<5 m G

= '*>[email protected] G 7

¤ 7<5 m G ,

f;/ ') 0(: 7X! % . fI 1 -; f $ ¢ -; j + 1 -;9lC-; £ f ¢ &-21 m 9. EDM &-21 H '#" ( m 3 3 ¢ f -21 0Q3hf9

-; j &-21 ) m m (8 , m 10. D &-21 H ' ( 7! 3 ¢ fI &-!&4 0 f -; j + 1 -;9lC-; . m ¢ , ) &-21 ¢ ¢ a 11. k &-21 3hf & 1 M 3hf & -21 f;/ F 7 G f9 -21 ¢ a 12. k 1 3hf9

& &-21 k -21 0f;/ F 27 G f9 ¢ 3hf9 O& -21 ¢ ¢ a 13. k -21 3hf9

& 1 k &-21 3hf9

& -21 f / 43V -; £ -;&-21 8.

5

= '*>[email protected] = '*>[email protected] = '*>[email protected] = '*>[email protected] f;/ ¤ 0f / ¤ 0f;/ ¤

14.

15.

16. 17. 18. 19. 20. 21.

22.

23.

24.

k a 1 ¢ h 3 f & a 1 k -21 ¢ h 3 f & -21 f;/ f9 f / ¤ a 1 ¢ 3hf9

& -21 -21 ¢ 3hf9

& &-21 f / £ 3 m ¢ 3hf9

& -;&-21 f;/ ¤ , ) k 1 ¢ 3hf9

& &-21 &-21 f;/ 0 ¡ = '8>[email protected]

k &-21 ¢ f9

& P f;/ ¡ k -21 ¢ f9

& Pk &-21 f / £ F3 k a 1 ¢ f9

& Pk -21 f;/ £ k &-21 M ¢ f & &-21 f;/ F 7 G f -21

k a 1 k -21 ¢ f9

& &-21 F 7 G f9

7 G -;&-21

0f;/ ¤

f / ¤

¢ f & -;&-21

f;/ ¤

0f;/ ¢ 9f

& -;&-21

f;/ ¤

3 m , )

1.2.1. Derivatives with respect to the argument

2.

fI& EDM &-21 F

3PfI ; < If & ¤ m GIH f -;& -21 F < m G ¤

1.2.2. Derivatives with respect to the parameter 1.

&

M]! ¤

!;

= '8>[email protected] = '8>[email protected]

1.2. The Hurwitz Zeta Function ζ(ν, z) 1.

= '8>[email protected] = '8>[email protected]

k &-21 ¢ h 3 f9

& 1 ¢ h 3 f9 O& &-21 0f / ¡ k &-21 ¢ 3hf & &-21 f;/ ¡ k 1 ¢ f9

& &-21 f;/ ¡

2.

3. 4.

5.

6.

7 ' a $ £) 3 £ F '*G 7 - 1 3 $ ' " £! 3 £ ( 7X! " & -21 . ( 7! " ] $ (:7X! % " 3 ' ! ' + -21 F 'PG 3 O' ! " 1 7 Y & -21 ' F £) ' G ' " 43 £! ¢ = ' 7 1

F G a 3 ' ' £ 1A- 3 ¢ 3 £ ¢ - 1 C7( " ' ! 7 F G a C7( " ' !0' - 1 ( 7X! ' 7 ] ' " ' # ' " '#" + &-21 . F G 3 ' " ' (:7 3 £ ¢ C7 (* " ' ! 7 G a ' F

C7( - " ' ! 1 ( 7! " . 7 £ % ' & # ' " # ' " 3 ' " ' + &-21 F G 7( 3 '#" 3 £ ¢ C7( " ' !AC7<5n " ' ! F G a '

- 1 C7 ( " ' ! £ ' & ' " ' C7(* ! ( 7X! " "" 5_7X! . 7 # ' " ] ' " # ' " ] ' "

+

F G

& 2 1 '

7 (* " ' 7( " ' 3 £ ¢

31

! =

@[email protected]

= = @ #C7 [email protected]

= = @ C7 [email protected]

= = @ C7 [email protected]

= = @& [email protected]

1.3. The Exponential Integral Ei (z) 1.3.1. Derivatives with respect to the argument 1. 2.

" & 2- 1 m ]! ¢ ¢ 3PfI& 3 &-21 3 -; - &$ % 3 ¢ -; - -; fI& &-21 :

= '8>[email protected] B = '8>[email protected] B

$! "!$ #

" & -21 m ]! ¢ GH Q 3 3 -21 - R $&% 3. EDM &-21 F 3 9 3 ¢ 3 ¢ -21 -R -; F mG 4. &-21 " & -21 ( 7X! "'& $&% f 43fI& f m ]! 5. 3fI& 0 m 6. EDM &-21 R F]3 GIH 43fI -;&-21 R F]3 m G 43 ¢ f &-21 -; & -21 3 ¢ F G m 3 ¢

-;&-21 43fI& ¤ 7. -21 - P 3PfI& a - R ED -21 R F3 m G9H H 43 ¢ 4 F]3 m G ¤ 8. EDM 1 f 3PfI& ¤ 9. V- 3PfI& m m 10. ED - R PED &-21 R F]3 GIHOH f -;&-21 F]3 G ¤

m

= '8>[email protected] B = '8>[email protected] B = '8>[email protected] B = '*>[email protected] B

1.4. The Sine si (z) and Cosine ci (z) Integrals 1.4.1. Derivatives with respect to the argument 1. 2.

3. 4.

5.

')(:7X! % -; 0fI& m EDM &-21 F GIH 7! % 43 ¢ ' (8 D ')(:7X! % -; 0fI&

&-;-2 1 3AfI& 3 V -

m G 3 #- R R -; &-21 F 3 -; 43 AfI& - &-21

&-;-2 1 AfI&

= '8>[email protected] B

m ;&- -2 1 F 9G H &-;-2 1 AfI&

= '*>[email protected] = '8>[email protected] B

m EDM &-21 F G9H 4 3 ¢ ')O(8 7! % D R ;- F3 m G #- R ;- F m G9H &2- 1 &2- 1 & <3 0& 3 ¢ FO63 ' G & 3 F Q3 ' G &

7

= '*>[email protected]

' 7 ' & 3 )£ ¢ 3P

¤

1 !<

6.

&

0& 4 3 ¢

F Q3

' G

F Q3

;

' G

0&

7 0& 3 '%& 3 !£ 3 ¤ 1

1.5. The Error Functions erf (z) and erfc (z) 1.5.1. Derivatives with respect to the argument 1. 2. 3. 4. 5. 6.

7.

W W m' ¢ 0fI& 4 3 &-21 #- &-21 0fI& W ' (87! % m 0f /

1 ; 1 -; 0f9

& &-21 ] ' " W ( 7! &-2 1 0f;/ '#" - &-21 0f;/ 7 ( 7! ' 7 f G D f;/ H

F ; & 2 1 W W m m m'

9 G H DM &-21 F 3 -;&-21 V- &-21 F G W "' m m ] ' ED F GIH 3 " - &-21 F G 7 7 m m D &-21 F GIH , )

W '6(*7X! % m G H 4 3 ¢ -21 - 1 -; ) 8. EDM &-21 F 9 &-21 W W W W 43 AfI AfI& 9. D fI& H W W 3 ' & 8 3PfI 3 W W W W m m AfI -;&-21 F G 10. D &-21 H W W. 3 '%& 8 3 f -;&-21 + W W ( m ! ' 7 7 _

H G 0 f 11. D / F 3 f9

G ¤ F L m

= '8>[email protected] B = '8>[email protected] B = '8>[email protected] B = '8>[email protected] B = '8>[email protected] B = '8>[email protected] B = '8>[email protected] B m ,

= ' > [email protected] B

£ ¤ -;&-21 / fI

m

¤ -;&-21 ) ,

! #

13. 14.

15. 16.

17.

18.

19. 20.

21.

22.

23. 24.

" (

W 7 _ m ¤ m ' 7 , ED &-21 F IG H F G -;&-21 ) 3 W W 7 D &-21 V- D 0f;/ HOH F G f -21 0f;/ ¤ W W m ED a 1 - PED &-21 F GIHOH F 7 G f9 -;&-21 F m G ¤ W W 7 D a 1 k -21 f / H F G 3Pf9

0f / ¤ W m ED &-21 ED &-21 F GIHOH F 7 G 43f #- W F m G ¤ W W W W 3 ' & 8

H ED 0fI& 3PfI -;&-21 / £ fI ¤ W W. W W m ¤ m 3 '%& 8 ED &-21 F GIH f -;&-21 + -;&-21 ) , W W "' £ £ ¤ D &-21 0f;/ H -21 A - &-21 f / W m ED -21 F GIH "' ¤ 3 ¢ £ -;&-21 W + . - &-21 ) f * , W W D -21 V- D &-21 0f;/ HOH 43V -; £ -;&-2 1 0f / ¤ W W m ED a 1 - ED -21 F GIHOH 43V -; £ F m G ¤ W ED 1 &-2 1 f;/ 0 H ¡ = '8>[email protected] B W W D a 1 - D -21 0f / HOH f9 f / ¤ W

12.

m

P

W

25.

ED &-21 - ED &-21

W

:1

m F G9H H f -;&-21

m F G ¤

1.6. The Fresnel Integrals S(z) and C(z) 1.6.1. Derivatives with respect to the argument

fI& 0 " m ')(: 7! % 1 -; D 1 -; 43 AfI& <3 - &-21 m 2. EDM &-21 gF GIH 43 ¢ " m ')(: 7X! % -!&4 D R 1 -; F]3 m G 3 #&-21 1.

0 fI& ') (:7! % " m 1 -; D 1 -; 43 AfI& &-21 m 4. DM &-21 F G9H 7X! % 43 ¢ " m ')(: -!&4 D R 1 -; F]3 &-21

1 -; H &-21 AfI&

3.

# -

m 1 -; F GIH &-21 ' >[email protected] =*

1 -; H &-21 AfI&

m G

R

= '*>[email protected] B

= '*>[email protected] B

m - R & 1 2- 1 -; F IG H > [email protected] = '*?

1.7. The Generalized Fresnel Integrals S(z, ν) and C(z, ν) 1.7.1. Derivatives with respect to the argument 1.

2.

3.

fI ; ') 0 3 (: 7X! % f# -; -; 3AfI& 3 - -; AfI& &-21 &-21 m D &-21 F GIH 43 ¢ &-21 ')(: 7! % f V - -21 D R -; F3 m G 3 - R ;- F m &-21 &2- 1 0fI ') (:7X! % 3 f# -;

&;- -2 1 3AfI&

,9N

-

&;--2 1 AfI&

= '*>[email protected] G9H = '*>[email protected] = '*>[email protected]

4.

m EDM & -21 F G9H 7! % m 43 ¢ &-21 ')(: fV - 2- 1 D R -; F 3 G 3 & -21

m - R & -;-2 1 F 9G H ' >[email protected] =*

1.8. The Incomplete Gamma Functions γ(ν, z) and Γ(ν, z) 1.8.1. Derivatives with respect to the argument

fI& 0 3 ¢ fV -; - & -;-2 1 fI& m 3 ¢ 3 ¢ f# - -21 -R -; F m 2. EDM &-21 F GIH &-21 ¢ < ¤ 43 - -; fI& 3. - fI& 0 m -21 F < m G ¤ a 4. EDM -21 F G9H ¢ 3 3PfI )3 < fI& ¤ 5. fI& 0 m ¢ m 6. DM &-2 1 R F G9H 3 f -;&-21 R F T3 _ 7 m ¤ ' % f N '65_5_ !7 L 7. &- fI& 0 1 1J m 43 ¢ ' % f# -;&-21 N '65_5_7 ! 7 8. D -21 R F G9H ` 1 1^ a - P - fI& f fI& ¤ 9. m f -;&-21 a 10. ED &- -R PED -21 R F GIHOH

= '8>[email protected] B = '8>[email protected] B

1.

11. 12. 13. 14. 15. 16. 17.

G

G ¤ ¤

m F G ¤

fI& 0 ; 43fI fI& ¤ m m ED &- R PED a -21 F GIHOH 3fI -;&-21 R F G ¤ -21 - < &- fI& 0 43 ¢ ¢ 3 -;&-21 fI& ¤ m m ED &- a 1 -R PED -21 R F GIHOH 3 ¢ ¢ 3 ; F G ¤ &- V- fI& 0 f ¢ 3 - fI& ¤ m m ED a -R PED &-21 R F GIHOH f ¢ 3 -;&-21 F G ¤ fI& 3Q 3 ¢ f V -; - & -;-2 1 fI& = '8>[email protected] B a

< -

,

18. 19. 20. 21. 22. 23. 24. 25. 26.

L1 ,

m ED &-21 F G9H - fI& m ED a -21 F fI& 0

43 ¢ &-21 3 ¢ fV - -21 -R -; 0fI& = '8>[email protected] B &-21 ¢ _ ¤ 3 - -; - fI& m GIH -21 F < G ¤ ¢ 3 ; 43fI T3 _ fI& ¤ m m D &-21 R F GIH ¢ 3 ; f -;&-21 R F T3 < G ¤ '65_7 m &- fI& ¢ 3 ; f J 5_! 7 L ¤ '65_7 m D -21 R F GIH 3 ¢ ¢ 3 ; f -;&-21 ^ 5_! 7 ` ¤ m m ¤ ¢ _ ¢ 3 fI& 3 " 1 -; - A &-21 F G m m m . ED &-21 F ¢ 3 < GIH " -!&4 -R + &-21 F OG ¤

1.8.2. Derivatives with respect to the parameter 1.

]]!

! ( 5 7 5_7 L ¤ & 63 N J _

1.9. The Parabolic Cylinder Function Dν (z) 1.9.1. Derivatives with respect to the argument 1.

2. 3. 4. 5. 6. 7.

m ' m fI& 0 F]3 G J $ L £ 43 ; &- F :G - 0fI& ¤ F3 )m G ' $ L3 £ A F m G a fI& ¤ & J W W W W D fI& H 43fI 43 -; fI& ¤ W W W W D - fI& H 3PfI - a 0fI& ¤ W W. W W. m m D &-21 + F GIH f 3 ; -;&-21 + -; F G ¤ W W. W W. m m D &-21 - + F GIH f -;&-21 - + a F G ¤ W W D &- -21 A f / H £ -; 3 ; - -21 A - 0f / ¤ , ,

W . 4 3 a 8. EDM + -21 - A 0 f;/ H W . m + F G9H 9. D 3 £ -; 43 W a . - + . F m G9H 10. ED - + 1 £ -;

£ -; I+ -21 . - W A a 0f;/ ¤

W . m -; a + - F G ¤

W . m . -;&- + a 1 - + a F G ¤ W 3 £ -; -;&-21 V- W A ¤ H 11. D -21 V- A ; f / - W . m £ -; -21 - W + . ¤ 9 G H 12. ED &-21 - + F - W 3 ¢ £ -;&-21 O/ -;&-21 F f " G ¤ H 13. ED -21 - A ; f / - &-21 W . m £ -;&-21 O/ F m G ¤ 9 G H F 14. ED - + - &-21 W 15. ED A 0f / H W '65 $ ! 43 ¢ A & -21 '%&" f &- 3 ; ) ' ( ! ! $ 9 $ % &- -; a 0f;/ = '*>[email protected] W 16. ED V- A f;/ H '65 $ ! 43 ¢ - W A & -21 '%&" f &- a f;/ = '*>[email protected] ) ' ( ! ! $ & $ % & W W a 17. D 1 V- A D -21 A 0f;/ H H W m F 7 ( G , - A 0f / ¤ ) W . W . m 18. D &-21 - + P D &-21 + F GIHOH W . m m F 7 ( G , -;&-21 - + F G ¤ ) W W 19. D &-21 - A D A 0f / HOH F]3 G m -21 - W A f;/ ¤ , )

,93

20.

21.

22.

23.

24.

25.

26.

27.

28.

W . W . m a ED 1 - + PED &-21 + F GIHOH W . m F]3 G , -;&-21 - + ) W W D &-21 A D - A 0f / HOH W 7 m F 5_ G 3 , -21 A ) W . W . m D a 1 + D &-2 1 V- + F GIHOH W . 7 m F 5_ G 3 , -;&-21 + ) W W ED a 1 A ED -21 #- A 0f;/ H H F ¢ G 3 m W A , ) W . W . m ED &-21 + ED &-21 - + F GIHOH F ¢ G 3 m -;&-2 1 W + . , ) W W ED a 1 - A ED &- -21 A f;/ H H 3# -; 3 I+ -21 . -; - W A W . W . m D &- -21 - + D + F GIHOH 43V -; 3 ; - + a 1 . - W + .

W W . a D - -21 A D + -21 - A f / HOH 43# -; ¢ -;&- -21 W A W . W . m . a a a

+

+

ED 1 EDM - 1 V- + F GIHOH 43# -; ¢ W + .

1.9.2. Derivatives with respect to the order 1.

P1 ,

m F G ¤

f;/ ¤

m F G ¤

0f / ¤

m F G ¤

m F G ¤

£ -;&-21 - W F G

W #$ 7 #7 7 ! Y 3 C 3 £ F 3 G F G 3h N

&

,!5

M ]!

f;/ ¤

f / ¤ m F G ¤

2.

W 7 ¢ £ 3 4 3 & -21 - $ &-2- 1 ) , W " W #$ 1 $ W L Y 3-" 1 - 3 , J

N 1 $ !5

1 -21 ) 1 1 J L" W " W ' ( L 7 7 ¢ £ 3 3 &-21 F G - J $ L J F 3 1 J L" W M]! £ -;&-!&4 - a

a 1 F G W #$ 1 7#7 7 ! Y C £ 3 F3 3 G 3 F G N

&

7 ( " W ¢ £ £ ¢ $ (:7! % 'T( $ 5_7X! % F 3 G 3 -;&- -21 - W 7 1 ¢ £ £ ¢ -21 -!&4 3 &-!&4 - $ W &- ) , &- a 1 ) , 1 $ #$ W "" ! Y "" N

3 / 1 -2 1 - ) 3 , 1 L)" 1 1 $ 5 1 J

W ¢ £ 3 43 &-21 / -;&-21 ) 3 , '65_7 W #$ W W '& ' % ¢ ! a 1#- 1 N 1 'g5 3 V- F]3 3 3 & & J L'

7 #7 #$

&

& & N & ! 1 -1

& £ £ £ &4 C -21 ) , 1 ) , £ £ £ £ 3 C 1 ) , 83 C 3 -2 1 ) , =

G ¤

7 G ¤

3.

1.10. The Bessel Function Jν (z) 1.10.1. Derivatives with respect to the argument 1.

m ' fI& 0 F G 3 ¢ J $ L

, ;

¤

2 0fI&

@CB

+ (

2.

3.

4. 5. 6. 7. 8.

9.

" % $ m ( ! ! % a 0 fI& ¤ m ' -;&-21 43 ¢ J $L 2 F G ¤ m ¢ ' $ L F G ¤ F G f;/ J &. I+ -; 2 f / ¤ 7 m ¤ F G &-21 f / ( 7! ' m ¤ F G &-21 0f /

" 43& -; ')( (m ] $ ! ! % &

EDM &-21

m m F GIH F G

7 f;/ 0 F]3 G m f / 0 F G

. I+ a 1 a 1 0f / 0 . I+ a 1 -;&-21 0f / . - + a & a 1 0f;/ / *F m G a 1 a 1

f / 0 F m G + -!&A . & -21 fI - $&% '6')5 ( $ $ ! !

10. 11. 12. 13.

15.

m m F G -;&-21 F G ¤ -

2

0f /

= '*>[email protected] F m G ¤

7 ' m ¢ ED &-21 F IG H J $ LF G F G &m m m . ED -21 F GIH F G - + -21 F G ¤ ( 7X! ' m m m ¤ . EDI+ &- a 1 F GIH F G &-21 -;&-21 F G 7 m m m . EDI+ &- -;&-21 F GIH F G &-21 -;&-21 F G ¤ m . EDI+ &-21 a 1 F GIH m ¤ 43 ¢ / 8F m G a 1 -21 a F m G G F 1

-;&-21

m ! '#" &-21 &-21 fI& 0 &-2 1 = '8>[email protected] B m

14.

N1

,9:

16. 17. 18. 19.

20. 21. 22. 23.

24. 25. 26.

27.

( 7! ' £ = '8>[email protected] B D -21 R &-21 F 9G H If &-21

- R m ! '#" &-21 fI& &-21 fI& 0 &-21 £ fI& ¤ m ! ']" &-21 0fI& &-21 fI& 0 &-21 £ fI& = '8>[email protected] B m ! '#" ¢ a &-21 fI& 1 -; fI& 0 3 1 &-21 £ fI& = '*>[email protected] B m ! '#" ¢ &-21 0fI& 1 -; fI& 0 3 &-21 £ fI& ¤ m ¤ m ( 7X! ' £ m D -21 &-21 F GIH fI &-21 - m m m ( 7X! ' £ ED -21 &-21 F GIH = '8>[email protected] B fI &-21 - ' m 0f;/ F G 43 ¢ J $ L a f;/ -; a 0f;/ ¤ m ']" . £ &-21 & -21 0f;/ = '8>[email protected] B I+ &!- & &-21 f;/ m ']" . &-21 1 -; 0f / 3 I+ &-!& &-21 £ f / = '8>[email protected] B &-21 &-21 0f;/ 1 -; 0f;/ '#" . m £ '*>[email protected] I+ &!- & 1 -; f;/ = m '#" . £ ¤ &-21 0f / a 1 0f / I+ &-21 a 1 f / ( 7X! ' m m . a D -21 & -21 F GIH

+ f F & 2 1 1 &-21 G '*>[email protected] = % ' & ( X 7 ! m m . D -21 1 -; F GIH f &-21 - + a 1 &-21 F G = '*>[email protected] m m D -21 &-21 F G 1 -; F G9H ' m ¤ ( 7X! f &-21

- + a 1 . 1 -; F G

m

28.

29.

30.

, <

+ (

31.

32.

m m ED -21 &-21 F G a 1 F G9H ' ( 7! m ED &-21 F GIH F3 m G -; -21 43 ¢

N1 ,

m . f &-21

- + a & a 1 F G ¤

'

J $L

m a F G

F m G ¤ a ;-

1.10.2. Derivatives with respect to the order 1.

2.

3. 4.

"" ' ' % ¢ 0& ¢ & -21 $&J % ' L ( $ ! 0 & ¤ £ & <3 £ & 0 = = @ B VC7 [email protected] * 1

£ & £ & 0 = = @ B VC7 [email protected] * -21

£ & a 0& 3 43 ¢ £ & a " 1

-;&-21 & " 1

' % ' % & -21 L

L J ' ( $ ! a 1 0& 3 F G ')J ( $ ! % $ & $ % 1 £ ¢ £ ¤ % &- a 1 & -21 & <3 3 &- - -;&-21 0& 1 - & 0

M]! M]! M]! M]!

Y 2- 1 M]!

£ & 1 -; & <3 3 ¢ £ & &-21 & ""' 1 -; ' % & -21 ( L ' % " 3 $&J % ')

( $ ! 1 - & <3 ')( $ ! % $ 1 ")"'& £ Y -21 % 3 ¢ a 0& -; 1

-21 & 3 3 ¢ a &- -21 & 1 M]! ¢ &-21 £ a 1 / -;&-21

6. " 1

')( $ ! % L 7 G 3 F 3 Y &-21 J

X D F F ) ' * ( ! 5 * ( 6 ' 5 $ % $ 1L $ $9% 1L J J ,9L

5.

£ & 0 ¤ 7 GG

Y

¢ £ &-21 / -; a

Y DXF F

£ & H " ) ' ( : ( X 7 ! $ % L & -21 1 &$ % ')(* $ (:7! % $ 5 J $ * ( '65 &L J J 7

G 3 F 3 GG £ 3 & £ & H

£ &

1L

= '*>[email protected]

1.11. The Bessel Function Yν (z) 1.11.1. Derivatives with respect to the argument 1.

2.

m ' 0fI& F G 3 ¢ J $ " 43& -; ')( (m ]$ ! ! % &

3.

m m D &-21 F GIH F G

4.

5. 6. 7.

m f / 0 F G . I+ a 1 a 1 0f;/ . I+ a 1 -;&-21 0f / . - + a & -;&-21 0f / 43 ¢ a 1 / 8F

( 7X! ' m F G &-21

0fI& ¤ m F G ¤

f / ¤

m a m m OG 1 a 1 F G ;- &-21 F G ¤

f / 0 F m G I+ -!&A . & -21 If - $9% '6')5 ( $ $ ! ! - 2 f /

m F m G I + -; . -21 F m G ¤ 9. D -21 F G9H ( 7X! '%& m m . &-21 -;&-21 F m I G H G F 10. EDI+ &- a F 1

8.

¤

2 0fI& " % $ m ( ! ! % a ' -;&-21 43 ¢ J $ L 2 . I+ -; 2 0f / ¤ 7 m 3 F G &-21 f;/ ¤

L

,9P

= '*>[email protected]

G ¤

+ (

11.

. EDI+ &- . D + &-21

12.

13. 14. 15.

16.

17.

m F -;&-21 m F

-;&-21 3 / *F

7 m G9H F G &-21 -;&-21

m F G ¤

G9H

m a 1

m G -21 a 1 F G ;- &-21

m ! '#" ¢ a &-21 1 -; fI& 0 3 1 &-21 m m ! '#" D -21 R 1 -; F GIH 3 - R m ']" . ¢ a &-21 1 -; 0f;/ 3 1 I+ &!- & 1 -; £ f;/

m F G ¤ = '8>[email protected] B = '8>[email protected] B = '*>[email protected]

&-21 &-21 0f;/ 1 -; 0f;']/ " . 3 ¢ a 1 m '8>[email protected] B £ I+ &-!& & -21 f;/ = &-21 0f;/ a 1 0f;/ '#" . 43 ¢ a 1 m £ ¤ I+ &-21 -;&-21 f;/ m m m '#" . ED -21 1 -; F GIH 3 - + a 1 1 -; F G = '8>[email protected] B m m D -21 &-21 F G 1 -; F G9H ']" m ¤ . 3 m - + a 1 &-21 F G m m ED -21 &-21 F G a 1 F G9H m ¤ 3 7 f &-21 - + a & . F -;&-21 G &-21 &-21 0f / 1 -; 0f ']/ " . 3 ¢ a 1 m £ '8>[email protected] B I+ &!- & &-21 f / =

18. 19.

20.

21.

22.

&-21 1 -; 0f;/ & -21 0f;']/ " . 3 ¢ a 1 m £ I+ &-!& &-21 ;f / 3 N

= '8>[email protected] B

, 3 8 , 3 8

23.

24.

25.

26.

&-21 &-21 0f;/ &-21 0 f;/ ( 7X! ' . f &-21

I+ &-!& 1 -; &-21 0f / -;&-21 0f / ( 7X! ' . f &-21

I+ &-21 a 1

£ f / ¤ £ f;/ ¤

a 1 0f;/ 1 ;- 0f;/ '%& ( 7X! f &-21

I+ &-21 . a £ f;/ ¤ 1

m m D -21 &-21 F G &-21 F G9H ]' " m ¤ m a . - + 1 1 -; F G m m ED -21 1 -; F G &-21 F G9H ']" m ¤ 3 m a . - + 1 &-21 F G m ¤ m m m '#" a . D -21 &-21 F G -;&-21 F GIH - + & a 1 F G m m ED -21 a 1 F G 1 -; F G9H ']" m ¤ . 3 m a f

+ F & 2 1 & a 1 G

27.

28. 29.

1.11.2. Derivatives with respect to the order 1. 2.

" "' ' M ]! % 3 ¢ & ¢ & -21 $&J % ')L ( $ ! & ¤ M]! 3 0& 43 ¢ M! ¤

1 1

2<1

1.12. The Hankel Functions Hν(1) (z) and Hν(2) (z) 1.12.1. Derivatives with respect to the argument 1. 2.

. m . + f / 0 F G I + -;

. m m EDM -21 + F G9H F G I + -;

31

. 0f / 2 . m . -21 + F G +

= 7][email protected] = 7][email protected]

+ (

3. 4.

. I+ a 1 DM &-21

. ( 7X! m . U + a 0f / F G & -21 + -21 1

. . + 1 0f;/ + f;/ H ¡ 1 -; 1 -;

!, ,

= 7 ][email protected] = '8>[email protected]

1.12.2. Derivatives with respect to the order 1.

2.

3.

4.

" "' . ' % & -21 L . + & $&J % ')( $ ! + 0 & = 7][email protected] "" ' ' % ¢ a + . & <3 3 ¢ & -21 $9J % ')L ( $ ! + . & = 7@CB 3 8 M]! * Y 1 + -2 1 . 3 ¢ a 1 £ & <3 £ & 43 ¢ = 7@CB 3 8 M]! * Y -2+ -21 1 . £ & 3 ¢ a 1 £ & 0 3 ¢ = 7@CB

3 8 M]! 3 ¢ 3 8 M]! 3 -;

1.13. The Modified Bessel Function Iν (z) 1.13.1. Derivatives with respect to the argument 1.

2.

m fI& 0 F G J 43& -; ')( (m

3.

m DM &-21 F GIH

4.

f;/ 0 F]3

5.

6.

f;/ 0 . I+ a 1 a 1

'

¤

2 0fI& ]! " m ]! " $ ! % &- % $ ( ! % - a 0 fI& ¤ m m ' F3 G -;&-21 J $ L 2 F G ¤ 7 ' m ¤ G G F]3 G ;f / J $L F & F mG I+ -; . 0f;/ ¤ 2 7 m f / 0 F G &-21 0f / ¤

$L

3,

7 m . I+ a 1 -;&-21 f / 0 F G &-21 0f / ¤ a . 8. - + & a 1 ;f / 0 m ¤ / 8F Om G a 1 a F m G F 1

-;&-21 G 9. k 0f / F m G I+ -!&A . & -21 3PfI - $&% '6' 5 ( $ $ ! ! - f / = '*>[email protected] 2 m m m 7 ' $ L F G F G ¤ G 10. D &-21 F G9H F 3 J &m F]3 m G - + . -21 F m G ¤ 11. ED -21 F G9H ( 7! ' m m . &-21 -;&-21 F m G ¤ I G H G F F 12. EDI+ &- a 1

( 7X! ' m m . &-21 -;&-21 F m G ¤ I G H G 13. EDI+ &- F F -;&-21

m . 14. D I+ &-21 a 1 F G9H m ¤ 3 ¢ / *F m G a 1 -21 a F m G F G 1

-;&-21

m ! '#" 15. &-21 &-2 1 = '8>[email protected] &-21 0fI& m ! " '#" -;&-21 £ ¢!!£ fI& ¤ 16. k &-21 - a 1 fI& 5n'65 5n'65 1 L £ J 17. k &-21 fI& fI # -21 5_ 7X! 1 N 1 5_7 m 1 ¤ ! ' ( X 7 ! m £ fI &-21 - R '8>[email protected] I G H F = 18. ED -21 R

&-21 m ! " '#" £ ¢) m ¤ m ¢ G F 19. D -21 V - R a 1 H 43 7 m 20. ED R F GIH 5n'65 #$ 5n'65 1 ¤ 3 ¢ £ fI # - -;&-21 J 5_ 1 7XL! N 1 1 5_7 ! 7.

3 3

+ (

21. 22. 23. 24. 25.

26. 27. 28.

29. 30.

m ! ']" &-21 0fI& &-21 0fI& &-21 £ fI& ¤ m ! '#" &-21 0fI& &-21 0fI& &-21 £ fI& m ¤ m m ( 7! ' £ ED -21 &-21 F GIH fI &-21

- m m m ( 7! ' £ D -21 &-21 F G9H fI &-21 - m ' V f / F G J $ L a 0f / -; a 0f / ¤ m '#" . £ &-21 V& -21 f;/ 0 I+ &!- & &-21 f;/ m '#" . £ &-21 V1 -; f / 0 I+ &-!& &-21 f / &-21 &-21 f;/ 1 -; f;/ ']" 0 . m £ I+ &!- & 1 -; f;/ m ']" . £ &-21 0f / a 1 f / 0 I+ &-21 a 1 f / m ED &-21 F GIH F3 mG -; -21 ' $L a F m G a -; KJ ' ( 7! m m . D -21 V& -21 F G9H f &-21

- + a 1 &-21 F G ( 7! ' m m . a

+ ED -21 1 -; F G9H f F & 2 1 1 &-21 G

31

= '8>[email protected] = '8>[email protected]

= '8>[email protected] = '8>[email protected] = '*>[email protected] ¤ F m G ¤

31.

= '*>[email protected]

32.

33.

m m ED -21 &-21 F G 1 -; F' GIH m ( 7X! f &-21

- + a 1 . F 1 -; G m m ED -21 &-21 F G a 1 F GIH ' ( 7! f &-21

- + a & . a 1

34.

395

= '*>[email protected] = '8>[email protected] m

F G ¤

35. 36.

37. 38.

39. 40.

41. 42.

0fI& F m ED &- - R PED a &-21 R F GIHOH F 7 G 7 &- P - 0fI& F 3 a - P -

7 G £ fI - 0fI& ¤ m £ fI - ;- &-21 -R F G ¤ G 3 £ fI - fI& ¤

m ED a R ED &- 2- 1 V-R F GIHOH F 7 3 G 3 £ fI -;&-21 R F mG ¤ 7 £ &- - < 0fI& F 3 G If - - 0fI& ¤ m a I G O H H D #- R D &- -21 R F F 7 3 G £ fI -;&-21 -R F mG ¤ 7 a P - - 0fI& F G 3 £ fI fI& ¤ m a D &- R P D -21 -R F GIHOH 7 m F G 43 £ fI - -;&-21 R F G ¤

1.13.2. Derivatives with respect to the order 1.

2. 3. 4.

"' ( ' M]! % L 3 ¢ a 1 & & -21 J % 'T

( &! 0& ¤ 7 M]! £ £ @ B C7 [email protected] = = * 3 & #- & 1

£ £ ¤ * D & 3 & H M]! 7 3 £ & a -;&-21 & a 1 " & 0

1 ' ( 7X! ' % & -21 ( 7X! 3 £ & a 1 0& 643 ¢ $9% ')( $ ! F ,G -; a 1 & " ( L ( ! ' % -21 3 " ')J ( $ ! % $ % £ Y 3 ¢ a 1 & &- 1 ¢ -; &-21 0& -21 £ & 3 43 &- &- a 1 & -21 & 1 - £ & 0 ¤ 3;

+ (

5)

( 7! ' £ 5. & 00 ;- &-21 & a 1 " & 0&3 a 1 & ' ( 7X! Y £ & £ & 43 ¢ % & -21 $&% ')( $ ! F G -; a 0& 1

" ( L (]! ' % -21 3 " ')J ( $ ! % $ % £ Y 3 ¢ a &1 &- 1 ¢ -; &-21 0& -21 £ & 3 43 &- &- a 1 & -21 & 1 - £ & 0 ¤ M]! 7 £ & <3 £ & 00 6. -;&-21 & a 1 & 0 1 -; ( 7! ' 3 " "' £ & £ & &-21 & " ' % & -21 ( L ' %

-21 L 3 $9J % ')

( $ ! 1 - & " ')J ( $ ! % $ % 1 £ Y 3 ¢ -21 &- -21 ¢ & -; a 1 0& -21 £ & £ ¤ 43 &- &- -21 & -21 & 1 - & 0 M]! 7. " 1

" ')( $ ! % ( 7 ! £ &-21 O/ -;&-21 & -21 J $ L (*'65 ) ' * ( ! 5 9 $ % $ % $ 1 1L J L J

7 7 Y D £ F F G 3 F 3 G;G 3 £ & V- £ & H " " ')( $ (:7! % ( 7 ! & -21 £ J $ L (*'65 &!- &4 / -; a 1

) ' * ( : ( X 7 ! 5 9 $ % $ % $ & 1L J L J

7 Y D £ F F G 3 F 3 GG 3 43 £ & - £ & H ¤ 7

£ & 3

1.14. The Macdonald Function Kν (z) 1.14.1. Derivatives with respect to the argument 1.

2.

m ' fI& 0 F]3 G J $ L 2 0fI& ¤ " " 43& -; ') m ( ]! $ ! % % ( $ m ( !! % a 0 fI& ¤ &- 3 :

$

m m m ' F GIH F G -;&-21 J $ L 2 F G ¤ F3 m G I+ -; . ¤ 4. f;/ 2 0f;/ 43 ¢ '% & f &-21 - ¤ a . ; f / 0 5. I+ 1 a 1

( 7! ' m a . a 1 -;&-21 a F m G ¤ G 6. - + & a ; f / 0 F 1

1

7. f / F3 m G I+ -!&A . & -21 f - $&% '6')5 ( $ $ ! ! - '*>[email protected] 2 f / = m F mG - + . -21 F m G ¤ 8. EDM -21 F GIH m ¤ m . 7 F m G a 1 -21

9 G H 9. DM + &-21 a F F G a 1

1

¢ £ £ ¤ 10. k &-21 a 1 0fI& 3 / 6 fI -;&-21 -;&-21 SEa 1 )SEa fI& 11. k 1 3 ¢ SEa _ / g £ fI &- S -21 &- S -21 £ fI& = > '[email protected] B S -; 12. k &-21 )SEa fI& 43 ¢ S 1 / 6 £ fI - S -21 - S -21 - S -21 £ fI& = > '[email protected] B S -; S fI& 13. k - -21 )SEa 3 ¢ SEa 1 / g £ fI - S -21

- S -;&-21 - S -;&-21 £ fI& ¤ S SEa 1 - )SEa fI& 14. k 1 3 ¢ SEa _ / g £ fI &- S -21 - &- S -21 £ fI& ¤ S 15. k &-21 - )SEa 1 fI& 43 ¢ S / 6 £ fI - S -21 - S -21 - - S -21 £ fI& ¤ SEa S 16. k - -21 # - )SEa fI& 3 ¢ S 1 / g £ fI - S -21 - S -;&-2 1 #- - S -;&-21 £ fI& ¤ SEa m 17. D -21 R )SEa 1 F GIH 43 ¢ SEa / 6 £ fI - S -21 S -; - S -21 F m G = >_'[email protected] S -; 3.

EDM &-21

3<

+ (

18.

ED

20.

D

21.

ED

22.

ED

24. 25.

26.

m F 1 IG H ¢ S / g £ fI - S -21 S - S -;&-21 S m &- S -!&4 R )SEa 1 F G9H 3 ¢ S ? / g £ fI &- S -21 -;&-21 &- S -21 F m G = S -; m -21 V-R )SEa 1 F GIH 3 ¢ SEa / g £ fI - S -21

S -; - R - S -21 SEa SEa &-21 - R )SEa F m G9H 1

3 ¢ SEa / g £ fI - S -21

S - R - S -;&-21 SEa m &- S -!&4 -R )SEa 1 F G9H 43 ¢ S _ / g £ fI &- S -21 -;&-21 V- R &- S -21 S

ED ES a & -21 R ) SEa 3

19.

23.

5)

28.

29.

> '[email protected] B m F G ¤ m ¤ G F : m F G ¤

m ' 0f;/ 0 F]3 G J $ L a f;/ -; a 0f;/ ¤ . &-21 & -21 f;/ 0 43 ¢ / f &-21 + &-!& &-21 £ f;/ ¤ &-21 0f;/ a 1 f;/ 0 43 ¢ / f &-21

I+ &-21 . a £ f;/ ¤ 1

m ED &-21 F G9H F m G -; -21 ' $ L a F m G F m G ¤ a -; J m m . D -21 & -21 F G9H / f &-21 - + a 1 &-21 F G ¤ m m D -21 &-21 F G a 1 F GIH / f &-21 O - + a & . a F m G ¤ 1 &-21 &-21 f / &-21 f ']/ " 0 . m £ '*>[email protected] I+ &!- & &-21 f;/ =

27.

m ¤ G F 8

3 L

30.

$

m m ED -21 &-21 F G & -21 F GIH ' m ( 7X! f & -21

- + a 1 . '8>[email protected] F & -21 G = - &-21 fI& 0 7 3 ¢ F 3 G F 7 G -;&-21 - 0fI& ¤ m ED - R PED -21 R F GIHOH 3 ¢ F 7 3 G F 7 G -21 V- R F m G ¤ < &-21 - fI& 0 7 43 ¢ F 3 G F 7 G -;&-21 0fI& ¤ m ED R PED -21 - R F GIHOH 3 ¢ F 7 3 G F 7 G -21 R F m G ¤ &- S -21 - SEa 1 )SEa 1 fI& 0 3 £ fI - S -21 #- )SEa fI& ¤ 1

m ED SEa a 1 - R PED &- S -!&4 R )SEa 1 F G9HOH 43 £ fI S -;&-21 -R )SEa F m G ¤ 1

a E S a S 1 < - -21 - )5n SEa'!1 % fI& 0 43 £ fI SEa 1 )SEa 1 fI& ¤ % m ED &- S -21 R PED a2S -21 -R )SEa 1 F G9HOH 5n% '! % 3 £ fI - S -;&-!&4 R )SEa F m G ¤ 1

a E S a S 1 - P - -21 )5hSE'a ! 1 % fI& 0 % £ fI SEa 1 - )SEa fI& ¤ 1

m ED &- S -21 - R PED a2S -21 R )SEa 1 F G9HOH 5n% '! % £ fI - S -;&-!&4 V-R )SEa F m G ¤ 1

- &-21 )SEa 1 fI& 0 5n'! % (*'! % -;&-21 - )SEa fI& ¤ 1

31.

32.

33.

34.

35.

36.

37.

38.

39.

40.

41.

3 P

+ (

42.

5) ,

m ED - R PED -21 R )SEa 1 F GIHOH '! % m 5n * ( '! % -21 - R )SEa 1 F G ¤

1.14.2. Derivatives with respect to the order

M! ¡\¤ M! 2. M! 3. M! 1 " 4.

1 M! 5. &-21 " " ' ' % & -21 L J ' ( ! $&% $ 1.

"' ' % & -21 L ¤ J ')( &! 0& %

= = @ B C7[email protected] " O 43 £ & £ & <3 £ & 0 ¤ 43 ¢ 43 £ & &-21 & 1 -; 0& " ( L ' % -21 0& 3 43 ¢ / ')J ( $ ! % $ &- -21 0& 1 !& -21 % -; a 1 0& -21 £ & ¤ 3 ¢ £ & <3 £ & 00 6. &-21 & 1 -; 0&" ""' ( L ' % ' % & -21 L ¢ J ')( ! ')J ( $ ! % $ &- -21 & $9% $ -21 & 3 3 / 1 !& -21 £ & ¤ % -; a 1 0&

2 1 M! 7. " a 1

( 7! " ')( $ ! % L 3 ¢ a 1 £ &-21 &4

-;&-21 &-21 J $9% ')(* $ ! % $ 5 1 L $ (*'65 1 L J J 7 7

Y D - F F G 3 F 3 G;G 43 £ & H " ( 7! " ')( $ (:7! % L & -21 ¢ £ a a J 3 1 &-!&4 &4 -; 1

$9% ')(* $ (:7X! % $ 5 & L $ (*'65 1 L J 7 J

Y D - F F G 3 F 3 GG 3 3 £ & H ¤

5N

L

H

1.15. The Struve Functions Hν (z) and Lν (z) 1.15.1. Derivatives with respect to the argument 1.

2.

H fI& 0 " ' ( $ ( 1L m

& - ) F3 G $&% J ')(* $ ! % L
F]3 G $9% J ')(* $ ! % & - ' ( $ L F]3 G J & 5 1L 7 m a -21 m ¤

Y F nG H a 0fI& 3 F nG -21 J 5 Q( 5 1 L F m G J

m . H f;/ 0 F G I+ -; H -; f;/ ¤ m . - H f / 0 F3 G - + a H a 0f / $ 5 1L ( 7X! ' m a & 2 1 3 F G &-21 -21 5nJ ')( $ 5 F m G ¤ 1L J

m m . + a 1 H a 1 f;/ 0 F G &-21 ¤ m m m . EDM &- -21 H F G9H F]3 G - + a -21 H -; F G ¤ m m m . D I+ &- H a 1 F G9H 43 ¢ F G &-21 -;&-21 ¤ m m m . DM a -21 H F G9H F G + -; -21 H a F G 5 1L $ 7 m a ¤ &-21

& 2 1 J G 3 F -;&-21 5:' ( $ 5 F G 1L m J

L 0fI& " ( 1L m

& - ')( $ F]3 G $&% J ')(* G nG L - 0fI& ¤ L F ] F 3 $ ! % J &-

3. 4.

5. 6.

7. 8.

9.

5)

+ (

!; ,

" ( J 1 L & ¢ ')( $ L F 3 G 4 3 10. F 3 G $&% ')(* $ ! % J & - 5 7 1 Y F m G L a fI& F m G a -21 -21 J Q( L 5 F]3 m G 5 1L J

m . L 0f / F G I+ -; L -; f / ¤ 11. F mG - + a . L a f;/ 12. - L 0f;/ $ 5 1L 7 m a & 2 1 G J F G &-21 -1

5n')( $ 5 1 L F 3 m J

m m . ¤ a &-21 13. I+ 1 L a 1 0f;/ F G m F3 m G - + a . -21 L F m G ¤ 14. ED &- -21 L F GIH -; m . 3 ¢ F mG &-21 -I&-21 m I G H F 15. EDI+ &- L a 1

m F3 mG I+ -; . -21 L a F m G a 16. ED -21 L F GIH 5 1L $ ( 7X! ' m a &-21 F3 m G F G &-21 -;&-21 5nJ ')( $ 5 1L J

¤

¤

¤

¤

1.15.2. Derivatives with respect to the order 1.

2.

H M! 7

&

H M! 3 7 & -21

J

1 J L

1

3 0&

!¡!¡

]' " "' &

& L)" &- -21 F G D '])" " ' % & -21 ¢ 3 5,

> @CB = Q

<!¡

7 3 F 3 GIH "" ' L > @CB $&J % ')( $ ! H- & = Q

3.

4.

5.

M]!

3 ¢ a 1 0& #' " ; - "' ¢ 4 3 &

! _!¡

H

L

H

"" ' ' % & -21 ( L 3 J$&% ' ( $ !

H M! 1 * Y C £ & 3 £ & 0

£ & 3 £

H

- & = Q> @CB

0& = = @ B #O [email protected]

H M! *

1 Y £ & 3 £ & 0&3 £ & <3 £ & 0 = = @ B # [email protected] H M]! £ & 3 £ & 0 a 0& a 1

£ 1 £ ¢ 3 & 3 0& -I&-21 0& H a 1 & <3 a 1 & 0 7 a 7 7 £ £ F G 1 F G D C F 3 GIH " ' % & -21 ( L ' % a & -21 1 L)" 3 F G $&J % ' ( $ ! - -21 0& 3 F G 1 $9% J ') ( $ ! ']" L & -21 J 1 L" ¢ J ') ( $ ! % F G 3 3 " -21 J L &- -21 J L &

1 ; ' ( ! 3 / 8F G $&% $ % £ £ Y 43 ¢ a 1 a 0& 1 ¢ 1 - & <3 -21 1 £ - & 3?43 - -21 & -21 0& 3 -21 -21 £ & ¤ ¢!¢ 7 L M! = Q> @CB \¢)\¢ & <3

6.

7.

8.

9.

3 & <3

L M! & <3 7

¢)\¢! \¢)\¢!

53

= Q> @CB

¢)\¢ \¢)\¢

¤

+ (

10.

11.

12.

13.

14.

15.

16.

3 0& 3

!; ,

¢)\¢) \¢)\¢)

¤

( O! ']" "' 0&

¡

7 1 J L " F G &- -21 D 3 F 3 G9H 1 J L '#)" " ""' ' % & -21 ( L J % ') ( $ ! L 0& = Q> @CB 9 $ ( ! '#" "' L M]! ¢ 3 &

¡\ !

; - ""' ' % & -21 ( L 3 $&J % ')

( $ ! L- 0& = Q> @CB 7 7 L M]! £ £ 0& 3 F

C ,G - & <3 * 1

3 43 £ & <3 £ 3P& = = @ B C7 [email protected]

L M]! 4 3 ¢ 7 & -21 3 43 ¢

L M]!

L M]!

1

-1

*

*

£ & 3 £ 3

0&

£ & <3 £

£ & <3 £

& 0 3 £ & <3 £

3

L M]! £ & 3 £ & 0 a 0& a 1

1 £

£ 3 & <'%3 & & 0 -;&-21 0& L a 1 & <3 ( 7X! a 7 F G 1 F G D C £ £ F " ' % & -21 ( L J ')( ! F3 G $9% $ - -21 & ¢ ' % a & -21 $9% J ') 1 L ( " $ ! 43 F G 1

5 5

& 0&3

0& 7 ;- &-2 1

3 IG H

C

¤

& 0 ¤

J

'#"

E

& -21 ¢ J 1 L " ¢ ') ( $ ! % F G 3 3 " ( L ( &-21 J &- -21 J L ¢

1 ; , G ' ( ! % 43 / 8F $&% $ Y a & & <3 £ -21 £ & 1

1 1 3 - -21 0& -21 & <3 £ -21 -21

L M]! £ & <3 £ & a 0& 1

-I&-21

£ £ & <3 & 0 -;&-21 & "" ' " ( ' % & -21 ( L ' % L -21 ( ! 3 $9J % ')

( $ ! a 1 & / ')J ( $ ! % $ % 1 £ £ Y a & 1 - & <3 -21 &

&- 1

2 1 3 -;&-21 0& £ 1 - 1 - & <3 1 -

L

J

17.

£ & ¤

£ & ¤

1.16. The Anger Jν (z) and Weber Eν (z) Functions 1.16.1. Derivatives with respect to the argument

" ( 1 J L & - ')( $ ; L F J 0& F 3 , G $9% ')(* $ ! % J ! Q( Y J & <3 3 ¢ -21 F " ( 1L & - ')( $ ]F 3 ,G -; $&% J ')(* G < L F 3 $ ! % J

1.

2.

Y 3. 4.

F 7 G

J 0f;/ 3

G \ \ ]F 3 ,G ( 5 7 G F G

\ \ F , G 5 5_7 G F]3 G

¤

! -21 ( ¢ ¤ J a 0& 3 4 3 F ¢ ' ¤ , $ L 4 3 J &

2 J m . F G + -; J -; 0f;/ ¤ m ! . & -21 ' ( $ ( 5 7 G G F3 m I+ -;&-21 F F m G

5;

+ (

:1

m - J 0f;/ F 3 G - + a m ! 3 F G m - + a a

. J a f;/ ¤ . & -21 ')( $ 5 5_7 G F]3 m G 1 F m F3 m G - + a . -21 J F m G 6. EDM &- -21 J F GIH -; ! ¤ m . & -21 ')( $ ( 5_7 a a G F m G 3 F G m - + 1 F m F m G I+ -; . -21 J a F m G a 7. D -21 J F GIH ! ¤ m . & -21 ')( $ 5 5_7 G F]3 m G 3 F]3 G m I+ -;&-21 F " ( J 1 L & - ')( $ ; ) ' ( G G 9 $ % 3 L 3 8. E & 0 F F F $ ! % \ \ G J 7 Q( ( 5_7 ¤ Y E & -21 3 ¢ 3 ¢ XF G G F " ( J 1 L & - ')( $ ; G L F]3 ,G $&% ')(* F 3 F 9. $ ! % \ \ ,G J 7 ( 5 5_7 ¤ Y E a & -21 ¢ 3 ¢ a CXF G G F F 7 G 43 ¢ ' $ L E & ¤ 10. 2 J F m G + -; . E 0f;/ 7 F m G + -;&-21 .

11. E f;/ 0 m -; ) ' ( ( $ 5_7 Y & -21 3 ¢ 43 ¢ XF G F m G ¤ 5.

12.

- E f;/ 0 F]3 mG - + a Y & -21 3 .

7 m . a 0 f;/ m F G - + a a 1

¢ 3 ¢ XF ')( $ 5 5_7 G F G ¤ m

E

5:

13.

14.

J ν (z )

m m F G9H F]3 G - + a m " 3 '&(' & F3 G m Y & -21 3 ¢ 3 ¢ m ED a -21 E F G9H F m G I+ -; . -21 E a F m G Y & -21 3 ¢ 43 ¢ D &- -21

Eν (z )

. -21 8

E

XF

E

m F ;- G

')( $ ( 5_7 G F m G ¤

m 3 " ']" 8 F3 G m ')( $ 5 5_7 ¤ G F m G XF

1.16.2. Derivatives with respect to the order 1.

2.

3.

4.

J M! 7 & -21 3

' % & -21 1 J L" 1 J L '#)" "

#' ")"

L $9J % ')( $ ! 0&

&

H

( 7X! ' & -21

& 2 1 F , G 3 '])" " ' J M! % 43 ¢ &-21 & -21 $&J % ')L ( $ ! & -I ( 7X! '

E M! 0& ']")" ' % & -21 L J ')( ! $&% $ 3 H 0& 7 & -21 ¢ ')( $ ¢ 3 3 XF ']")" ( L ' E M! % &-21 $&J % ' ( $ ! H -I 7 & -21 ¢ ')( ¢ 4 3 43 XF 5<

7 2- 1 5_7 G

¢ F ')( $ 5_7 G F G ¤

-; & 5 7 ¤ & -21 ')( $ _ G F G F H

1L J

F ,G - -21 1 )L "" J

a -21 7 5 ')( $ 5 7 ¤ F]3 G 1 n

& 3 ¢ & a -21 7 $ 5_7 G F G 1 5n')( $ 5 7 ¤

+ (

!<

1.17. The Kelvin Functions berν (z), beiν (z), kerν (z) and keiν (z) 1.17.1. Derivatives with respect to the argument 1.

2.

3.

4.

5.

6.

7.

f;/ 0 F mG I + -; . D

F

'

f;/ 0 ' m . G + -; D

2

0f;/ <3

2

0f;/

. I+ a 1

. I+ a 1

. I+ a 1

. I+ a 1

' '

7 m a 1 f / 0 3 F G &-21

O')(:7X! Y D F f " G FOf " G O ')(:7! F f " PG 7 F m G &-21

a 0f;/ 1 O')(87! Y D FOf " PG F f " PG ')(:7! F f " G 3 ( 7! ' & m

/ F G &-21

-;&-21 O0')f (8 7! Y D FOf " G F f " G ')(:7X! F f " G 3 ( 7X! ' m &-21

-;&-21 0')f;/ (:7X ! F G Y D F f " G FOf " PG O')(:7X! F f " G

2

0f;/ H ¤

2

0f;/ H ¤

. k - + a & a 1 0f / O'65_7X! Y D m 3 a 1 F

F f " G9H ¤

F f " G9H ¤

F f " G9H ¤

F f " G9H ¤

/ F Om G a 1

m m a 1 F G -;&-21 F G m G -;&-21 F G9H 5L

#

" "! #$" #$$!

O'65 7!

m m D a 1 F G -;&-21 F G m m ¤ a 1 F G -I&-21 F GIH / 8F Om G a 1

a . 8. k - + & a ; f / '65_1 7 ! m m Y D a 1 F G -I&-21 F G m m 3 a 1 F G -;&-21 F G9H '65_7! m m G D a F G F 1 m -;& -21

m ¤ a 1 F G -I&-21 F GIH m m 0 I f & 9. EDM &-21 A F

fI& G9H &-21

& 2 1 # ' " O')(87! £ '8>[email protected] B m ! / fI H = &-21 D m m 10. ED &-21 A F

0 I f & 3 &-21

&-21 fI& G9H '#" ! O ) ' 8 ( 7 ! m £ '8>[email protected] B / fI H = &-2 1 D m m

11. D &-21 F

&m -2 1 fI& m 3 &-21 fI& G9H # ' " ' (87! m ! £ fI / £ fI &-21 D / O')(87! £ fI / £ fI H ¤ / 3 m m

12. D &-21 F

&m -2 1 fI& m &-21 fI& G9H ] ' " O')(:7X! m ! £ fI / £ fI &-21 D / O')(87! £ fI / £ fI H ¤ / m ']" . f < 3 f I +

13. k &-21 / / & ! & <1 O')(:7X! O')(:7X! <1

£ £ Y D H

&-21 f;/ <3 &-21 f;/ '* >? = [email protected] 3

5P

+ (

!<

m ']" . 0 f f I +

14. k &-21 / / & ! & <1 O')(:7X! O')(87! <1

£ £ Y D H

&-21 f / & -21 f / '* >? = [email protected] 15. 0f;/ F mG I+ -; . 2D ' 0f / 3 ' 0f / H ¤ 2 2 16.

17.

18.

19.

20.

21.

22.

0f;/ F mG I+ -; . 2D ' f;/ ' 0f;/ H 2 2 . + a 1 a 1 0f;/ 43 ¢ ' & f &-21 - / A Df " O'65 ! H . I+ a 1 a 1 0f;/ 43 ¢ a 1 '% & f &-21 - / A D f " O'65 ! H ( 7! ' m a . a k - + & a 1 f;/ F O G 1

O'65_7X! m m Y D a 1 F G 3 a 1 F G9H '65_7X! m m G F 3 £ F a a 1

1 G ( 7X! ' m a . a k - + & a 1 f;/ F G 1

'65_7! m m Y D a 1 F G 3 a 1 F GIH '65_7X! m m £ G F F a a 1 G 1

. k &-21 &-21 f / &-21 0f / 3 ¢ f &-21

I+ &-!& ')(:7! 7X! £ f;/ <3 O')(: £ Y D

&-21

&-21 f;/ H k &-21 &-21 0f;/ 3 &-21 0f;/ 43 ¢ / f &-21

I+ &!- & . ')(:7! 7X! £ f;/ O')(: £ Y D

&-21

&-21 f;/ H ;9N

¤

¤

¤

¤

¤

¤

¤

#

23.

24.

" "! #$" #$$!

k &-21 & -21 0f;/ &-21 f;/ f;/ 4 '#" 3 &-21 0f;/ O')(:

& 2 1 X 7 ! m . £ f /

I+ &-!& D

& 2 1 O')8 ( 7! £ ;f / H ¤ 3 & -21

k &-21 &-21 0f;/ &-21 f;/

']" &-21 0f;/ ') (: &7-2! 1 0f;/ 4 m . £ f;/

+ &!- & D

& 2 1 O')(:7X! £ ¤

&-21 f / H

1.17.2. Derivatives with respect to the order

M! ' % & -21 M! ' % & -21 M! M! M]! ' % & -21 M]! ' % & -21

1.

2.

3.

4.

5.

6.

3

0& 3 ""'

L $9J % ')( $ ! D

"" '

& <3

L 9$ J % ')( $ ! D

& ] $ (* '!

& ] $ (* '!

& <3

& <3

&

3 & <3 & ¤ & <3 & ¤ & "" ' $ ( '2! L $9J % ')( $ ! D 3

""'

# $ * ( '!

0&

# $ (* '!

$ (* '!

& H ¤

& H ¤

& H ¤

& H ¤

&

$ (* '! L $9J % ')( $ ! D ;

$ * ( '!

+ (

M! a 1

7 a 0& C £ & 3h

1

7 3 a 1 & V 1 N

!< ,

7.

7#7 ( #$ N ! 1 &

#$ ( 1

& ! & 1 &

( #$ 1 ¢ £ ! 3 / -;&-21 & -;&-21 0& 1 N 1 & 1 ( #$ & ! ¢ 43 & -;&-21 0& 3 -;&-21 0& 1 N & 1 ""'

')( $ ! ' % & -21 L ')( $ ! J ')( ! a a 1 0& 1 & H $9% $ D " ' %

L -21 ')J ( $ ! % $ % &- a 1 0& 3 $ (* (:7X! 1 £ & $ (* Q(:7X! £ Y D -21 $ (* (:7! -21 & H £ & &- a 1 & &D

2 1 $ (* (:7! 3 £ & H

2 1 $ (* Q(87! £ 3 43 ¢ a a -;&-21 0& D & 1 $ (*Q(:7X! £ & H 1 - Q(:7X! $ (* a a ¢ £ 3 3 -;&-21 & &D & 1 $ (* (:7! £ @ B 3 & H = 1 M! M! 8. a a 1

1

( #$ 7 1 a & # N ! 1 & & 1 & 1

7 #7 ( #$ 7 £ ! a 1 & C & 3h N & 1

; ,

#

" "! #$" #$$!

¢ £ 3 / ;- &-21 & <3 ;- &-21 0& 1 N ¢

3

#$ 1! ( 1 & 1 ( #$ & N ! 1 &

')( $ ! a & H 1

& ;- &-21 0& -;&-21 & 0 1 ""' ' % & -21 L ')( $ ! 3 $&J % ' ( $ ! D a & <3 1 " ' %

L -21 ')J ( $ ! % $ % &- a 1 0& $ ( Q(87! 1 $ ( (:7! £ £ & H Y D & < 3

-21 $ (*Q(:7X! -21 £ 3 &- a 1 0& D

-21 & $ (* (:7! £ & H

2 1 $ (* Q(:7X! £ & 3 3 ¢ a a -;&-21 & &D 1 $ (* Q(:7X! £ 3 & H $ (*1 Q- (87! ¢ £ a a 43 -;&-21 & &D & 1 $ (* (:7X! £ @ @ B & H = 1 M! 9. 1 -; 7 #7 ( #$ 7 £

! C

& <3h N & 1 -; & 1] #$ 7 1! ( 3 1 -; 0& V 1 N 1 & & &

#$ 1! ( £ ¢ 3 / &-21 & &-21 0& 1 N 1 & 1 ( #$ & ! ¢ 3 & &-21 0& 3 &-21 0& 1 N & 1 "" '

' % & -21 ( L ')( $ ! ')( $ ! 3 $9J % ')

( $ ! D 1 - & 1 - 0& H

;93

+ (

Y D

!< ,

" ' %

L -21 ')J ( $ ! % $ % 3 ¢ a &- -21 0& $ (* Q(:7X! 1 $ ( (:7! £ £ & & H 1 1 $ (* (:7! £ ¢ & 3 a &- -21 & &D 1 $ (*Q(:7X! £ 3 & H (*1 Q (8- 7! $ ¢ 3 43 -; a 1 & &D £ &

2 1 $ (* (:7! £ & H

2 1 $ (* Q(:7X! 3 43 ¢ -; a 1 0& D £ &

2 1 $ (* Q(:7X! £ @ @ B 3 -21 & H =

#$ 1! ( 0& V 1 N 10.

1 1 -; & & & 1 -; 7# 7 ( #$ 7 £ ! 1 -; & C & 3 N & 1

( #$ 1 3 3 ¢ / £ &-21 0& 3 &-21 0& 1 N ! 1 1 & ( #$ & ! ¢ 3 & &-21 0& &-21 0& 1 N & 1 ""'

' % & -21 ( L ')( $ ! ')( $ ! J ')( ! & <3 0 & H 1 1 $9% $ D " ' %

L ')J ( $ ! % $ -2 1 % 3 ¢ a &- -21 0& 3 $ ( Q(87! 1 $ ( (:7! £ £ Y D & <3 & H 1 1 $ (* Q(:7X! £ 3 43 ¢ a &- -21 0& D & 1 $ (* (:7! £ & H 1 $ (* Q(:7X! 3 3 ¢ -; a 1 & D £ &

2 1 $ (* Q(:7X! 3 -21 0& H ;!5

M]!

7

#

¢

" "! #$" #$$!

$ ( Q(87!

43 ;- a 1 & &D $ (*Q(:7X! 2- 1

£ &

-21 £ & H = @

@ B

M ]! & -21 0& 3 &-21 & £ &-21

3 & &-21 & 43 ¢ C £ & 1 -; 0& 3 3 ¢ 1 -; & ( #$ 1 3 &-21 0& 43 ¢ 1 -; & 0 1 N ! 1 & & &

¢ &-21 0& &-21 0& 43 1 -; 0& #$ 1! ( ¢ 3?43 1 -; & 0 1 N 1 1 &

3 7 C &-21 & <3 £ / £ & &-21 & <3 &-21 & ( #$ & ¢ ¢ ! 3 1 -; & 3 1 -; 0& 1 N & 1 7#7 ( #$ ¢ ! 7 &-21 0& 3 43 1 -; & 0 N & 1 ""'

' % & -21 L ')( $ ! ')( $ ! J ' ( ! -21 & -21 & H $&% $ D " ( L ' % -21 J ')( $ ! % $ % &- -21 0& 43 ¢ a -; a 1 & 3 $ 1

5 P5 ! 5 P5 ! £ & <3 $ Y D £ & H

2 1 2 1 ¢ 3 a -; a 1 0& 3 &- -21 0&

5 P5 ! 5 P5 ! £ & $

£ & H Y D $

-21 -21 @CB = M! 3

12. &-21 & &-21

£ ¢ ¢ C & 0 &-21 0& 43 1 -; 0& 43 1 -; 0& ; ;

11.

+ (

3

¢ &-21 0& 43 3 &-21 0& 3

L1 1! (

1 & & &

#$

1 -; & <3 &-21 & 0 1 N ¢

0 & 4 3

&-21 1 -; 0& #$ 1! ( ¢ 43 1 -; & 0 1 N 1 & 1

¢ & &-21 & &-21 & 3 3 1 -; 0& #$ & ! ( ¢ 43 1 -; & 0 1 N & 1

7#7 ( #$ ¢ 3 7 43 1 -; & &-21 & 0 N ! 1 & ""'

' % & -21 L ')( $ ! ')( $ ! 3 $&J % ' ( $ ! D -21 & <3 -21 & H " ' % L -21 ')J ( $ ! % $ % 3 1 Y 43 ¢ & 3 ¢ -; a 1 &

& 2 1 5 5 ! 5 5 ! £ & $ Y D $ £ & H

2 1 2 1 ¢ ¢ M3 &- -21 0& 3 43 -; a 1 0& $

5 P5 ! 5 P5 ! £ & <3 $

£ Y D -21

-21 & H @CB = M]! M]! 43 ¢ a 1

¢ a 1 ¤ 13. 0 & ! 3

& 2 1 &-21

M! 1 -; ¢ a M! ¤ 43 1 &-21 0& 43 ¢ 14. 1 -; &-21

1.18. The Legendre Polynomials Pn (z) 1.18.1. Derivatives with respect to the argument 1. 2.

j S 0 fI& £ 3 ¢ f S a ;- 1 fI& 5n' ! % m ¢ G 3h9f

O -; SE 1 a -; fI& F ( * ' ! % 1L' J

;9:

= > I' @CB = > '[email protected] B

& '

-21 ¢ h 3 f & &-21 j f;/ 0 7 43 ¢ F G f 63hf -;&-21 j F 7

m m 4. DM Q3hf &-21 j F

GIH 43 ¢ F 7 G f9 Q3hf9 O -;&-21 j F

m a ¢ a f / 0 5. 1 3hf9

& &-21 j 3 ¢ O'6 5_7 F 7 1 G f9 a 1 -;&-21 ¢ 3hf9 O& -;&-!&4 j F 7

m 5 m a . 6. D - + 1 Q3hfI &-21 j F m G9H F 7 G f

-;&-21 63hfI -21 j F " m

5 m 7. D fg3h& &-21 j F m G9H 43 ¢ F 7 G f fT3h& -21 j F " m

5 m S S 8. DM fg3h& &- -21 j S F m GIH ( % '2! % f + S -; . 0fg3h& - S -21 j S F 5 m G = > -; m SEa 1 . 0fg3h& SEa j S F 5 m GIH 9. DM - + m n 5 ' ! % 43 ¢ SEa a 1 . fg3h& S j SEa F 5 m % f

+ m 5 m ¤ j f

10. D 0Q3hfI 2j F m GIH D F " m G9H E( m S 11. D 0 3hfI& j S F ( m GIH " ' O( m !% C7 ( ( ! ! (* > S . ' '! % 0 3hfI& + -; j S -; F ( m G = m (8 S S 12. D &- -21 0fg3h& j S F m ( m GIH " ' ' ! % m S- -21 fg3h& + S -; . j S F m (8( G = > ( O ' ! % m m -; 1 ( L ' J

. E S a j

+

S 13. ED 0 3hf F ( m GIH 1 43 ¢ 5n% '! % 3hf9

- + SEa a 1 . j SEa F ( m 3.

; <

G ¤ G ¤ G ¤ G ¤ G ¤ '[email protected] B G ¤

'[email protected] '[email protected]

G ¤

+ (

14.

15.

m

ED 0fg3h& -21 j F#" ¢ 3 GIH F m

a 1 D a 1 j a 1 F " ¢ 3 GIH 3 ¢ 'g 5 7 F

L1

7 G 0 fg3h& -;&-21 j ( m, ¤ ) * 7 G 63hf9 -;&-21 j E( m-, ¤ )+*

k &-21 0fI)3 ¢ -21 j / ¢ 3 fI F 7 G ;- &-21 fI 3 ¢ -I&-21 j F 7E( 7 G ¤ m

7 ¢ 17. D &-21 3hfI& j

F 7( m GIH 3 ¢ F 7 G -;&-21 j / ¢ 3hfI ¤

S 18. ED 0 3hf j S F ( m GIH

( !O% '! % 3hf9

S -; j S > '[email protected] B F

- ( m G = a S 19. 1;D 0 3hf9 O j S:F ( m GIH

( O')! % (:7X! % 3hf S -;&-21 j S > 'g5 [email protected] F ( m G =

- &-21 m 20. ED -;&-21 0f9 3h

j F (8 GIH m

7 43V F G f - &-21 f9 3h

j F m G ¤ m (8 m S S 21. D &- -21 0f 3h j S F m (8 GIH

m (*! % '! % f9 - S -21 0f9 3h

S -; j S > '[email protected] F m (8 G =

- m a S S 22. 1;D &- 0f9 3h O j S:F m (8 GIH

! % m 3 (* a S ')(:7X! % f9 1 - -21 0f9 3h

S -;&-21 j S - &-21 F m (8 G = > '[email protected] ¢ ¢ a S 23. &-21 3 f9

& 1 < 3hf9

& &- -21 j S 0f;/ ! (* % '! % F mG -21 ¢ 3hf9

& - S -21 j S > '[email protected]

- 0f;/ =

16.

;9L

24.

25.

26.

27.

) ' * '

m ED 0Q3hf9

a 1 ED S 0 Q3hf9

&- S -21 j S:F GIHOH m (*! % '! % F mG S ;- 63hf9

- S -21 j S > '[email protected]

- F G = m 5n D P D fT3h& S j S F m (* G9HOH D (*% '! % H fg3h& S -; j S F m 5(8 G = > '[email protected] -; m m 5 D P D SEa fg3h& - S -21Oj S F m (8 GIHOH D 5n% '! % H f S fg3h& - S -;&-21 j SEa F m 5(8 G ¤ m m 5 ED PED fT3h& - S -21 j S F m (8GIHOH ¤ D 5:% '! % H fg3h& - S -;&-21Oj SEa F m 58 ( m G

1.19. The Chebyshev Polynomials Tn (z) and Un (z) 1.19.1. Derivatives with respect to the argument 1. 2. 3.

4.

5.

6.

ST fI& 0 £ &-21 3 ¢ f S m 5 EDM -1 ] fg3h& F m (8GIH 3 m 5 S

0f9 3h

S ) m (8 , ( !O% '! % f9

5 S S

&- -21 0 3hf9

S ) ( Y - S -21

> '8>[email protected] 0 I f & = -; 7 ¢ F G -;&-21 0fg3h& ¤

3h

S -; m m , 3 f9

S -;

m 5 S -; ) m (8 , ! % ( O'! % f9 5 m S -; ) ( m ,

= > '[email protected] B = > '[email protected]

m

D0 f 3h S S F m (8 GIH m (*! % '! % 0f 3h S -; S > I' @ B F (8 G = m

- m

a 1;D0 f9 3h

S S:F m (8 GIH ! % m (* ' 5 R7 @CB O')(:7X! % ; f9 3h

S -;&-21 S - &- F m (* G = > g ;9P

+ (

m

ED0 f9 3h

SEa 1 SEa 1 F m (8 GIH m (* 5_'g75 ! % 7! % f9 3h O S ;- a 1 S m (8 a F

- 1 S 8. ED0 3hf9 O - S:F ( m G9H ')(:7X! % . 3 5n(: 7X! % 3hf9

- + SEa SEa F ( m G S S 9. EDM &- -21 3hf9

S F ( m GIH

(*!O% '! % f9 - S -21 3 f9

S -; S ( m F

- a S S 10. 1;D &- 0 3hf9 O S F ( m GIH

! % 3 (* ' (87! % f9 a

- S -21 3hf9

S -;&-21 S - &- F

P1

7.

= > '[email protected] G

= 5n'*>[email protected] G

= > '[email protected]

( m G ' [email protected] = > 6

ED &- S - 0 3hf9

SEa 1 SEa 1 F ( m GIH ( O5_'67X5_! % 7X! % f9 - S - 0 3hf9

S -; a 1 S a F ( m G

- 1 = > '[email protected] B m 5 S 12. ED &-21 PED fT3h& S:F m (8GIHOH £ - (*!O% '! % -21 ] fg3h& S -; S F m 5n * ( G = > '[email protected] B m -; m 5 S a S 13. D 1 D &- -21 fg3 S F m (8 GIHOH (*! % '! % F m G - S -21 fg3h& S -; S F m 5(8G = >_'[email protected] -; m m 5 S a 14. ED 1 PED -21 0fg3h& - S8F m (8GIHOH £ - £ fT3h - S -; SEa F m 5(8 G ¤

m SEa &-21 fT3h& - S S F m 5(8 GIHOH 15. ED &-21 PED m £ - £ f S -21 0fg3h& - S -; SEa F m 5(8 G ¤

m £ fI a 1 0fI& = > '[email protected] 16. ST0fI& S -; 11.

: N

17.

18.

5 m n S ; f9 3h

S ) m (* , 5n ! % (* '65n ! % ;0f9

m S S &- - f9 3h

S ) m 5nO! % (* '65nO! % f - S - 0f

T n (z )

3h

S ;- S ;- 5n (* , 3h S -; S -;

U n (z )

m 5 ) m (8 ,

= > '[email protected] B

m 5 ) m (8 ,

= > '[email protected]

m

D;0f h 3 S S F m (8 9G H m (*O5_'675 ! % 7! % ; f9 3h O S -; S F

- m (* G SEa 1 SEa F m (* G9H 20. ED;0f9 3h

1 m 5:O! % m (* O'65n ! % ; f9 3h

S -; a 1 S - a 1 F m (8 G S 21. D 0 3hf - -21 S F ( m GIH 43 ¢ 5n% '! % 3hf9

- + SEa . -21 SEa F S S 22. ED &- - 0 3hf9

S:F ( m G9H

( O5_'67X5_! % 7X! % f9 - S - 3hf9

S -; S

- F ( m G S - & 0 3_f SEa 1 SEa F ( GIH 23. D &- ! m

1 ( O58' O58! % O! % f9 - S !- & 0 3_f9

S -; a 1 S a F (

- 1 19.

24.

25.

26.

m 5 D & -21 P D &- S 2- 1 fg3h& S S F m (8 G9H H m S m 5nO! % * S ( 'g5:O! % F G - -!&4 0fg3h& -; S -; F m m 5 D a 1 D 0fg3h& S S F m (8 GIHOH 5:O! % m £ - (* O'65n ! % 1 fT3h& S -; S -; F m m 5 D a 1 D SEa a 1 0fg3h& - S - S F m (8 GIHOH 5:O5_'67X5_! % 7X! % F m G SEa 1 fg3h& - S -;&

:1

= > '[email protected] B = > '[email protected] ( m G ¤ = > '[email protected] m G = >:'[email protected] B

5 (8G

= > '[email protected]

5n (* G

= > '[email protected] B

m 5 ¤

ES a F m 8 ( G

+ (

27.

,9N1

m 5 ED &-21 PED 1 0fg3h& - S - : S F m (8GIHOH ' 5 7! % ¤ £ - 5n 5_g S -;&- ES a F m 58 7 ! ( G % g f h 3 & m

1.20. The Hermite Polynomials Hn (z) 1.20.1. Derivatives with respect to the argument

% £ * ( '! % fI S -; 0fI& = > '[email protected] 1. ST fI& 0 2. -21

7

ST0f / 4 3 ¢ SEa £ S _ F 3 G -;&-21 S-;&-21 0f9

& ¤ 3. SEa 0f;/

1 3 ¢ SEa £ SEa 1 ? F3 7 3 G fI -; a 1 -; a 1 0f & ¤ S ! % S S T f;/ 0 (* '! % - S -21 S - 0f;/ = > '[email protected] 4. &- -21 S 5. &- -!&4

SEa 1 f / 0 5_7X! % (* '65_7! % - S -!&4 S - a 1 f;/ = > '[email protected] B m % m ( '2! % 43 £ fI -;&-21 S -; F G = > '[email protected] 6. &-21 S F G m S S:F m GIH 43 ¢ ( !O% '! % S -; S = > '[email protected] 7. EDM

- F G SEa 1 SEa F m GIH 8. DM

1 m 3 ¢ (* 5_'67X5_! % 7X! % S -; a 1 S > '[email protected] B a F

- 1 G = 7 m m ¤ ¢ £ _ S S 9. DM &-21

F 3 G -21 S-;&-21 ) ,

S F GIH 3

m 10. ED &-21

SEa 1 F G9H 3 ¢ S £ SEa 1 _ F]3 7 3 G fI -!&4 -; a 1 m ¤ S ) , W W W W 11. ED - S 0fI& H 3PfI - SEa fI& ¤ W W W W m m S F G9H f -;&-21 - SEa F G ¤ 12. D &-21 - :,

13.

14.

15. 16. 17.

18.

19.

20.

W

# . '/

ST0f;/ H 3 ¢ S £ S -I&-21 - W -;&-21 f9

& ¤ ES a W ED V- SEa 1 0f;/ H 3 ¢ S £ SEa 1 fI -I a 1 #- W -; a 1 0f & ¤ SEa ' W W ( X 7 ! D SEa &-21 - ST f / H ' S -21 - SEa 0f / ¤ W W ( 7X! ' S E S a ' V- SEa a 0f;/ ¤ D V- SEa 1 0f;/ H

1 W m D &-21 - S F GIH 3 ¢ SEa £ S -1 - W -;&-21 m ¤ SEa ) , W m ED &-21 - SEa 1 F GIH 43 ¢ SEa £ SEa 1 fI - &4 - W -; a 1 m ¤ SEa ) , W m S D - -21 - S F G9H 7 ' - S -;&-21 - W SEa F m G ¤

W m ED - S -21 - SEa 1 F G9H 7 ' - S -;&-21 #- W SEa a F m G ¤

1 ED -21 -

1.21. The Laguerre Polynomials Lλn (z) 1.21.1. Derivatives with respect to the argument 1. 2. 3. 4. 5. 6.

(Q(*')(:7 '65_7 c & c 0& c & <3 c a 1 0& ¤ Sc 0fI& 3PfI Sc a -; fI& c Sc fI& 0 3 ¢ 3 d 3 c -; &- S -21 Sc 0fI& 43Pd 3 - S -21 m m D &-21 S F GIH f -I&-21 S -; F G : 3

¤ c a 1 0&

= @ B = > '[email protected]

¤ Sc -; 0fI& Sc -; fI&

= > I' @CB = > '[email protected] B

+ (

m m EDM &- c -21 Sc F GIH 3Pd 3 - c -21 Sc -; F G ¤ m ¢ m S 8. DM Sc F GIH 4 3 43Pd 3 S -; Sc -; F G 43fI - c a fI& ¤ 9. - Sc 0fI& S n 5 ' ! % c -;a fI& ¤ 10. c - Sc 0fI& % c -; - SE a2SEa - Sc fI& 0 5n% '! % c a2S - SEc a fI& ¤ 11. c m m ¤ a 12. &-21 - R Sc 0 f - R Sc F G 5:'! % m ¢ G 3 % - c -21 -R SEc -;a 13. &- c -21 -R Sc F m S 14. - c - -21 #-R Sc F G 3 ¢ 5n% '! % - c - S -;&-21 -R

, ,

7.

= > '[email protected]

m F G ¤ m ¤ SEc a F G

1.21.2. Derivatives with respect to the parameter 1.

-/' M! & -21 7 ')( $

= @ B

c &

1.22. The Gegenbauer Polynomials Cnλ (z) 1.22.1. Derivatives with respect to the argument 1. 2. 3. 4. 5. 6.

7.

S c 0fI& £ fI 0d2 S c a -; 0fI& = > '[email protected] &c a2SEa &-21 c S 0f;/ 0d2 &c a2S -21 c S a f;/ ¤ c a2SEa &-21 c SEa 1 f / 0 d2 c a2S -21 c SEa a 1 f / ¤ &- S -21 c S 0f / 0d2 - S -21 c S a - 0f / = > '[email protected] &- S !- &4 c SEa 1 0f;/ 0d2 - S -!&4 c S a - a 1 f;/ = > '[email protected] 5n'! % 5nO')(:7X! % % (87! % % SEa &-21 ¢ 3hf & 4c -21 c S 0f;/ £ -; % 7 Y C7 ( &! ' S -21 ¢ 3hf9

& c -;&-21 c SE-;a f / ¤ n 5 ' ! n 5 O 6 ' _ 5 7X! % % % 5 7! % % SEa ¢ 3hf9

& c -21 c SEa 1 0f;/ £ -; % 7 Y C7 ( &! ' S ¢ 3hf9

& c -;&-21 c SE-;a a f;/ ¤ 1 :95

0 ' /

£ fI d2 -;&-21 c a F m G = > '[email protected] S -; m S c F ¢ 3 G9H 9. (* 6( ! ' m C7 ( &! ' &- c fT3 £ & c -;&-21 ES c -;a F ¢ 3 G ¤ 5 m 5 £ m n ( G

d2 f 0fg3h& -;&-21 S c a -; F m * 10. D 0fg3h& &-21 S c F m (8 GIH = > '[email protected] B m 5 11. D&c -21 fg3h& &- c S c F m (8 GIH &! ( ( 43 ¢ 5n% '! % Q(*J O 1 '! ( L ' c -;&-21 fT3h& &- .c c -; F m 5(8 G ¤ &(' SEa m m 3 ¢ S d2 S -; c a F m G S = > '[email protected] 12. D c S F GIH

S - SEa 1 c SEa F m GIH 3 ¢ 0d2 S -; a 1 c S a a F m G 13. ED

1

- 1 = > '[email protected] B m S 14. D - c - 0Q3hf 4c -21 c S F GIH

7 3 £ -I 5n% '! % 5nO(8')7(:! % %7X! % % C7 ( &! ' - S - c Y 0Q3hf c -;&-21 c SE-;a F m G ¤

m S 15. ED - c - -21 0Q3hf9

c -21 c SEa F G9H

1 43 £ -; 5n% '! % 5n 5_'67X5_! % %7! % % C7 (7 9! - c - S -21

' Y 0f9 3h& c -;&-21 c SE-;a a F m G ¤

1 5 m S S 16. D 0fg3h& &- -21 S c F m GIH ! ( C7 (* Q( ! ' 5 m . C( 7 9( > '[email protected] Q( ! ' (9! ( "' f

I+ S -; fg3h& - S -21 S c F -; m G = a2SEa &-21 S c F 5 m GIH S 17. ED - c - 0fg3h& c m n 5 ' ! % 43 ¢ SEa . fg3h& c a2S -21 SEc a F 5 m G ¤ %

+ f c m S 18. ED 0 3hf9

- c - S c F ( m GIH 3 ¢ 5n% '! % 3hf9

- c - + SEa . c F G ¤ SEa ( m 8.

m EDM &-21 S c F IG H 43 DM &- cI fg3 £ & 4c -21

£ -; 5n% '! % C7

:;

+ (

, , ,

1.22.2. Derivatives with respect to the parameter 1.

2.

' / M! 5n')(* $ d2 3 d2 c & $ ( 5n')( $ ! &c - & 1 = = @ [email protected] 7 D F d 7 G 3 Fd G 3 £ £ d2 £ £ d £ H c & '#")" $ 5 9! & -21 7,5hC( 7X! £ ')( $ ! P5 $ 5:'! c & = @CB

3.

4.

£ £ d2 <3 £ d2 c & 7 (&! " 3 P5n'! " F $ $ 5:Q(:7G 1 7 5n')(* $
£ 3 ¢ c a & &-

&- &

c

= = @O [email protected]

= = @& C! @MB

1.23. The Jacobi Polynomials Pn(ρ, σ) (z) 1.23.1. Derivatives with respect to the argument 1. 2. 3. 4. 5. 6.

7.

8.

5 65n'65_7 . ¤ a . 7<5 j + 1Rl & <3 j + l & 5n')(*'2 . . + -21Rl 0& 7(8 j + l & ¤ 7 . a . ¤ j + l & <3 j + l 1 & '65n'2 #7 6(85n ' ! j + l V-21 . & <3 65n j + l . 0& 7 8 ( . m . ¢ j S + l fI& F G j S + -;a 9l a 0fI& = . M ¢ fI& j S + l 0fI& 3fI 3_3 ¢ fI V-; j + a 9l-; S . ¢ S M fI& &- -21 j S + l 0fI& 43fI 4383 ¢ fI - S -21 j + a 9l . 0fI& = S -; . ¢ fI& a a2SEa j S + l fI& f ¢ ¢ fI& a a2S j + a 9l S

. j + l & 5n'! 3 ]7 * ( j 5 g5n'65

7(*

: :

= @ B

.

> '[email protected] 0fI& ¤

.

> '[email protected] B 0fI& ¤

&' 3 465 78

. M ¢ 3hfI& ¢ If & j S + l5n 0fI& 43 £ If % '! % ¢ 3hfI& -; ¢ fI& -; j + -;9lV-; . fI& ¤ SEa ¢ SEa a ¢ fI& j S + l . fI& 10. M 3hfI& 43 £ fI 5n% '! % ¢ 3hfI& a2S ¢ fI& -; j + lV-; . fI& ¤ SEa . ¢ a 11. M fIQ3 j + lC-;&-21 fI& m 5_7 ¤ 5_5 77! ! ' f fIQ3 ¢ a 1

, ' ) * . k a 1 j S + lM1 - S 5 -;7 ! ¢ ( fI(:

7X ! 12. ( (*'65_ 7! ( ' - a 1 j + -;9lM1 - SEa . ¢ fI

¤ S . m 13. ED &-21 j S + l F G9H - ¢ F3 mG -;&-21 j + a 9l a . F mG = > '[email protected] B S -; . m 14. &- -21 $ f9 j S + l F G . m f 4 3 83 - -21 0 fI -;2j S + a 9l V-; F G ¤ . m S S 15. D 0 fI &- -21 j S + l F GIH f 3:3 S -; 0 fI - S -21Oj + a 9l . F m G = > '[email protected] B S -; S fI a a2SEa j S + l . F m G9H 16. ED - - V- -21 0 . m 43fI ¢ - - V- S -;&-21 0 a a2S j S + a 9l F G ¤ I f . m a 17. ED - - &-21 Q3hfI 0 fI j S + l F GIH . m £ fI 5n% '! % - - a &-21 0Q3hfI -; $ fI V-; j SE+ a -; 9l V-; F G ¤ . m SEa a S 18. ED - - V- -21 0Q3hfI fI j S + l F GIH £ fI 5n% '! % - - V- S -21 0Q3hfI a2S fI V-; j + l -; . F m G ¤ SEa . ¢ m 19. ED &-21 j S + l F 3 G9H F mG ¢ -I&-21 j + a 9l a . F ¢ 3 mG = > '[email protected] B S -; 9.

:<

+ (

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

,931

. m m . ED S j S + l F ¢ 3 G9H 43 ¢ 3 g3 S ;- j S + -;l a F ¢ 3 G = > '[email protected] B m . ED - - V- S -21 j S + l F ¢ 3 G9H 43 ¢ ¢ - - V- S -;&-21 j + l a . F ¢ 3 m G ¤ S m . ED &- -21 j S + l F ¢ 3 GIH 3 ¢ 3 ¢ - -21 j + -;9l a . F ¢ 3 m G ¤ S m . ED S 0fg3 £ & j S + l F ¢ 3 G9H m . £ 3 83 S fg3 £ & -; j S + l V-; F ¢ 3 G ¤ m . ED S 0fg3 £ & &- S -21 j S + l F ¢ 3 G9H . m f 4 3 83 S -; 0fg3 £ & - S -21 j S + -;a 9l F ¢ 3 G = > '[email protected] B m . ED S 0fg3 £ & &- S - -21 j S + l F ¢ 3 GIH m . 43 £ 3 g3 S 0fg3 £ & - - S -21 j S + -;9l F ¢ 3 G ¤ m . D - - V- S -21 0fg3 £ & j S + l F ¢ 3 GIH £ 5n% '! % - - - S -21 fT3 £ & V-; j + l-; . F ¢ 3 m G ¤ SEa m . D &- - V-21 fg3 £ & j S + l F ¢ 3 GIH £ 5n% '! % &- - -21 0fg3 £ & V-; j + -;9l -; . F ¢ 3 m G ¤ SEa m . ED - - V- S -21 0fg3 £ & a2SEa j S + l F ¢ 3 G9H £ 5n% '! % - - - S -21 fT3 £ & a2S j + -;9l . F ¢ 3 m G ¤ SEa . m 5n ED a 0fg3h& j + lC- -;&-21 F m (* G9H 5 5_77X! ! ' f D a 1 F " ¢ 3 G9H ¤ ' m . 5 m D - -;&-21 63hf9 2j + lC- -;&-21 F E( m GIH 3 ¢ 5_5 77! ! ' - -;&-21 D a 1 F " ¢ 3 mQG9H ¤ ' : L

K

E

D

1.23.2. Derivatives with respect to parameters

' 3 465 78 M]! . £ ¢ <3 - ¢ 0 j + l 0& 5 g5n $ 5_7 g5 $ 5_7! ']")" . & -21 ')( $ !9 5 65 $ 5n'65_7X! 5 65 $ 5_7X! '#")" j + l & = @CB ' 3 465 78 M]! . £ ¢ <3 ¢ 0 j + l & 2. 5 g5n $ 5_7 5 $ 5_7X! '#")" . & -21 ¢ &- ')( $ ! 5 g5 $ 5n'65_7! 5

g5 $ 5_7! ']")" j + l & = @CB 4 3 ' 3 465 78 M! 5_7! ' 3. 7 . 5_7 7X! 43 ¢ a 1 ( 5: 'g(*5 '7! "! F $ 7 5 7 $ G F (: + a l & G j ' " &1 = = @ [email protected] ' 3 465 78 M! 5 g 5n'65_7! ' 4. g5 7! ' 5 g5n $ 5_7

5 g5_7! " .

5 g5_7X! ' &-21 ')( $ !# 5 g5 $ 5n'65_7X! 65_7! " j + l & @ = = [email protected] 1.

1.24. The Complete Elliptic Integrals K (z), E (z) and D (z) 1.24.1. Derivatives with respect to the argument 1. 2. 3.

4. 5.

7 ¢ 3hf9

& P K f / 0 F G 9f ¢ M 3hf9

& k &-21 K f / 43 ¢ k M ¢ 3hf & &-21 K f;/ 43 ¢ F 7 G

7 ¢ 3hf9

& &-21 < E f;/ 0 F]3 G k ¢ 3 f9

& &-21 P &-!&4 E f;/ 3 : P

K 0f

/ ¤

7 F G -;&-21

K f

/ ¤

f9 ¢ h 3 f9

& -;&-21 K f / ¤ 7 m ' F G 7 ( m E 0f;/ ¤ "' ¢ F]3 7 G 7K ( E 0f;/ ¤ m

+ (

, ;

k ¢ 3 f & -21 P ¢ 3 f & &7 -21 E f;/ 3 ¢ F G F G f9 ¢ 3 f9

& -I&-!&4 E f;/ ¢ ¢ 7. k 3hf9

& 9M 3hf9

& &-!&4 E 0f;/ 3 ¢ F]3 7 G F 7 G f9 ¢ 3hf9

& -;&-21 E f / ¢ ¢ F 7 G F G f E f;/ a 8. 3hf & 1 3hf & -1 E 0f;/ ¢ ¢ a 9. M 3hf9

& 1 k &-21 3hf9

& -1 E 0f / 3 ¢ F 7 G -;&-21 E 0f;/ 7 ¢ a 10. 1 3hf9

& D f;/ 0 F G F G f9 D f;/ ¤ 7 7 ¢ F3 G F G f9 D f / ¤ 11. &-21 3hf9 O& D f / 0 ¢ 12. k; 3hf9

& P &-21 D f;/ 43 ¢ F 3 7 G F 7 G f9 1 -; D f / ¢ a 13. k -1 3hf9

& P 1 D f;/ 3 ¢ F 7 G F G -;&-21 D 0f;/ ¢ a 14. k 1 M 3hf9

& &-21 D f / 3 ¢ F 7 G f ¢ 3hf & -I&-21 D f;/ 6.

¤ ¤ ¤ ¤

¤ ¤ ¤

1.25. The Legendre Function Pνµ (z) 1.25.1. Derivatives with respect to the argument 1.

2.

D j

D j

m ! ")" £ F m IG H 3 If 0f 3h -I $9% T3 ¢ 4383 0f9 3h O &--; £ m ! ")" F m GIH 3 fI 0f 3h -I $9% Y 0f9 3h O -; &-

Y

<9N

L ']")" ')( J $ 5 ! ']")" a + 1A- . F Ggj - F m m ( L'#")" J ')( $ (! '])" " a + a 1 . F Ggj a F m m

G ¤

G ¤

&

ED0 f9 3h O j F m IG H . 43 ¢ T 3 ¢ 43:3 ; 0f 3h + -; j -; ¢ 0f 3h - + a . j a F G ¤ 4. ED0 f 3h - j F m G9H 4 3 m a . 5. DM &- + -21 fT3h& j F#" m GIH £ -; f 43:3 ; - + a . -21 fg3h& + -; . j -; F#"

-; . a a 6. EDM + -21 0fg3h& - j F " m GIH 43 £ -; f

I+ a -21 . 0fg3h& - + a . j a F#" a . a 7. EDM + - -21 fT3h& j F#" m GIH £ -; f )3 ¢ I+ - -21 . fg3h& + -; . j -; F#"

a . a 8. EDM + - -21 fT3h& - j F " m GIH 43 £ -; f

I+ - . -21 0fg3h& - + a . j a F#" -; m 9. EDM 63hf9 j F " GIH 3 £ -; 3:3 ; I+ -; . Q3hfI + -; . j -; #F "

-; m Q3hfI - j F " QG9H 10. ED £ -; I+ -; . 63hfI - + a . j a F " -; m . a 11. ED - + 1 0Q3hfI j F" QG9H 3 £ -I T3 ¢ - + a a 1 . Q3hfI + -; . j -; F "

a m a . F " GIH 12. ED - + 1 0Q3hfI - j # £ -; - + a a 1 . 0Q3hfI - + a . j a F " a 3.

F m G ¤

¤ m G ¤ m G ¤ m G ¤ m G m ¤ G m ¤ QG m ¤ QG m ¤ G

1.25.2. Derivatives with respect to parameters 1.

j & 3

5_7 3

1

j & £ a 1 $ J 5n 1 L' " ! % $ ¢ 3 a 1 & = = @ #B [email protected] &<

+ (

, ; ,

j & -;&-21 3 j & ¤ j 3. & &-21

)(* '65_7 3 j G ¢ 3h

& 3 ¢ £ &-21 F )( O'6 5 $ 5_7 ' &" -21 £ 65n '65_7 Y $ L G G9H

3 D F F J

- a 1 L" J Y j -; 0& 43 £ & - ¢ 3h

-

l -21

5 Q(87! Y -21 % $ $ ( Q(:7X%! % £ & - ¢ 3h

j a -;&- & ¤ -21

7 7 D F 3:3 G 3 F 3 G 3 HTj 0& 4. &-21

']" ¢ 3h

1 ( L ' 1 5 L ' J

J

a ' ) * ( O 6 ' 5 &- -21 J - & L Y $ L 3P £ F G 63 3 % ')( $ ( ! J & 5 Q 8 ( 7 ! 7 Y j a & 43 £ & - ¢ 3h O - -21 % $ $ ( Q(:7X%! % F 3:3 G l -21

- Y £ & - ¢ 3h

j - a a & ¤ -21

3 ¤ j j 5. & -;&-21

0& &-21

6. j % & ! £ 5n ¢ £ C j 3& O! ] j 0& 3 £ £ 5_7X!A 5:O! 3 - 4 R0 j a 1 & j a 1 3P& ¢ 3h j 1 a 0& j 1 a 3& 0 ¤ 7 (8 " 7. D j 0& H DC3 £ C £ &Q3 ¢ 3h

1 1

_F ¢ 3h G F ¢ 3 7 (8 G F ¢ 3h G F ¢ 3 7(* G9H = C7 ]! @CB 2.

< ,

6 !

1.26. The Kummer Confluent Hypergeometric Function 1 F1 (a; b; z) 1.26.1. Derivatives with respect to the argument

m !' m ! m 5:' ! @ VB B 7 @MB 1. D N == ! ' 1 N 1 J 5n' L 1 1J LH m ! m 5n' @ VB B 7 [email protected] a 2. EDM &-21 N == 1 1 J L H 0fI -21 1 N 1 J ! L m ! m ! ¢ ¢ @ VB B 7X@MB 3. DM -21 N == 1 1 J L H 3 3O -;&-21 1 N 1 J (*' L m ! ( m ! ' m ! @ VB B 7 @MB H ! n 5 ' L 3 ' - 1 N 1 J == 4. ED

- 1N 1 J L m m (*' 5. DM -2 1 V

- 1 N 1 J ! L H 43 ¢ ¢ 3O -;&-21 #- 1 N 1 J (*!' L ¤ m ! m (*' a H K3hfI - -21 - 1 N 1 J L 6. EDM - &-21 - N 1 1J L ! @ B B 7 @CB == m ! m !' m 5n' ! ¤ ! 7. &-21 N 4 3 N ; & 2 1 ` ' 1 1^ 1 1 ^ 5n' ` m 5:' m! ¢ 43 0fI - -; 1 N 1 ¤ 8. - 1N 1^ ` ! m! ¢ m! ¤ 9. & 3 O - 1 N 1 ^ (* ' ` 1N 1^ ` m! m! ¤ ( ! m ! ' 10. &-21 #- C N ; & 2

V- C N 1 ` ' 1 1^ 1 1 ^ 5n ' ` m (*' m! ¢ 11. - V

3 3hfI - -; # - C 1 N 1 ¤ - 1 N 1 ^ ` ! g5 1 1 ( ' 6 ' 5

6 5 1 L' L' 1 a 12. N1 1 43V J J ! '%& 7 1 N 1 5n'6

5 ! ! = 7 ! = @ @CB & ( (*' '65 & W ! 7 " WU ! a !

1N 1 3# 1A- 1 N 1 5n' 13. ' ! ! = 7 ! = @ @CB <93

+ (

( (*'65 1 a

-21 1 N 1 14. ! 5 1 43 ¢ ¢ 3 £ O a - - &-21 N = * ( '

1 1 ! * ( ) ' ( a 1 15. - N 1 1 ! 43 ¢ £ 3 £ O a - - &- N ( ( 1 ! (* ' =

1 1

m m 5n'65 £ a a !

a 16. 0 I f 1 N 1 g5 1 1N 1 1

! =

m m 5:' £ a !

0fI 1A- 1 N 1 & ( ! 17. N 1 1 &

= W m 5n')( 1 a 18. - N 1 1 m! g5 ( 1! 3 ¢ £ J 1 L ' J ! 1 L ' - W N m 5n m '%& 7 1 1 m '6 5 = m 5n'6

5 ( & W a 19. - N m! 1 1 ( m ( & £ J & L ' J ! 1 L ' 1A- - W N m ' 1 1 m 5n' ! = '6

5 ( 1 W a 20. -21 - N m! 1 1 (')( 1 3 ¢ ¢ 3 £ fI a - &- -21 - W N

1 1 m (*' ! = '6

5 ( 1 W 43 ¢ £ 3 £ fI a a

- #- 1 N 1 21. m!

* ( ) ' ( W 1 Y - &- - V- N = * ( ) ' (

m 1 1 !
,9:1

7 @ @CB ! =

7 @ @CB ! =

7 @ @CB ! =

7 @ @CB ! =

7 @ @CB ! =

7 @ @CB ! =

7 @ @CB ! =

7

! =

@ @CB

6 !

m ! - 1N 1 1

m W 3# a F 7 3hf G - 1N 1 a m W a

- 1 N 1 !&

3V F 3hf G 1A- - W N m 1 1 &

22.

a

23.

W

( ' * g5 ! 1

7 @ @CB = ! =

(*')( (

7 @ @CB = ! =

!

1.26.2. Derivatives with respect to parameters

m ' 5! 7 L H SEa D1N 1Jg 5n% ' ' % ! %

1.

a 1 ¢ <3 ¢

S-; 3P&

;- C £ & <3 & -; 43& SEa SE a 7

3 $ SE;- a & - 43& £ 43 ¢ - -21 & 1 % ' % ! S " 3 5:'2! % -; J $ L $9%

( 7! Y C $ ¢ -; 43& <3 &-;- a 4 3& ¤ 1 m ! 1 N 1 1 (*' SEa

1

£ 43 ¢ _ a 1 ' $L F]3 7 G 3P& - J 1 ( 1 J S L J L ' 7 #7 ( ( 7X! 7 a Y % F 3 G S - 3P& N & ( $ ! (

7 ¢ £ -;&a -21 3P& G H 3 43 D F C SE S 7 7

$ -21 3 3 ¢ SEa $&% ( $ ! J LF 3 G -;&-21 4 3& ¤

2.

< ;

+ (

, <

3.

4.

5.

m ! 1 N 1 & (*'

&4

'& (]! ( & J L'& '65_7 D1N 1J m ! LH

7 #7 ( ' G 3& - N ! ( $ ¤ $ L F 3 J

7 1A- 43& fg3 ¢! & $ &-- a 1 43& --21 -21 0& ( ! " m 5 $ (:7 m 5 $ (:7 1 ¢ 3 0fg3 $9% m 5 $ (:7X! &- 3P& N J m 5 $ m 5 $ ( L ! m= 7 9BB B@CB ')( 5_7 m G 3 H D 1 N 1 J m ! 5 L H a . D £ ¢ <3 F +

& 1 7( (*' E- a 1 1- - ¢ a Y N '6 G 7( ' 1 1 5_ 7 ! 3 1 -; F ! 5 $ ( 7X! ' ' % ! " ' & -21 $ 5 1;- E- L 1 ¤ J $&% ! ')( $ ! 43& N a _ 5 7 $

1 1 E- 1 ! J L 7 ')( 5_7 m ( G 3 H D 1 N 1 J m ! 5 L H a . D £ ¢ 3 F + &- 1 a ( ' % "' Y N E- 'g5 1 7 !

1 1 E- a 1 L J $ 5 ; 1 E 3 L & 2 1 £ Y 1 J

f O ! ) ' ( ! & $ % $ ¡_ 3

& $ 5 1;- - ¤ Y N 1 1 $ 5_7 ! ( 1A-

6.

1.27. The Tricomi Confluent Hypergeometric Function Ψ(a; b; z) 1.27.1. Derivatives with respect to the argument 1. 2.

m m 5n' ED J ! L H 3 fI J 5n!' L 5 ' m m n ! L D a &-21 J ! L H fI 0g f 3 ¢ 2- 1 J <9:

= = @ VB B 7 [email protected] = = @ VB B 7 @MB

3. 4. 5.

m EDM -21 J ! L H 4 3 ¢ 0fg3 m ! ED # - J L H 43 V- m ! a EDM - &-21 - J L H 3

¢ -;&-21 m !(*' L J m ! 5n' L J ¢ - -21 - m (*' ! J

= = @ VB B 7X@MB @ VB B 7 [email protected] B ==

m 5:' m &-21 ^ ! ` fI -;&-21 ^ 5:'! ` ¤ m 7. - ^ ! ` 3 ¢ fI 0fg3 ¢ - -; m! m! ¤ ¢ 0 g f 3 8. & ` ^ ^ (* ' ` m m! 9. &-21

- C ^ ! ` - C -I&-21 ^ 5n ' m! m (*' - -; - ^ 10. - - ^ ` 6.

!

L

= = @ B VB 7 [email protected]

m 5n' ! ¤ ^ `

` ¤ !

` ¤

1.27.2. Derivatives with respect to parameters 1.

m! ' LH D J 5n 3h O # 5n ! ' L &-21 J 7 & -21 ')(:7 3 ! KJ $ L 3 ¢ 5 $ ! 7 #7 &" D 5 $ (:7 N ( ! J m! D J 5_7 L H 3 - O 3 ¢

' (87 $ L O - - J % ! 5 $ ! 7 L 5 $ 1 N 1 5 $ 5_ J ( $

2.

3.

< <

+ (

K a2S 7 ES a 2- 1 0 & - a2S 2- 1 - 2- 1 3P& 3 1

Y

4.

, < ,

0& - 2a S - S 2- 1 43& = ' >[email protected] m ( 7 ! ( 7 ' 7 ! ' G F3 ' 5 1 6 J $ L & ( ' L " S

J

Y - a2S -21 &

& <3

5.

7 #7 7 Y £ N ! 3 / <3 F 3 G 3 F 3 3 ]

&

a2S 7 3 a2- S -2- 1 & 1 Y / 1 - 43& <3 N 1 P5 ! 1 1 -21 1

J L 2 a S 7 7 5 ( ] ! $ ¢ L 43 F G F G S F ( $ 1 J J L 43 ¢ - -; D m ! L H a a J 1 Y 3 ¢ ¢ 0fI C 3 # "'& $ 5_7 $ 5 7 ! L C 7 ( ! _ 5 X 7 ! 3 N J $ ( m 5: $ 5n m "'& $

3 43fI 3 ¢ 9l - -21 43& ( 7X! " $ $&% J L43fI - 3P& - 0 f

7 G £

7 3 G ¤

D

6.

Y

J

5 7

/ -21

!

( 7 ! % D C £ £ F GIH L H a &4

3P& a2S 7 - a2S -21 3P& / <3 - S -21 3P& a2 1 <9L

¢! 3&

' J $ L 1 - 0 & -21

¤

( ! % -21

Ψ(a; b; z )

' $ L 3P& J 5 P5 #$ a2S ( ! 1! ( 1 Y % 5_7! a2- S 43& N

P5

& P5 &

a2S ( 7X!

- S -21 43& ¤ 3 C V& - a2S -21 43& a2 1 ( 1L ( ( 1L ' 5 7 ' $ 5 !% J J H

D J L 0 7. ! % 5 1 L ' J $ L & (*' L " ! &4 -; J

a2S J

7 7 Y D £ F G 3 F 3 3 G9H J $ 5 ( L 7 Y / -21 - -21 43& / <3 - -21 43& 1 - 0& -21 1 ( % ( 7 7 1L' ' a2S $ 5 ( $ $ !% L L ! % J 5

1 L ' J & (*' L " J J J

5 1 5 1 ( #$ ( X 7 ! ! Y # 5 & 5 & J L 5_7X! N

3 / -21 ] C V& - -21 43&

-21 ( 7! ¢ ¤ Q( 43 / -21

P 3 & - -21 m! n 5 ' H L S 3? $ 0 J ! 5:' L 8. D J m " ( SEa &-21 5n')(:7 ¢ $ L 3 - (87! % J (]! " ( SEa &-21 5n')(:7 ¢ $ L 3 3P& 3 (:7X! % J a2S 7 Y & <3 & 0 - a2 S - S 43& <3 -21 - 0& a2- S - S 43& -21 -21 - -21 1 = >[email protected]

<9P

+ (

9.

g ' 5 1- m! 3 D J m 5 L H a 0. O '65_

7! + 1A-

" " & -21 ' ( ! £ $ & & &-21 -; $&%

,9L1

3

£) 3 < 3

¤

1.28. The Whittaker Functions Mµ, ν (z) and Wµ, ν (z) 1.28.1. Derivatives with respect to the argument 1.

2.

3.

4.

5.

6.

7.

&- -21

A

l 0 fI & ¢ 7 F 3 G - -21 A 29l -21 A l 0fI& 43 ¢ 3 £ f

- + a 1 . A 2 l -;

- -21 A l 0fI& (g5 ¢ J 5_7! 1 L ' f

- - + a 1 . A ' 2 l a

-21 A 7 l 0fI& 3 ¢ F 3:3 G f

- + a 1 . A -; l 2 -21 A 7 l 0fI& 3 ¢ F 3 G f - + a 1 . A -; l 2

&- -21 A l 7 0fI& F 3:3 G F 7 3 G - -21 A -;9l -21 - A l 0fI& 3 ¢ f - + a 1 . #- A a l 2

0fI& ¤ 0fI& ¤ fI& ¤ fI& ¤ fI& ¤ 0fI& ¤ 0fI& ¤

1.29. The Gauss Hypergeometric Function 2 F1 (a, b; c; z) 1.29.1. Derivatives with respect to the argument 1. 2.

m !' !' m m 5n' 5:' ED N 1 J L H ! ' N 1 J 5n' L ! ! m m 5n' L EDM a &-21 N 1 J L H fI 2- 1 N 1 J ! ! L N

= = @ #B B @MB = = @ #B B [email protected]

3. 4.

! $

m D -21 N 1 J L H 43 ¢ ¢ 3 -;&-21 N 1 J ! m ED0 ¢ 3h& a &-21 N 1 J L H ! 3 ¢ m ! ' ! ( ! ' ¢ h 3 & -21 N 1 J '

m (*' L

= = @ #B B [email protected] B

5 ' m n 5n' L

= = @#B B @CB

!

!

m ED0 ¢ 3h& a - N 1 J L H ! m @VB ( m ! ' ! ( ! ' ¢ h a

; n 5 3 & N '

1 J ' ! L == a m ¢ - N 1 LH 6. EDM -21 3h&

J ! (*' 3 ¢ ¢ 3 -;&-21 ¢ 3h& - N m (* @VB

1 J ' ! L == ¢ a - N m L H 7. EDM -21 3h&

1 J ! ' (*' 43 ¢ ¢ 3 -;&-21 ¢ 3h& a - A-; N m (*(* ' L = = @ VB

1J ! m 8. EDM - a &-21 ¢ 3h& a - N 1 J L H ! 3hfI A- -21 ¢ 3h& a - -; N m (*' L = = @VB

1 J ! m m 5n' : 5 ' ¤ m !' !' ¢ ! 9. &-21 N 4 3 N ; & 2 1 n 5 ' '

1 ! 1

1 ! 1 m m 5n' ¤ 10. - N 1 1 43 ¢ fI - -; N 1 ! 1 ! m m 11. &

- N 1 ¢ 3 - N 1 (*' 1 ¤ 1 ! ! m 12. - Q3 ¢ a &-21 N 1 ! 1 m ! ' ! ( ! ' - -; Q3 ¢ -21 N m 5:5n' ' '

1 ! 5.

L1

B @CB

B @CB

B [email protected] B

B @CB

1 ¤

+ (

13.

m N 1 ! 1

A- - &-21 Q3 ¢ a -

43 ¢ ( m ! ' ! ( ! ' ' a

m A- - -21 0Q3 ¢ a - A-; N 1 n 5 ' 1 ¤ !

¢ 3 - 0 Q3 ¢ - N m (*(*' ' ¤

1 ! 1 m &- - 63 ¢ a - N 1 ! 1 ¢ 3 &- - 0Q3 ¢ a - A-; N m (*(*' ' (*' ¤

1 ! 1 m - 63 ¢ a - N 1 ! 1 43 ¢ 3hfI - 0Q3 ¢ a - A-; N m (*' ¤

1 ! 1 ('65 1

a N 1 ! '6

5 g5 1 5n'6

5 43# F 7 G F 7 G !! ' & 7 N '& 7 1 5n'6

5 ! 7 @ & [email protected] = ! = (')( g5 &

a N 1 ! '65 & 5:' 43V F G F 83 7 G !! '%& 7 1A- N '%& 7 5n '

1 ! 7 @ & [email protected] = ! = (*')( g5 1 43 ¢ ¢ 3 £ a

A-21 N 1

a ! 5 1 7 @ [email protected] Y A- &- V-21 N ! =

1 (*' ! =

16.

17.

18.

19.

¢ - a N m 14. - 63

1 ! 1

15.

,9P1

L,

20.

(*')( a

A- N 1 !

1

! $

( 1 (*')( ! = 7 ! = @& [email protected] 5n')( 1 7 7 a ¢ a 43# F G F G 21. 3h

&-21 N

1 ! ( ( !'& 1 Y ! '%& 7 ¢ 3h -;&- -21 N n 5 6 ' 5

7 1 ! 7 @ & O [email protected] = ! = 5:'g5 ( & a a ¢ a 22. ; 3h V-!&4 N

1 ! ( & 43# F G F 83 7 G ( ! ! ' 1A- ¢ 3h

-;&!- &4 N ' 5n'

1 ! = 7 ! = @& [email protected] '65 ( 1 a a ¢ a 23. A-21 3h

- -21 N

1 ! (')( 1 (*' 3 ¢ ¢ 3 £ ¢ ( '

a - &- -21 3h

- -;&-21 N 1 ! = 7 ! = @& [email protected] 'g5 1 a a ¢ 3 ¢ £ 3 £ a 24. A- 3h

- 1 N 1

a ! ( ' ( g5 1 (*')( Y - &- - ¢ 3h - A-;&- a 1 N (*')(

1 ! = 7 ! = @& [email protected] m a 25.

N 1 1!

m 5n'65 5n'6

5 £ a 0fI O 51

a a N 1 !

= 7 = @& [email protected] ! 3 ¢ £ 3 £

a A- &- V- N 1

L 3

+ (

26.

27.

28.

m m 5n' 5n' £ a

N 1 &

fI O 1A- N 1 & ( !

! 7 @ & 7 = ! = m 7 ¢ a a

3h

-21 N 1 1 £ a F 3hf G a !

* ( ' (*' 7 m Y F 3 G a ¢ 3h a - &- V-21 N 1 1 5 ! 7 @ & = ! = m ¢ a a

3h

-21 N 1 & £ F 3hf G F 3 G !

m (*')( (*')( Y 1A- ¢ 3h O a - &- V!- &4 N

1 & ( !

7 @& = ! =

,9P1

[email protected]

[email protected]

[email protected]

5 ( #$ m m ¢ a a a 1 29. 3h -21 N

1 1

! 5 5 ( #$ m m 1 3# fI ¢ 3hfT3 , ¢ a a 3h O -;& -21 N 1 5 1

g ! 7 @& [email protected] = ! = m ¢ a a 3h

&-21 N 1 30. ; ( m ¢ Y 1A- 3h -;&- V-21 N

1

31.

¢ a a

3h N 1 43V a

m ( g5 1 #$ 3# 0fI ¢ 3hf-,

& !

( g5 #$ m 1 = 7 ! = @ [email protected] ( & !

7m 1!

7 '6

5 65_7 m F 3 f G a N 1 g5 1 ! 7 @& [email protected] = ! = L95

! $

1.29.2. Derivatives with respect to parameters 1.

2.

3.

4.

5.

'65_7 m D N 1 J m 5 ! L H 5 7 ! " . a a O; ¢ 3h& -;&-21 ¢ h j + 1Rl -;&-21 ¢ 3 £ & 3 & 5 5 ! 9 $ % $ m &5 5 $ m 5 5 $ 5 $ 5_7 ¤ m Y N ! & J m 5 5 $ 5_7 m 5 5 $ 5 7 L '65_7 m ¢ 3h& - -21 m 5_7X! " 43& ¢ 3h& - D N H L P 3 I f ;

1J ! $&% 5 $ ! 5 $ 5_7 5 $ 5 $ ¤ 7<5 Y j + a 1Rl a -;&-21 . F 7 (8 G N m 5 $ 5 7 5 $ 5_! 7 ,-Q 1` & ^ & V7 1 N

1 ! 0 #ME7 (:7X! ¢ £ / 1 ¢ / ¢ 3 £ / ¢ 3 / <3 ¢ 3h& ¤ m D N 1 J 5_7 L H ! O - ¢ 3h& 1A- ¢ 3h& -21 ¢ 3hf O X 3hf ¢ < 3 ¢ 3hfI 7 7( m V7 ( 7 7( m V7 ( m V7 ( ¤ ¢ 7 ( ( 7 8 ( L 3 C7 ( m ! & N J ( m ( m 7 (8 L m 3h& N 1 J m ! ! mm 5 ! ( LH D N 1J m 5

a . ( 5_7 ')( 5_7 + &- 1 ') DC3 G GIH £ ¢ < 3 F 3 F a 5 a Y N E- '61 5_7 (E - 1

1 ! 7 ']" ' % C7,5 ! 3 " '#" 8 " ¢ ')( 5_7! 3 E- a 1 L 5 E- a 1 L J

J

7,5n')( 5n' 7 (*')( Y 3 F G O F G F G ¢ &

#$ a Y N 1 - 7 E ( - 1 E -

1 ! a1 ( (*' 7 1 (*') ( ')( 5_7 G F ' G G 4 3 O F F L;

+ (

a a 1 5 E- a 1 #$ Y N

5_7 1

1 ! a 1 $ 5 " $ 5 1 E - L Y &-21 J

$&% ! ')J ( $ ! $ 5 Y N

1

3 N1

4

-;

5 1;- E- L

1;- E- $ 5 5 ;1 - E-

$ 5_7 ! (

¤

1.30. The Generalized Hypergeometric Function p Fq ((ap ); (bq ); z) 1.30.1. Derivatives with respect to the argument 1.

m !95n' m ! m !' !' N ^ !95n' ! ` ¤ N ^ ! ! `

m ! ( N ^ ! ! ` 3 ¢ 3 -; 2a S N a2S m !! g g ^ = m ! N ^ ! ! ` ! ' ' ( % ! % m '#" a N a m !9!95n5n')')( ( ! ']" 1 1^ - -21 $ (*'65 5_7X! ' m ! " a $9% ! " -; =

2.

3.

5_7! ( ! 7! ` (*'65_ ')(:7 ')(*' ( &B B [email protected] B

'65_7 ! 7 ` ' ( 5_

!95n' '65 m ! 7 m !' m 1 £ !

N ^ ! `

F G ! ' a 1 N a 1 !95n'

1

! m ! a

1 N ^ ! ! ` !95:'g5 7 '65 ! m £ a 1 F G m ' & a N a !' & 1 1 !95:'g5 7 & !

4.

5.

')(:7' ( ')( 9BB B@CB

L :

#$

¤

&

#$

¤

1.30.2. Derivatives with respect to parameters

" m !" m m ! ( ! ¢ a 1 N ^ ! ` 3 $&% ! " ¤ ! (' m m ! a N a

1 ^ ! ! ` ( ' m m ! 3hfI 3 3hf a N ^ & ! ` ! " ' % '65 ( m !A ( m (*'! " & -21 ( 7X! $ (* $9% ')( $ !A $ ( m 5 ! 3 m ( 5 7! ' ')( $ ( $ m 5:' ( $ m !95 m ! Y & !95 & L P 3 & a a N ` ! 1 ^ J ! ( ' & m ! 0 a N a ( &' & ! m ! ` ! ` a N a ! O 3 1 1^ 1 1^ ! ' % & -21 ! " ( $ & m ! ')( $ ! a 1 N a 1 ! ` !' $9% ^ ! m ! m ( m a N

^ !! ` a

D F 3 G 3 F G9H a N m ! (*' 5:'

! !

" " 5n $ (*' a 1 L & -21 L £ J 5n &-21

' $&% ')( $ ! J a 5 $ (*' 1 L L J

J ! a (*' 5: $ ( ' #$ m 1 Y a N a a 5 $ (*

' ! & 1 1 !

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m m m ( N& ^ m 5_7 ! m ` 3 K F 5_7 5 7 G 1

7 Y _ K _ 5 7 5 , 5_75 5 75 7 ) 5_75_7 K F 5_ 7 5_7 G ¤

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m m & ! \¢ 3hfg )¢ ) 3hf 3f 3 K 3 h 3 f 3 3 ¤ Y & \ ¡ ) \ ¡ £ £ 3 3 f 3 f mm 5 m 5 ( m 5 5 1 10. N & m m (K5_7 ! 3 D F ,G F 3 3 ¢ G £ CH #! ( ( 5_7 L a ( 5 5 5 £ Y N 1 7 J 5 5

-21 / & ! L \¢ J

¢ 3 3 3 3 ¤ Y & \ ¡ ! ¡ ¢ ¢ 3 3 3

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m ( D 1 N J 7! #7 L H m ! 1N & & 7

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J 1 Q( 5 & L F G J

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J ] m 5 L H + &- ')a (1 . 5_7 ')( 5_7 G F G9H D £ ¢ <3 6 3 F #$ a ( - 1 ! Y N 5

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! £ 19. N 0f f & + 1A- .

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J L J

5_7 Y I+ a 1 . F 3 G N a a

& &

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P N

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mm 5 ! 5 23. N 5 a

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mm! ( D N J m 5_7 m 5 L H a . - 1 ') ( 5_7 D ')( 5_7 3 6+ & G £ ¢ H 3 F a a ( #$ E 1 E 1 ! Y N

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mm 5 1! N 5 7 _ 5 7 5

m m m

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& $ 5_7 $ 5 &- E- $ 5 &<- E-

mm 5 1! ')( 5_7 3 29. N 5 7 _ 5 7 5

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#$ mm 5 1! 30.

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a 5 5 _ 5 7 g ' 5 7 E- 1 E-

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& $ 5 7 $ 5 1;- E- 5 $ ( a 5 5_7

31.

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N &

P95

Chapter 2

Limits 2.1. Special Functions 2.1.1. The Bessel functions Jν (z), Yν (z), Iν (z) and Kν (z) 1.

2.

3.

4.

5.

6.

7.

a 1 IF G a 1 F G

& 7 ¤ 0& W 7 9/ ¤ 09/ ' 3 ¢ D F O G a 1 & 7 ¤ '

G H

F

63 -;&-21

& * F O'PG a 1 0& 3 a 1 0& ¤ F O'PG a 1 0& 3 a 1 0& ¤ 7 B

&

& M! a 1

a 1

(:7 B F G -21 0& "

2.1.2. The Struve functions Hν (z) and Lν (z) 1.

2.

. + a 1

. + a 1

F G F G

W 7 ¤

/ - F G W ¤ 7 L 0 / F G H

2.1.3. The Kelvin functions berν (z), beiν (z), kerν (z) and keiν (z) 1.

F G a 1 2D

& P;

7 & H ¤

+ #

2.

(:7X! F G 2- 1 D

, 5

0&

(87!

2.1.4. The Legendre polynomials Pn (z) 1.

2.

3.

4.

5.

6.

7.

8.

9.

7 1 O j F 5_ 7 G C 7 (8]! 1 -; £ & -; j ¢ & 7 ]1 ¤ + 1A-; . £ & -; j 0/ & 7 V- \]W ¤ ! ¤ ¢ j ' G 43 1 F O]! ¤ ¢ 43 1 j a 1 F ' G j F¢ ' G / £ ¤ ¢ j ) * ' , 0& ¤ ' j F ' 5 G 0& ¤ 'g5n 0& ¤ j , ' g ' : 5 ! , )

2.1.5. The Chebyshev polynomials Tn (z) and Un (z) 1.

2.

3.

4.

5.

6.

7 FQ 3 3 ¢ G & 7 £ -; ¢ & 1 ¤ 43 ¢ F 'PG £ & ¤ 43 ¢ a 1 F 'PG £ & ¤ F ¢ ' G / £ ¤ ' 5 £ ¤ ) ' (8 , &

P :

& H

3-"

¤

=

[email protected]

[email protected] =

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

£ & -; -; ¢ ' ) * ' F ' 5

7 ]\ W ¤ / & - ¤ ,

G

¤

FO63 3 ¢ G & F £ 3 £ £ £ ;- ¢ & 1 ¤ 3 ¢ F ' G £ & ¤ 3 ¢ a 1 F ' G £ & ¤ 7 & ¤ ¢ ' F ' G 7 ' 5 & ! ¤ ' ) ' (8 , \]W £ & -; -; / & - ¤ 7 & ¤ ¢ ' ) * ' , 7 ' ' F ' 5 G ¤

3 ¢ G -21

= [email protected]

2.1.6. The Hermite polynomials Hn (z) 1.

2.

3.

7 '2]! ' & F3 ' G F ' ' ( '% &! '!

a 1 F

\]W ¤ - G £ 1 £ & ¤ £ 1 £ & ¤ ' G

2.1.7. The Laguerre polynomials Lλ n (z) 1.

2.

c

- c c F ' G - c £ / c " ' ' % F G d -; c d 3 / dK P<

= @ B = @ B

+ #

( "' ( 8 a ' % F G -; c F ( G

3.

, L

= = @ [email protected]

2.1.8. The Gegenbauer polynomials Cnλ (z) 1.

c

0& ¤ '

d 2 - 1 c 0 &

(O]! ' ¤ ' % 2. d -; c 0& c 5_7 C7 (8! " / (&! 3. = [email protected] 1A- c& c F G 7 4. d -; c F G ' % 0& ¤ c £ £ & - c F 7 G ¤ 5. 3 & -; c -; c -; & - c 7 ¤ £ ¢ £ 6. 43 & -; c -; c & & c F ]1 - c G ! ¢ ( &! ¤ 7. 43 1A- c c F ' G

O]! ¤ ¢ ' ( &! 8. 43 1A- c c a F G

1 ¢ ¢ a F ' G 9. 43 1 d -; c d 3 c -; c -;

7 F Q(:7 G -21 £ / Kd & = = #[email protected] @CB ¢ ¢ a F ' G 10. 3 1 d -; c 0d 3 c -; c a -;

1 £ d & = = #[email protected] @CB 7 ( Q(:7X! ]" / K ( W #$ ' 7 C7 (&! 1 N 1 ! ¤ 11. F G c c -; F ' G 1

1A- c c F ' G ( &! £ & 1 - c & ¤ 12. ' 5

c 2 1

2.1.9. The Jacobi polynomials Pn(ρ, σ) (z) 1.

2.

. -; j + lV- ;- I& ]F 3 . j + l a F ¢ G F 3

,G - V-;&-21 F G ¤ ¤ G P L

5n . g j + lC- a F (* G . -; j + l F G - j + l-; . F ¢ ' G - j + lC-;&- -21 . F

3.

4.

5.

6.

-21 j + 1 lC-;&-21 . ¢ ) -21 j + 1 lC-;&-21 . )

7.

8.

- j + lC-;&- 9. - j + lC-;&10. - j + l 11.

¤ "' & ' % & ¤ g5 7 ¤ 7 5_7! N 5

5 1 1 7!

¤ ¢ ' G

G G F A F a 1L J ¤ F G ' , ¢ ' W F G ¤ ,

4 . ¢ J L G ¤ F ' G ! A F S - -21 . F ¢ ' G 5 % 5_7X! S F G ¤ . ¢ -21 F ' G F 5 G £ ¢ ¤

2.1.10. Hypergeometric functions 1.

- 1 N 1

2.

-R N

m! 5 ^ 1 5

(

` m 5 1 (

!

W

- & W

P P

= @ B - &

= = @ [email protected]

Chapter 3

Indefinite Integrals 3.1. Elementary Functions 3.1.1. The logarithmic function 1.

2.

3.

4.

5.

6.

Z

Z

Z

Z

Z

' m 5 !

7 m

3 '65_7X! m F]3 G a 1 f O ' % ( 7X! " 3 m ')( $ ! % &- f O a

m 5 !

m Z ' & ' m 5 !A 5 #! '

5 #! Z m 5 ! 7 m 5 ' 5 #! 3 ' 5 #! ' ' ' Z '#" m 5 !

' m 5 ! ( &

m 5 ! m 5 !( & ' 7 ' m ( 5 7 Z m 5( ! m 5 ! 3 m m 5 ! m 5 #! Z m 5 !

m 5 ! (' 5 #!

('%& '

m Z (' 5 #! m 5 !

' 7 Z m 5 ! 5 #!

(' & '

3 ' ' f O 5 #! Z m 5 !

m 5 ! '& ' (' 5 #!

7 m Z 5 #! ' (' m 5 ! 3 ' (' 0f O m 5 ! Z m 5 !

% ' & '

m 5 ! m 5 !' &

Z m 5 ! 7 ' ' 5 ! ' m

3 m m 5 ! ' f m

Z

m 5

N1

m F ¢ G ¤ !

= '8>[email protected] B

!

= '8>[email protected] B

9 = '*>[email protected]

I = '*>[email protected]

O

= '*>[email protected]

Z

7.

m 5 0f O m 0 f O 7 Z 3

& -21 0f O 3 m

31 , Z

m 5

f O ¤

3.2. Special Functions 3.2.1. The Bessel functions Jν (x), Yν (x), Iν (x) and Kν (x)

4-4# $ Z 3 a a 1

1. 2.

3.

4.

5.

6.

= " £ a a 1 ( 9$ 7% ! F G a a ¤ & - 1 Z £ (*'65_7 G &- a a 1

F ( '65_7 Z £ * G F a ¤ '& Z ¢ a 1 5_7X! £ X a 1 ¢ ¢ 3 ¢ X a 1 1 £ 3 1 a 1 ¢! ¤

3 ) ¢ ! ¢ ! ¡ \ ) ¢ & Z 3 3 £

£ ¤ 3 1 Z 3 £

£ ¤ 3 1 7 Z " ! " ! ! F !6G 3 N F !6G = ! [email protected]

7.

Z

8.

Z

9.

Z

10.

Z

! ! F !G

']" = [email protected] [email protected] ' & = " '#" = [email protected] [email protected] ' & 3 =

7

&-21

! F V! G ! ! D " !H ¤ ! ¤ !H ' 7 ¤

2- 1

N ' O' D

N,

= ! [email protected]

11. 12. 13. 14. 15. 16. 17.

18. 19. 20.

21.

22.

23.

24.

7 -; &-21 a 1 3 ' -; ¤ Z " ! ¤ !

Z '& ! 3 ¤ !

' !& ! Z !& [email protected] ']" = O' D " ! H ¤ 2 -21 0 = " [email protected] '%&

Z 7 ! ! ! 3 ¤ D !H Z 7 [email protected] 7 '& ! ! ¤ = " ! !& !

" '#" Z = [email protected] ! '%& ' ! !7 ( ! ! ) 7 ! ! Z 3 8 3 8 3 N ! , ! ) ! ,

) 7 Z 3 8 ! 3 8 ! ! ! ,

N ! , ) ) 7 Z 3 8 ! 3 8 !

N ) 3 8 ! , = 3 8 [email protected] ) 3 8 ! , Z & -21 (65_7 G - a a 1 £ F Z (65_7 £ G F " ' (

! Z 1a - L' J

(65_7 Y Z a 3 £ &- F 3 G -; &1 Z V <3 # 0

#

1 3 D # <3 #1 3 7 Z " ! " ! ! D ! H 3 N D ! H Z

N 3

! ¤ !, = ! [email protected] = ! [email protected] = ! [email protected]

-; a ¤ a a ¤ - 1 7

1 H ¤ = ! [email protected]

25.

Z

26.

Z

27.

Z

28.

Z

29.

Z

30.

Z

31.

Z

32.

Z

33.

Z

34.

Z

35.

36.

37.

38.

31 ,

7 &-21 2- 1 ' ¤ 7 -; &-21 a 1 ' ;- ¤ '& ! ! <3 ¤

" ! !

7 ! 3 N D ! D !H

'#" = [email protected] [email protected] '%& ' ! = = " ! 3 3 !

!& ! 3 !

: &-21 2- 1 3 Z -; : &-21 a 1 ( 7X! '%& Z a 1

Z

- I B ;

9 B ;

¤

3 £ X 1 0 ! !H

¤

= ! [email protected]

7 ! !;( ! [email protected] !#G ¤ ¤

¤

7 ' : ¤ 7 ' 3 -; : ¤

43 ¢ a 1 3 ¢ ¤ 1 I B ! 7 5 5 Q(:7 9 3 ¢! :3 ¢) I 3 ¢! ) 3 ¢ 3 6 65 5 Q(:7 1A- 0 = 7 g5 ( P5 7 I 3 ¢) ¢) ; 9 3 ¢) - ¢ 65 ( P5_7 1A- 0 =

39.

9 ¢) 3 ( ¤ N95

@CB @CB

3 E5 7 3 ( 2- 1 S 2- 1 7 9 ¢) 3 3 ¢ S £ 4 3 ¢ 2a S 1

40.

3 2- 1 0 S = >[email protected]

3.2.2. The Struve functions Hν (z) and Lν (z) 1.

2.

Z

& -21 7( £ F 7 & -21 3

( G "" '& 1;J g5 (* $

- 2- 1 & " " - L" ! ( $ 5 1 L 7 J ( ( Z £ G F ;- L -; ¤ " ( Z

L & -21 2 &- a 1 L &- a 1 J ')( $ ! % L - -21 " ' % ' & & -21 ¢

L 3 " 3 ')( $ ! % O'T(* J $ 5_7! ( $ 5 1L J ' ] " !& £ ¤ a 43 &- 1 L -;&-21 <3 * ( '65 1 L J

Z ¤

H

H1 H 3 Z L 3 L ¤

L

1 L

-

L

3. 4.

3.2.3. The Airy functions Ai (z) and Bi (z)

4-4# $ f f - %03 $ ,- $ '= Z 1. &-21 3 3 ¢ &- 3 ¢ R 3 £ 2. 3. 4.

&-!&

= = @ [email protected]

= = @ [email protected]

Z

Z 2 3 Z 2 3 £ &

Z

= = @ [email protected] £

Z

= = @ [email protected]

N;

5.

Z

6.

Z

7.

Z

8.

Z

9. 10.

&

31 , 3

= = @ [email protected]

3 2

7 '65_7 D a 1 3 & -21 ' ¢ £ 3 R 3

3

Z

= = @&C7X[email protected]

7

2 3 7 Z 2 & 32 £ Z

' 3 3 ¢ &-

&-!& H = = @[email protected]

= = @ C7 !0!AB

3

= = @&C7 [email protected]

& &-21 7 ' (87 Z 3 & - & 3 &- & 3 ¢ R 3 £ & -!& &

£ Z @ [email protected] 3 & -!& &

==

Z

& &-21 7 3 3 ¢ ¢ &- & 3 7 3 3 £ R 3 &- 7 £ Z 7 £

3 &-!& & 3 3

11.

12.

Z

&

13.

Z

& &

14.

Z

&

&- & 3 £ &-!&

7 £ 3 R 3 R 3 V &- & Z X 3 R 3 V R 3 # &- &

= = @ [email protected] 7

3 = = @& [email protected] & 3 & 7 7 Z

3 & 3

= = @& [email protected] & &

3

7

15.

Z

& 3

&

& 3

& & 3

&

3

Z

= = @ O [email protected]

&

& &

£ 2

N :

3

& 3

&

= = @ O [email protected]

16.

Z

Z

18.

Z

19.

Z

20.

Z

21.

Z

22.

Z

23.

Z

24.

Z

25.

Z

26.

Z

27.

Z

28.

Z

7 a O ' 5 1 3

£ & -21 & & -21 3 ' 3 ¢ &- ' ¢ £ Z Z 3 R 3 &-!& 3 3 ¢ &-

= = @ [email protected] 7 a ' 1 3

3 &-21 & ¢ &- & 3 7 3 ¢ R 3 & -

&-21 3 3 Z 3 ¢ R 3 £ &- 3 3 ¢ X 3 £ R 3 V &-

Z 7 ¢ £ 3 R 3 R 3 &-!&

= = @ [email protected] 7 £ @ &

2 3

= = [email protected] & 7

7 F & 3 & 3 & 3 G @ = = [email protected] Z

3 &-21

= = @&C7 [email protected]

17.

7

7

7 7

)F 7 ' & 3 ' 7 & 3

= = @&C7 C! @MB = = @&C7 C! @MB

7 O '65 3 ' 3 £ &-21 7

3 2 7

F3 &

Z

&-21 &

= = @&C7 C! @MB = = @& C! @MB

Z

&-21

= = @&C7 [email protected]

3

a 7

3

£

2

N<

G

a 1 ¢

£ £ Z & -

3 G

= = @ O [email protected] = = @& 7[email protected] = = @& [email protected]

29. 30.

31. 32. 33.

34.

7 2 3 & 2 3 7 ' 7 Z Z 3 & -21 & 3 ' 3 7 Z 7 & & 3 &

& & 7 Z

& 7 ' 7 Z Z & 3 7X &-21 3 ' X 7 3 3 ' Z & 3 a 1 & 3 &-21 ' ¢ 7 Z # & 3 3 R 3 Z

35.

Z

36.

Z

37.

Z

38.

Z

39.

Z

40.

Z

41.

31 , 3

Z

3

&

&

3 2 &

&

3 & & 7 &

'65 Z 7X 7 '65_7X! 7 a 1 3

7 &

7

3

£

3

a 1 &

¢ Z & - &

= = @ [email protected] = = @& C! @MB = = @& C! @MB

a 1

¢

Z

= = @ C! @CB

&-

'

3 ¢ &- & £ Z &-!& &

= = @ [email protected] = = @& 7[email protected]

7 7

= = @& [email protected]

Z

= = @& [email protected]

&

& 3

7 &-21 & 3 ' ¢

3 3 R 3

= = @& [email protected]

£ &

' a 1 3 ¢ &- £ Z &!- &

= = @ C! @CB a & 3 a £ £ a 1 & ¢ £ ¢ Z R 3 a

' ¢ £ Z @ R &-21 = = [email protected] B

3

'

2 3 7

! ! F !#G N F

&

7

3

&

Z

! !#G N L

&

7

7

7

&

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= ! [email protected]

Z

42.

7

! EF

! !VG 3 N F

! ! G

= ! [email protected]

3.2.4. Various functions 1. 2.

(' ( !" 3 ' % (:7 3 ¢ ') ( $ ! % - K3 3 ¢! ¤ Z 3 £ 0 2 ¤ 2

" " ! 7 ! Z ! D ! H 3 ! N D < ! H = ! [email protected] 7 ( ! Z ( ! ( ! ! D ! H N D ! H = ! [email protected] 7 Z " " ! N D " " ! ! H ! D = ! [email protected] ! H

7 Z j j 3 )( !A 65 5_7X! Y j j <3 j j -21 <3 83 4 j j 0 ¤ 2 1 7 Z j 0 3 5_7 " ! ! ¤ Y Dj

j j j j H h 3 3 -21 -21 Z

3.

4. 5. 6.

7.

8.

Z

9.

Z

2 3 3 2 7 F 2 n3 :3 E

7 F

K

N P

¢

K

2

3 £ E ¤

G

%

K

:3

7

3

G

E

¤

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Definite Integrals 4.1. Elementary Functions 4.1.1. Algebraic functions

; $ # $ f

1.

¡¤

Z

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1

2.

Z

- !- &4 0fg3 - 2- 1 ¢ 3 0fg3

( ( / J ( ! 1 L F 3 G a 1 = m

3.

4.

5.

Z

Z

Z

( 7 !

!

( m ! @ B

( m ! @CB

3 0fg3

-21 0fg3 -21 ¢

¢ 3 m j ( m , ) ( m , )

3 0fg3 1 f -21 0fg3 1 ¢ 3 0fg3 1 £ -21 0fg3 -21 ¢

E

m , )

m , E )

=m

@ B

& ( m ! @CB =

& ( m ! @CB =

+

6.

7.

8.

m m 3 0fg3 -21 K ) , 3 E ) 1 0fg3 1 ¢ & ( = Z m 3 -21 ¢ 3 0fg 3 -21 £ K ) , -21 0fg & ( = Z

Z

10.

11.

12.

13.

14.

, m ! @CB m ! @CB

3 0fg3 -21

-21 0fg3 -!&4 ¢

9.

5)

5 m

K

m 5 m

=

Z

3 0fg3 -21

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3 1 0fg ) Z

Z

3 0f63 -!&4

1 0fg3 1 ¢

( m ! @ B

( m ! @CB =

=

( m ! @CB

m -21

,

3 ¢ & ( m ! @CB =

m m ( m ! @ B E , 3 K) , = ) " m ( ! " Z £ K m 3 £ D m <7 5 7 ( m ( !

, , ) ) & ( m ! [email protected] B = Z 7 5 m ¢ 3 m 7<5 7 ( m ( ! ( m m , ) & ( m ! @CB =

( m !

!,

15.

16.

m ( ! m ( m ! 3 E , = 7<5 7 ( m ( !

) " m ( ! Z m m f K) 7<5 7 ( m ( !

, 3hf D ) , & ( m ! = 5n ! Z 0fg 3 O £ - -21 / f 4a &4 5 3 4a 1 ¢ 0f6 L J

( ] 5n ( 7 5 m !

Y N 5 !

1 ! ( =

Z -!&4 fT3 -21 ¢ 0fg3

5 / £ F ¢ m G j m 5 m , = &5 m ! ) Z

@CB

@CB

17.

18.

19.

20.

21.

22.

Z

3 fg3 1 ¢ 7 m ¢ m F

@ B

@ B

1

m ¢ m G F 3 G

5 m 6( m

m 3 ¢ = 5 m ! @CB

5 m 6( m

= & ( m ! @ B

Z

3 0fg3 1

-21 ¢ 7 m / G f F¢ 3 *

Z

3 fg3 1

-21 fT3 1 ¢ £ 3hf O K m £ f 3 ¢ E m , , ) * ) *

Z

3 0fg3 1 £ &4

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3

= ( m ! @CB E

m , ) * = ( m ! @CB

+

23.

24.

25.

26.

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3 fg3 -21

1 ¢ 7 ¢ m F G

5 m m

6( m 3

-21 ¢ 0fg3 -21 *

¢

2 1

3 0 g f 3

-21 *

Z

Z

= &5 m ! @ B

m *

= ( m ! @CB

5 m 6( m

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27.

Z

28.

Z

3 0fg3 -21 £ &4

-!&4 fT3 -21 ¢

3 fg3 2- 1

1 ¢ m m ( m !

29.

Z

30.

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31.

5)

3 0fg3 2- 1 -21 ¢

K

m ¢ * 3

( m !

= & ( m ! @ B m * = ( m ! @CB

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¢ 3 F ¢ m * G -21

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Z

5

m , ) * = ( m ! @CB

= 5 m ! @CB = 5 m ! @CB

m 0fg3 -!4& m 5n

32.

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-21 ¢

33.

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3 fg3 -!&4

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£ 3 * ( m ! 0f 3

K

m , £ ) *

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¢ 3 !- &4 fT3 -21

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3 fg3 -

1 ¢

36.

37.

38.

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0fg3 -!&4

( m ! m (:7X! ( m ! m

Z

E

3 0fg3 -

-21 ¢

( m ( m ! m

= 5 m ! @CB

= 5 m ! @CB

( m

m , E ) * = ( m ! @CB

m *

= & ( m ! @ B

m *

= & ( m ! @ B

m 3 fg3 - ( m ! -21 fT3 1 ¢

Z

= ( m ! [email protected] B ( m! Z ¢

2 1 T f 3 3 0 g f 3

-!&4 ( m ! = ( m ! [email protected] B

m ( m ! ( m !

= ( m ! @CB

39.

Z

40.

5 1L C7( ! "

J _ 5 7X! N 1 5_7 m L C7<5n m 5 m !C<7 5n 5 !

J ! 2- 1 ( 7 [email protected] B =

3 -21 ¢

0fg3 -

1

Z

!;

+

5) ,

4.1.2. The exponential function

; $ # $ f

¡¤

W

£ -; - + a 1 . W + 0.

F G

1.

Z

2.

' F G G $ L 3 A F = & 2 1 J 1 W #$

Z ! -21 0fg3 -21 + - . f 4a -21 N Ia a 9 a

1

= Z m ¤ . W , + -

" )

# -

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F G

3.

m

" .

4.

Z

5.

Z

6.

Z

7.

Z

1 0fg3 -21

8.

Z

1 0fg3 1

9.

Z

10.

2

+

+ -

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Z

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+ -

.

+

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m

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W

m

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+ -

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@CB

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5 m m n m "

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W

@ B

m , 3 ¢ ¤ )

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m ) , ¤

W

m ¤ ) ,

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m

W m m ¤ ) , 3 1) ,

f) 3 7 , &4

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fI

fI

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0fg3 4a 1 / + - .

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5: n 5 ! 5 L 1N 1 5 J

m . 1 / + - f 3 ¢ *

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Z

14.

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m . -21 fT3 1 / + -

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£ - 2- 1 / f9 4a &4

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16.

17.

Z

18.

Z

+ &a

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£

5 ! '%& 5_7

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#$

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( [email protected] B

m ¤ , ) *

m m ¤ 1 F IG H D F G

m F G ¤ =

m ¤ , ) *

&

7

5 ! 5_7

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! =

@ @CB

= @ B

'27 '

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4.1.3. Hyperbolic functions

; $ # $ f

1.

Z

¡¤

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7 7 fg3 f 4 a F G W W #$ 5 I5 1 1! Y N = ( 7 @CB

& & 9a a 1 9a 1 5 7

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Z

3.

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m m 0fg3 1 F G ¤

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0f63

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L F * 1 m 3 D f L F 1 0fg3 1 0fg

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1 0fg3 -21

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Z

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4.

-21

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11.

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m m 0fg3 L F G ¤

m 0f63 L F G ¤

10.

fQ3 m

m G D £ L1 F * G9H ¤ m m G 3 £ L1 F GIH ¤

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m m ¤ ,

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F * 1

fg3

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m m ¤ G IG H #

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0fg3 4a 1

fg3

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18.

Z

G

m

L

m F1 #" G ¤

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fg3

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L

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m ¤ G F 2- 1 *

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@ B

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21.

Z

22.

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2

1

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m D £ f L F m G f 3 ] L F m G £ f H ¤ 1

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1 0fg3 -21

26.

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27.

Z

-21 fT3 -21

28.

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-21 fT3 -!&4

29.

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5) 3

m , ) *

m - ) * , ¤

m m m 0f63 D £ 1 F G f F GIH ¤

m m fg3 F G ¤

m fg3 F G ¤

fg3

m £ m m ¤ G G 9G H F D F f F * 1 1

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!,9N

m m ¤ F G

m ¤ G V

F * 2- 1

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30.

0f63

5n 5 O! £ - -21 / f 4 a &4 n ! 5 L 1 N 1 5 J

W #$

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31.

1

0f63

m D f L F " m G 3 / £ f L F " m G -21

32.

Z

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33.

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34.

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' (:7 £ a 1

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36.

'%& (:7 £ a 1

( [email protected] B

fg3 / f

L

L

m F1 " GIH ¤

m ¤ G F # " 2- 1

fg3

+ m F #" m G 1 m fg3 / £ F #"

£ 3

7 D + . F 7 G 3 + . F 7 3 G9H £ <3

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. 7 + a 1 F 3 G9H

m

m ¤ F #" G

G ¤

=

7 [email protected]

=

7 [email protected] B

4.1.4. Trigonometric functions 1.

Z

2.

Z

m ! m fI L fI 0fI L fI 0 5 7

2- 1 1

'

5

'

'

¤ !,

=m

@CB

+

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Z

4.

Z

5.

Z

- 0fg3

7 (:7

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7 (87

6.

Z

7.

Z

8.

Z

9.

Z

fI -21

O'65_7X! @ =

= '6 5_7X! @

Z

Z

12.

13.

Z

-21 fT3 -21

1

;

=m

!

[email protected] B

!

[email protected] B

=m

-21 f

@ B

'

¢ ; ¢ ¤

,

£ &4

N & &

fg3

' ¢ ¤

7 f 3 )

Y Z

f

=m

' ! ' ¢ ¤

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-21 f

£ 2- 1

¢ ;

f

' ! ' ¢¢L ¤

#

10.

11.

3

5) 5

f /

0fg3 f 4a W W 5 1 I5 1 ( #$ ! =m 9a a 1 9a 1 5_7

m m 1F G

fg3

3 Ff

fg3 / = = @ [email protected] 7 7 '

!

FOf m m G * m m m 3 G gF G 3 * !, ,

( 7 [email protected] B

=m

@ B

m F *G m

=m

@ B

14.

15.

16.

Z

Z

Z

-21

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fg3

£ " D m F m * G 3

0fg3

m F m G H F m G 1

1 0fg3 1

1 0fg3 -21

18.

Z

-21 fT3 -21

19.

Z

-!&4 fT3 -!&4

20.

Z

Z

m F1 GIH

=m

21.

- fT3 -

22.

Z

0fg3 4a 1

fg3

@ B

=m

@CB

@CB

@ B

H

F G

=m

0fg3

H

m F G

=m

m G

F * 1

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0fg3

@ B @ B

H

m GIH F1 *

=m

@CB

0fg3

m m G G9H = m @ B D

F

F * -2 1 &4 £ - -!&4 / f 4 a fg3

W #$ 5 5 L ( ( @ B

! m J 5 ! N =

! 1

& ] 5

Y

H

m m £ Df H F G 3 H1 F G9H m m

Z

=m

fg3

m

F m G D£ 3

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£ F m * G H F m * G F m * G D 3 1

m m TF *9G H

17.

!,93

+

m m m fg3 D / £ F #" G 3 / f & F #" IG H = m @CB

23.

Z

1

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25.

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-21 fT3 1

m fg3 f 1 1 F#" G

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27.

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28.

26.

fg3 / £

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29.

30.

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31.

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2

m fg3

fg3

H

m F1 #" G

£ F#" m G m m G H1 F #" G9H

@ B

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m F #" G

H

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@ B

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@CB

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f 4 a 2- 1 Z

@ B

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fg3

m H F #" m G 3 m

0fg3 m m D 1 F#" G H F #" G 3 F #"

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W W

( !

N & 1 9a 9a 1 a 1

m H F G -21

m

H

#$

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m G F 2- 1 *

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@ B

fg3

m £ m m £ G

D f H F 3 f ] H F 1 G f H !,!5

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32.

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fg3

£ " D m F m G

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1 0fg3 1

34.

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1 0fg3 -21

35.

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36.

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37.

Z

38.

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0fg3 4a 1

m fg3 F *G

m m G D £ 1 F G h 3 f F IG H / £ m F m G fg3

@ B

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fg3

1 F

*

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Z

m m TF G9H

m m m G 3hf F * G9H fg3 D £ 1 F * = m @CB m m fg 3 F G = m @ B

33.

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fg3

m G

F * -2 1

=m =m

@ B

@CB

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@ B

fg3 £ - 2- 1 / f9 4a &4

5: ( W #$ 5n ! ! Y = m ! ( [email protected] B 1 ] 5 5 L 1N

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Z m 1 fg3 Y D f H F #" m G 3 / £ f H F #" m G H F #" m IG H = m @CB -21 1

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41.

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fg3 / f !, ;

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m G F " 2- 1

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@ B

+

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44.

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fg3

+ m F #" m G 3 m F #" m G

1 m fg 3 / £ F #" G

=m

@ B

@ B

a 1 5_7V( ( #$

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7 a ! _ 5 7

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£ R ¢ f

" '#" ' % f 3 ; N 'g5 1 ]'6')5_( 7 ! ( m

1

=

C7<5 m ! @ B

=

C7<5 m ! @CB

X ¢ f

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N

5_7 L ( ! J

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f " 65_7X! 0f a 5_7 L f - 5_7 L J

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!,9:

a ( 1 1 '6 5_7 ! ( m

£ £ &N

a 1

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5_7( ( #$

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49.

50.

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SZ

f 0f

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3h

m ! m

f

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Z

Z

f £ £ N f &

#$

¤

W #$ 1 #7 ! ( ¤ & #7 ( V 7< 5

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W 7 ( £ £ 1 N 7 ( ! V 7< 5

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f f

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1

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m G ES-21 a 1

= m

@ B

7 5 (* $ 5 ' ( 7 #$ = m @CB

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+

55.

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57.

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60.

61.

62.

63.

64.

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V-

m 7 ( # , 7 5

& C7<5 ! = #7 &B B B@CB W ( "( SZ 7 ( #$ 7 ( ( X 7 ! ! a ¤ 5_7 N & -

m

&- & W #$

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m 7 (

1

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N 7 ( #7,5

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0f 1 V7 m f £ £ N ¤ ! 7 ( V < 7 5

m

f

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m 1 &- &

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N 7( ! #7, 5 m 1

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&

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66.

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f

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W a 7 N m 7 (

= m

@ B

Z

SZ

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SZ

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f

0f m ! m

f £ £ N f &

f / 0f m ! f f m

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/

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G =

@CB

W #$ 7 ¤ ! V7< 5

W #$ 1 V7 ¤ ! & #7 ( V7< 5

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5_7 #$ ¤ ! a 5_7

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= ( 7 &C7<5 ! ! #$ 1 1 V7 #7 ! ( ¢ 3 V- S N m & & #7 ( #7,5

#7 #7 ( #$ 1 ! 5_7 N

m & &- & a

m &C7<5 ! ! $ # 1 1 #7 #7 ! ( N m & & V7 ( #7<5

&C7<5 ! m ! X" J L F m G 3 ]M7 (:7X! 3 m

= m

@ B ¤

B B

@CB

m & 9a 1 L

5_7X! J L J

9a 1 #$ ( 7 C7<5 m ! @CB

=m ( m ! ! !

1 #7 9 a 1 N & 5_7 9 a & &

7,5 7( Z1 145. -21 7% ( 7( 0f

#$ 9a 1 #7 9a 1 m J 1 L ¢ N 3

5 7! L & & 5_7 ! ( m J

7<5 7( Z1 f

7 ( 7( 146. m f9 IF 7 7 £ ¢ 7 ¢ 7 f * f 3 * f )

Y

0f £ £ f

5) :

7 9a 1 9a 1 #$ & N 9 a & ! ( m = m ! ( R7 @CB

!< ,

,

f ¢ 3 £G = &C7<5 m ! @CB

Z

147.

148.

149.

150.

151.

_ 5 7X! (87

Z

-21 f9 3 2

-21 7 f 4a

Z

Z

Z

153.

Z

m

3 m ¢ 3 3 m F

f9 32

-21

-21 fg3 -21

F ¢) G N & m

7 m D 2 F GIH

m G

= m

@CB

7 #7#7 5_7 m #$ ! & 5 I5_7

( C7( m ! B ! !

¢ 3hf9

3 ¢ H = m ! C 7( m ! @ B

0 f9

m

f9 32

1

m D f9

<3 F 7 3 ¢ G m

f 4a a 1

152.

m (

m

=m

!

C7(

m ! @ B

0fg3

¢) ¢ N m

!

W W 7#7 #7 _ 5 7 I5_7 #$ ! & 9a _ 5 7 Ia a &

( 7 ( m ! B !

1 fg3 1 fg3

7 f9 3hf9

¢ 3 m 9f m ,

) , ) ( m ! @ B =m ! Z m m 3 -21 fg 3 1 fg

) , = m ! ( m ! @ B

!<93

+

154.

155.

156.

157.

158.

Z

5) :

1 fg3 -!&4 fg3

£ f 5( m ¢ 3 m 3 , m

) m= Z 3 -21 0fg 3 -21 fg =m

Z

-21 fg3 -!&4 0fg3

f 5 ( m £ ¢ 3 m , m m )

m

) !

!

m

)

,

( m ! @ B ,

( m ! @ B

Z

-!&4 fg3 -!&4

f 5( m

0fg3

m £ m ¢ , m ) 3

=m

=m

!

& ( m ! @CB

!

& ( m ! @CB

fg3 4a 1 fg3

W #$ 7 #7V7 5 " " m & 5 ! ! N &

5 L & ] 5 J

= m ! ( ! ( m ! @CB

Z

159.

160.

Z

1 fg3 -21 fg3

m m ¢ f £ h 3 f

) 3 , f ) , = m ! & ( Z m 3 !- &4 fg 3

-21 fg ) = m ! & (

!
,

m ! @ B m ! @ B

161.

162.

163.

5 m

fg3 m 6( m m ¢ & ( m ! m ) 3 , =m ! 7 Z 5 m ! 5 m ! 5 m

- 7 7<5 7 ¢ 3 £

= m C7( !

7 ( ! m Z -21 m 5 m (

7 #7V7 5_7 5_7 m #$ 5_7 L ! ] 5nO! f 4a J N n 5 L J = m ! ( ! &C7 ( m ! S Z " ¢ S m 3 V- #$ 1 #7V7# 7 #7 ! &C7 ( ! Y N = m

# 7 ( # , 7 5 ! &

" V7#7 #7V7 #$ Z 1 ! m N & V7 ( #7<5

m &C7 ( ! ! Z m 5 m ( m f £ ¢ 3hf <3

7,5 m C7 ( m ! 7( m = m ! Z

- fg3 -!&4

@CB

@CB

164.

165.

166.

1

167.

Z

168.

Z

7,5 7 (

m 5 m (

3 7 D f ] f9 3

@CB

@ B

B @CB

£ 3 ¢ ¤

m f

7<5 m ¢ 3hf9

<3 £ 7 ( m ]f 3 ¢ ¢ f & & H = m ! C7( m ! @ B !< ;

+

1

169.

Z

170.

Z

7,5 7(

5) :

V ¤

£ 3

5 m ( m

D 0 f O 3 4 3f O <3 7,7 5( m H 3 ¢ 3hf9

m m

m= ! C7( m ! " " 5 ! Z

f

5: L a 5n L - J

J

7 #7V7 a 5n #$ & ! Y N = ! C7 ( ! & - 5n a 5n

" " 5 ! Z 0f

f - 5: L a 5n L J J 7 #7V7 a 5n #$

£ & 0 f ! Y £ N

n 5 a f - 5n &

= ( 7 ! C7( ! SZ 7 f

f #$ 1 V7#7 #7#7 ! m f £ £ N m f & ]#7( V7<5

C 7( ! = 7

@ B

171.

@ B

172.

173.

1 7

174.

Z

175.

Z

7<5 7 (

@ B

@CB

3 £¤

1 2- 1 0f

7#7 #7 9a £ Y ' N & & ] 5n !

m 9 a & L 5:O! J 5n L #$ J 7 #7 5_7 9a & & £ m 3 N & & 5n 5: ! m ( 7 C7(

!<9:

!

#$

m! B

1 7

176.

Z

177.

Z

178.

Z

1 7

1

f & f

¤

0f 7 m f9 0f9 3 ¢ 0f9 0

1

Z

179.

&

C7 ( m ! B

C7 ( m ! B

f

7X Vf9 ¢ # f 3 & f9 3 ¢ 4 ¢ h 3 f9

¢ £ f 3 ¢ f9

m C7 ( ! m

1

180.

Z

181.

Z

&

1 -21

Y

182.

1

Z

183.

1

Z

¢ & N

0f

m L 5_7X! J I a & L #$ J 7 9a 5_7 #$ 1 3 & N 9a 9a ( m & & ! ( 7 C7,5 ! m B !

1 V7 5_7

& 9a & ! ( m

0f

m

)

¢ f 3 7,5 7<5 m 3 ¢

, C7<5 ! m

0f

m 3 f 7 m 7,5 m £ f f 3 ¢ 2

B

7

7 ¤

B

!< <

m

7,5 ,7 5 C7<5 m

m !

B

+

Z

G £ 184. F £ ¢ £ ) 3 7X , 3 7

5) ,

£ 3 7 £ ¢ 3 , 3 ) 7 ¤ 7X

&

4.2. The Dilogarithm Li2 (z) 4.2.1. Integrals containing Li2 (z) and algebraic functions 1.

2.

3.

Z

-21 0fg3 -21 0fg3 4

W #$ 7 V 7 # 7 _ 5 7 I5 _ 7

f 4a a 1 ¢) ¢ N ! Ia 5 7 9a a &

( 7 & ( m ! B m ! ! ( ( m Z

f ) 1 0fg3 2- 1 fg3 4

, 5 ( m m & ( m ! [email protected] B m 3

= ! ( ( m Z £ 3 4

) -21 0fg3 -21 fg , 5 ( m = m & ( m ! @CB 3

!

4.

5.

Z

-21 0fg3 -!&4 fg3 4 ¢ 3 ¢ 3 m 5 m *

( m

=m

!

& ( m ! @CB

" " m & 5 ! 0fg3 4a 1 0fg3 5 L

J

7#7 #7] 5 Y N ] 5 = m ! ! & ( m ! @ B &

!

Z

!<9L

+ #

6.

7.

8.

!

Z

-21 0fg3

Vf 1 m £ 3 ¢ * * m )

Z

m £ * 3 , = m ! ( m ! @ B

-!&4 0fg3 - 0fg3

¢ 3 ¢ 3 m ¢ ¢ 3 m 3 , m * * ) m=

Z

-21 0fg3 -!&4 £ 1 £ 7 3

)

£

!

( m ! @ B

0fg3

7 m * ¢ 3 , 3

7 7 m

) * ¢ 3 , = m ! ( m ! @ B

4.2.2. Integrals containing Li2 (z) and trigonometric functions 1.

2.

SZ

Z

7

f 0f

m m

1 #7 #7#7 #7 £ £ N #7( V7

= m ! C7 ( ! " " 5 ! f

0f a 5n L - 5: L J J 7 #7V7 a 5n #$ £ & 0 f ! Y £ N

n 5 a f - 5:

= ( 7 C7( ! f f

!

3.

@ B

Z

f

" " 5 ! N - 5: L a 5n L J

J

@ B

!<9P

7 #7#7 a & ] - 5:

= (

5n #$

a ! 5n 7 C7( ! @ B !

+

4.

SZ

N m

Z

6.

"

"

Z

5.

1 1 # 7#7 !

& - & a

1

& &

"

¢ 3 m

N m

#$

#$ 1 V 7#7 # 7V7 ! ¤ #7 ( # 7,5

-S

5) , 3

N

m 5_7 N

1 V 7#7 # 7V7 ! #7( # 7,5

7 #7#7 #7 & #$ ! &- & a m ! C7 ( ! B #$

m

!

&C7 ( ! B

4.2.3. Integrals containing Li2 (z) and the logarithmic function 1.

2.

3.

m & !

5_7X! 5 & L

J 7V7 5 7 7 5_7 5_7 m m ¢ Y N ! ! 3 N 5 n 5 5

&

&

& &

7 #7 #7 5_7 m ( C7( ¢ ! m = #' N & 5 & ! !

Z m 5 m ( F £ f 7 G / f I

7 ¢ &C7 ( f * m 3 / f I 3 f = m ! Z m 5 m ( -!&4

/ f / f g3 7<5 7( m £ / ¢ 3 f g3 £ m ) , m= C7( Z

-21

m 5 m (

!

m ! @CB

m ! @ B

m ! @ B

4.2.4. Integrals containing Li2 (z) and inverse trigonometric functions 1.

1 7

Z

/

X7

&

£ 3 £ £ ¢ £ ¤

L N

!>

2.

1 -21

Z

!>

m 5 & L

! 5 7J ! 5n

f

7 5_7 5 Y N 5n 5n & 3 ¢ N

m &

&

! 7 #7 #7 5 ¢ #' N & 5n /

7 #7 5 & n 5 m ! & = C7 ( m ! @ B m!

4.3. The Sine Si (z) and Cosine ci (z) Integrals 4.3.1. Integrals containing Si (z) and algebraic functions 1.

m m 0fg3 £ / f F #" G 3 £ m m m / f D 1 F #" G H F#" G 3 F #" G H1 F #"

Z

-21

/ £ 1 F#" m GIH

m

G

=m

@CB

4.3.2. Integrals containing Si (z) and trigonometric functions 1.

7

Z

£

m ( !

£

m

D fI

2.

3.

m ( !

Z

£ 3

f

4.

" " 5n ! - a & a a & J L J

Z O' ! 0f

m

m m F G m 0fI H F G

m m 3 F G m m m D fI <3 0fI H F G

7

Z

@CB

=m

=m

£

N L &

5 7 1 _

& & - a

W a & (

& ! a a &

#$

=

@CB

( @CB

W #$ ' 5 " '#" m '%& 1 #' 5 1 ! ( ' 5 7! % ' 5 7 ! N & ' 5 # ' 5 # O' 5 7 ¤ & &

L1

+

5.

Z

f 0f

Y

6.

7.

8.

9.

10.

11.

'

Z

!

f

&N

" " 5:O! a a & L - a & L J W

J

1 5_7 a & ( #$ ! = & & - a & a a &

f £ £ 0f

"' ' & '6 5_7X! % m '65_7X!

SZ

Z

7

Z

7

"

0f 0f

m

m

f f

Z

"

£ £ &N

£ 3

£

¢

m 3

m &N

-S

¢ f G

= m

F f

¢ f G

= m

&N

@ B

W #$ 1 1 #7 ! ( ¤

& & #7 ( # 7,5

W 1 1 V7 ! (

& & #7( V 7<5

L,

@ B

C7<5 m ! " 3 " m 5 7<5 m ! -

W a 1 ( #$ ¤ ! a & 'g5 7

a 1 a &

([email protected] B

W 1 1 7 ( #$ ¤ ! & & 7, ( 72 5

£ 0 m 5 7<5 m ! 3 " f C 7<5 m ! F

S Z "

' N &

£

V-

5) 31 ,

#$

= m

@CB

!> Z

12.

!>

" " C7 ( !

7( 5 7 ( ( Y F G F G &N

W #$ ;1 - a ! ( 1- # 7 (

=

1 1;- & &

[email protected]

4.3.3. Integrals containing Si (z) and the logarithmic function 1.

Z

-21

Y

2.

Z

Z

¢ N & m 5 m (

m 5 m (

&

3.

& m 9a 1 L 5_7X! J 5_7

L J W W W W #$

#$ 9a 9a 9a 1 1! ( 1 1! ( 3 N & 5 7 9a

& & 5_7 & &

= m ! ( [email protected] m £ f # f 0f O 3 0f O

1 ¢ 0f9 3 1 f O H 0f O 3 0f O H1 f O 0 = m @ B

m 5 m (

m ] 1 0f O £ f # f ]0f9 ¢ 0 f O 3 f9 m f 2 1 f O H 0f O 3 0 f O H1 f O 0 = m

@CB

4.3.4. Integrals containing Si (z) and inverse trigonometric functions 1.

1 -21

Z

0f

m 5_7 L ] 5_7! J 9 a & L J

3 £ f

m

2.

1

Z

&N

1 9 a 1 5_7 ! (

& & 9a & 9a &

W #$

=

( [email protected] B

£ 0f9 ¢ fI 3 £ f fI 1 ¤ f9 ] 1 fI H 0fI 3 0fI H1 0fI L 3

+

5) 31 ;

4.3.5. Integrals containing products of Si (z) and ci (z) Z

-21 £ 3 £ 0

" " V 3 ¢ 0

7<5 =( D F G 3 F 9G H

1.

2.

Z

3.

Z

-21 £ 3

@ B

£ 00 £ £ 0

£ - -!& 7<5

( 7 R7 @CB D F G 3 <3 F GIH = W 1 1 #$ ! m ! / E / fg3E = m @ B

N & & &

4.4. The Error Functions erf (z), erfi (z) and erfc (z) 4.4.1. Integrals containing erf (z) and algebraic functions 1.

fg3 f 4a F W W #$ 5 I5 ( Y N 1 9a 1 a I 1 a! 5 7 =m ! & & & 1

Z 4 a 1 0fg3 0fg3 £ - -21 f 4a W #$ ] 5 ( Y N 1 5 ! f ¡

&

Z

-21 0fg3 -21

7 7 G

2.

3.

Z

1

5 L J 5 !

0fg3

" m - W f m 0f ¢ m , , 1) )

( 7 [email protected] B

3

¤

L95

=m

@CB

4.

Z

! #

-21

" f - W m m , , 1) ) = m @CB

fg3

(

4.4.2. Integrals containing erf (z), erfc (z) and the exponential function 1.

-21 0fg3 -21 W + - .

Z

W

4a f .

=m

=m

4.

Z

-21 0fg3 -21

5.

Z

4a 1 0fg3 W / + - .

W

.

+ -

7.

Z

Z

1

-21

W/

W/

+ -

+ -

W

.

.

+ -

Y

6.

0fg3

F m G m

@ B @ B @ B

£ - 2- 1 Of9 4a 0fg3 W #$ 7 5 5 L ( @ B

! m J 5 ! N =

!

& 5

.

( 7 @ B

m 0fg3 F G

-21

Z

!

3.

=

W W #$ 1 ;5 1 ! a I a 5 7 1

m

=m

Z

+ -

0fg3

WW 0fg3 F 3 ¢ G

2.

7 5 7 7 F G & N & 9a &

0fg3

7 " D £ h £ f 3 ¢ W H 3 f

=m

W 0fg3 F 3 ¢ G

=m

L;

@ B

@ B

+

8.

W . -21 0fg3 -21 / + -

Z

5) 5) 3

m 0f63 m F #" G = m @CB

4.4.3. Integrals containing erf (z) and trigonometric functions 1.

Z

2.

Z

0f

'

!

f

7 ( m

#$ 1 #7 ! ( ¤

N & - & a

m !

"' m '%& '65 7! _ 5 7 J L

#$ a a ' 1 1! ( m ¤ N a ] '6_ & 5 7

Z

3.

' !

6 " ' m '%& '65 1 L ' 5 1 '65 1 ( ! ' % O'65_7XJ ! N

' 5 & ] '6 5_7 g

" " 5n ! - a & a a & J V7< 5 L aJ #$ L & ! 1 Y N =

a & & & - & a a &

" 5nO! - a & L a a & L J J #$ #7,5 a £ & ! ( 1 f = £ N

a a a & & 0f & - & &

"

" 5n ! W W

- a & L a a & L J 7#7,5 J a

#$

& ! Y N = & & & - a & a a &

0f

Z

4.

5.

Z

f

f 0f Y

6.

Z

f

m #$ ¤

( @CB

([email protected] B

L :

( @CB

7.

Z

! #

f 0f

Y

8.

Z

9.

Z

W a

-

-

W

" 5nO!

a a & L - a & L J

J 7 #7<5 a & #$ N ! = ( [email protected] B & & & - a & a a &

#$ 1 # 7 ! ( m 5_7 = m @ B

N &- & a

7 #7 m 5_7 N &- ! & a

= m @CB

W

f £ £ f

W

(

4.4.4. Integrals containing erf (z) and the logarithmic function 1.

Z

-21

m 5 m (

Y

&N &

1 9a

&

m &

5_7X! J 1 Ia 1 ! ( 5_7 Ia

9a 1 L

J L #$ m

=m

&

!

( [email protected]

4.4.5. Integrals containing erf (z), erfi (z) and inverse trigonometric functions 1.

2.

3.

1

Z

0f

m

1

Z

1

Z

2

3 ¢ ¤

0f m W W Y Vf f9 - m , f9 #f9 3 ¢ - m , 3 1) )

W W

W m m ¢ - 0f9 ) , f9 1 ) ,

f m f <3 £ L<

fT3

C

¤

¤

+

4.

5.

6.

1

Z

1

2

1

8.

Z

W W -

1

m Vf 3

V -

W

3 f9 P3

f9 O £

f

3 £H¤

C

W m m ¢ ) , 3 f9 1 ) , f9 P3

¤

W m m ¢ ) , 3hf9 #f9 1 ) , 3

f9 C £ fT3 43f9

0 ¤ 0f

m

¢

f

Z

f

m

1

7.

f

Z

D£ m

Z

W W

5) 5) :

W W

W

¤

0f

D £ - W f9 3 m

3Pf9

£

£H¤

f C3

4.4.6. Integrals containing products of erf (z), erfc (z) and erfi (z) 1.

Z

2.

Z

3.

Z

4.

Z

5.

Z

7

7

f

0f

m 5 5( m 3 7 m m m

7 m m D F G 3 F 3 G9H

=m =m

=m

7 W 3 f 3 ¢ G / E / fg3E F h

=m

@CB

¤ 7 W / fg3E F - f 3 ¢ G

/ E

@CB

L L

@ B

@ B

6.

7.

8.

Z

# 0

/ T f 3E 9f

7 Y D 0f <3 0f f L 0f <3 0f L f H m 1 1 1 =m Z m W W £ 3 E

f - 1 ) , =m / E / fg

Z

/ E

-21 0fg3 -21

@ B

fg3

W W #$ # 7 5 ;5

7 7 1 1 1! f 4a F G N & & 9a a 1 9 a 1 5_7

m

( 7 @ B = ! W £ f W m m @ B = / E / fg3E

, 1)

fg3

@CB

9.

Z

Z

10.

W

/ E / fg3E

W

¢ 0f

3

=m

@ B

4.5. The Fresnel Integrals S(z) and C(z) 4.5.1. Integrals containing S(z) and algebraic functions 1.

2.

Z

-21 0fg3 -21

& & m

*

4a 1 0fg3 " "

Z

0fg3

F

G &N

W W $# & 5 & I5 & ! ( 1 & Ia a & 9a a

=m

!

( @ B

fg3

W 5 5 11L ( #$ & 1 1 f 4 a &4 J 5 N & !5 & 1& 1& L J

m = !

L P

( 7 7 [email protected] B

+

5) ; ,

4.5.2. Integrals containing S(z) and trigonometric functions

1.

2.

Z

Z

O' ! f

Y

3.

4.

SZ

0f f

7

5.

Z

6.

Z

"

W ' 5 " '#" m ' & #' 5 & ( #$ 1 ¤ O ' 5 7! % ' 5 ! N !

& ' 5 & V ' 5 #O' 5 7

" ]" 65 L J

- a L a a L J W J & a a ( #$ ! = ( X @ B N & & - a a a

0f f

0f £ £ f

m

SZ

m

"

£ 1

£ ) , &N

"( 7E( m

f f

7

W #$ 1 & # 7 ! ( ¤

& V7 ( # 7,5

W #$ 1 & 7 ( ! 7 5 ¤

( E 7 &

W #$ 1 & #7 (

! & #7 ( #7,5

m @CB =

1

, &N )

1

m ) , &N

"

& ( L J

&- - &- a ( (* ( n 5 & Y F G F G & N &- - ! ( &

=

P N

W

#$

@CB

# 0

4.5.3. Integrals containing S(z) and the logarithmic function 1.

Z

-21

Y

m 5 m (

£ '

7 f 4 a 4& F G 4&

W W Ia & 9a & ( #$

& ! 3 N & & 9a &

N &

9a & L

5 J ! 9a L JW W 9a & ( #$ ! 9a Ia ( @CB =m !

4.5.4. Integrals containing C(z) and algebraic functions 1.

2.

Z

-21 0fg3 -21 fg3 f 4a -21 F 7 *

4a 1 0fg3

Z

7

G &N

1 5 1 ;5 1 9a a 1

m = !

0fg3

5 L £ - -!&4 f9 4a 1 J 5 1 1 L N & J

W W #$ 1! ( 9a 1 a

&

( 7 @ B

W 1 5 ( #$ ! 1 ] 5 1 1

m = !

( [email protected] B

4.5.5. Integrals containing C(z) and trigonometric functions 1.

2.

3.

Z

' !

0f

'! J %

L

' &

W ' 5 1 ( #$ g '65 7! 1 N '65 ! '65_ 7 ¤ " 65 & !

- a L a a L * J W J a & a ( #$

! = ( [email protected] B a a a -

0f f £ 1

0 f Y f £ & N 1

SZ 7 0f 0f

W #$ # 7 ( £ m 1 ! f £ F G 1 N ¤ ( 5 m # 7 # < 7

& f

P1

Z

+

4.

SZ

"

"

5.

Z

6.

Z

Y F

5) ; :

W #$ 1 # 7 ! ( ¤ N

& V7 ( # 7< 5

W #$

1 #7 ! ( 1 N = m @CB

& #7 ( V7< 5

" " 1 ( L #$ J 1 1;- - 1;- a W G &N 1 1;- &- ! (

= [email protected]

7 m * ¢ 3 V - S

7 m F G

7 (* (*

G F

7 (* 5n

4.5.6. Integrals containing C(z) and the logarithmic function 1.

Z

Y

-21

m 5 m (

£ ¢ N &

1 9a 1

m & 9 a 1 L

5 7X! J 9a & L * _ J W W #$ W W #$ Ia 9 a (

1 1! ( 1! 3 N & 9a Ia 9a & 1 &

m h7 @CB = !

4.6. The Incomplete Gamma Function γ(ν, z) 4.6.1. Integrals containing γ(ν, z) and algebraic functions 1.

& & " -21 0fg3 -21 fg3 4 m ' ; W #$ 5 I5 ( ! Y N = m 5 ! I5 ! & & 5_7 9a 5 9a a 1 5

Z 4a 1 0fg3 fg3

] 5 5n ! ( #$ £ - - -21 f9 4a a &4 5 5:O! N 5 7 5 5

5 5 L _ J

=m 5 ! Z

2.

@CB

P,

@CB

4.6.2. Integrals containing γ(ν, z) and the exponential function 1.

-21 0fg3 -21 + - . fg3 4

W #$ & & " 7 5 ;5 m ! ' ; & N & 5_7 Ia 5 9a a 5 1 m= 5 ! I5 ! @CB W 7 m Z ! . + - 0f6 3 4

, 5 1 L -21 ) J

= m @CB Z

2.

3.

4a 1 0fg3 / + - . fg3

Z

7 5 5 ! 5: ! #$ £ - - -21 f9 4a a 4& 5 n 5 5 L N 5_7 5 5 J

= m 5 ! ( A7 @ B ! Z . 3

-21 / + - 0fg ! 7 m G = m ! ( 7 X @ B F

5 1L * J

4.

4.6.3. Integrals containing γ(ν, z) and trigonometric functions 1.

2.

SZ

Z

9m

0f f

-

m

f f

W

£ 1 #7 ! ( £ & N & 5 7 #7 ( V<7 5

" " " 65: 5 7 ! a a a L a - a L #$

7 1 a J 5 5 5 7 J ! Y N =

_ 5 7 5 _ 5 7 a 5 _ 5 7 & & -

P 3

f

@CB

=

65 5_7!

@CB

+

3.

Z

0f f

W

5.

SZ

-

f f

a W

m

6.

-

-

-

8.

Z

Z

-

#-

-

a W

-

¢ 3

7.

#$ 5_7 ! 5 7 a 5 _

5 5? = 6

7X! O!

=

#$

@CB

@ B

=

#$ 1 # 7 ! = &m & N & 5 7 #7 ( #7,5

#$ 1 #7 !

&m & N & 5_7 #7 ( #7,5

= m

-S

@ B

1 #7 # 7 ! & m & N & _ 5 7 # 7 ( # 7<5

-S

0f £ £ f

£ 1 V7#7 ! £ & N & 5 7 # 7 ( V7<5

f f

W

¢ 3

SZ

5_7 L

-

9m

4.

" ]" 65n 5_7X! a - 5_7 L a a J J

7 a 1 5 5 Y N & & 5_7 - 5 5 7

SZ

5) :1 3

@ B

@CB

N &m & & 5 7 P95

1 #7 # 7 ! # 7 ( # 7,5

#$

= m

@CB

γ (ν, z )

4.6.4. Integrals containing γ(ν, z) and the logarithmic function 1.

Z

m 5 m (

-21

Y 2.

m 5

-21

Z

&N &

Y

&N &

m & 5 ! 5 5 ( m ! 5_7 5 5 1 5 5_7

m m (

5 7 5 5 m ! 5_7 5 5 1 5 5_7

5 !

5 5 1L

J

=m

5 !

5 !

@CB

& 5 !

!

J

=m

5

5 1L

@CB

4.6.5. Integrals containing γ(ν, z), erf (z) and erfi (z) 1.

2.

Z

"

Z

]0fg3

!

£ 0f O a &4 - £ f 3 £ )3 ¢ F - f GIH D 5 & J L =m

] fg3

Z

] fT3

#$

! & ! m a a 5 L f 4& 1 1 N 1 5 J

=m

3.

W W 7 ! ! a a 5 L f 4& 1 1 N a a J

#$

@ B

=m

@CB

@CB

4.6.6. Integrals containing products of γ(ν, z) 1.

Z

]0fg3 4 Y

1N

! ! 65 5n ! f a a 1 a W W #$ 7 ! =m a 5_7 a a &

P;

a

@CB

+

5) <

4.7. The Bessel Function Jν (z) 4.7.1. Integrals containing Jν (z) and algebraic functions 1.

2.

('%& !&

Z

5 ! ( & !&

&(' & ¢ ' 5 1 L 43 J $ L

5 m ( 7! ( 1 ( J L J Z

S

'

&\ - & - \ " #B 5: m !

. 0fg3 £ - !- & -21 / f9 4a + a & W #$ 5 5 5n L : 5 ( ! Y 5 J 7! 5 a L 1 N 5_7 5 a J

m= ! 5 ! ( [email protected] B . - 0fg3 - + a 0f63

m £ + a & . / f+ 1A- . -21 H G = m @ B F # " -21

. 0fg 3 + - 0fg 3

£ + &- . / f+ a 1 . -21 a F #" m G = m ([email protected] B ! 1

m 3 !- &4 0fg 3 £ 1 F " G -21 0fg = m ! ( [email protected]

0fg3 4a 1

3.

Z

4.

Z

5.

Z

6.

Z

7.

1

Z

-21

0fg3

7

£ £ D / 0f 3

0fg3

m F#" G £ / f

P :

m F #" G

m F#" GIH

=m =m

@CB

@ B

8.

9.

Z

-21 0fg3 1

Z

-21 0fg3 1 1

-!&4 fT3 -21 1

10.

m

G9H

=m

m F #" GIH

=m

1 F #"

@CB

0fg3

D / £

Z

0fg3

m D F #" m G 3

m F#" G 3 / f

fg3 m D ¢ 3

@ B

m F #" G9H =m

@CB

4.7.2. Integrals containing Jν (z) and the exponential function

SEa + &-21 . & + &-21 . V- F G

43 ¢ S £ -
Z

1.

4.7.3. Integrals containing Jν (z) and trigonometric functions 1.

Z

Y 2.

Z

/ E 43 ¢ SEa £ ES a 1 O/ S Y f + SEa a & . 3 ¢ $ L 43 3 ; S 3 m , - ) J .

] - + SEa a a & &SEa a a &4 f2 = m

SEa 0fg3

0f f

/ fg3

" ]" 5_7X! a - J

a Y N

& 5 7

65 5 7! 5_7 L a a J

a 1 a 5_7 !

a - 5_

7 a

P<

(

5_7 L W a

#$

5 7

f £ £ f

@CB

= 65 !

( [email protected] B

+

Z

3.

" ]" " 65 5_7X! a - 5_7 L J

J a a 1 a 5 7 ( Y N

7 a !

& 5 7 a - . 5_

f

Z

4.

£

-

5) < 3

0f

5_7X! a a W

#$

a 5_7

5_7 L = 65 !

( [email protected] B

W #$ " ']" " m '%&( 6 ' 5 ( 1 ' % '65 _ ' 5 5_ 7 ! '6 5_7 ¤ 5 7! 1 N 6

5.

SZ

-

-

6.

7.

Z

#-

-

£

Z

f9 3

3 ¢ SEa '

Y 8.

m SEa )

L J 5_7X! m ¢ 3 V - S

Z

W #$ 1 #7 ( ¤ ! N 5 7 # 7 ( # 7<5

& _

W $ # 1 #7 ! (

N & 5_7 # 7 ( # 7<5

= m @CB

F ¢ 3 J ( m ( %

0f £ f

L J _ 5 7X! m

'

f 3

G

G $ L &- F fT3 f 3 ;- m 5 m ( , S ;- a ) , =

F 43 ¢ E S a £ 3

P L

m @ B

f £ f G

¢ ' $ L]f &- E S a If S -; a O ¤ J

4.7.4. Integrals containing Jν (z) and the logarithmic function

1

1.

Z

2.

Z

7 f m fI <3 ¢ ¤

7

0 f O X £ f O <3 0f O £ 0f O X f O <3 f O = m m & Ia 5 ( Z L m L -21 m # 5 ! 9J a a J 5_7X! 1 L J W W 9a 9a ( #$ Y N 9a a I a ! 5_7 5_7 =m 5 !

& 1

Z m 5 m ( 1 C f O 3 0f O =m

1 Z m 5 m ( 7 ¢ 3 0f O

=m Z m 5 m (

& 7 f f O 3hf 0f O 3 = m ( m

f O

f

@CB

3.

4.

5.

6.

Z

7.

SEa -

0f

43 ¢ E S a £ fI S P < -21

)

P5

5 ] 5 ! 3

C,

=

@ B

@ B

@ B

@ B

W

m @CB

4.7.5. Integrals containing Jν (z) and inverse trigonometric functions 1.

1 -21

Z

f

9a a 1 L L 5 !J 5_J 7! 9a L N & J

P P

W 9a 9a a 1 ( #$ ! 9a 5_7 9a 5_7 5_7

= 5 !

@CB

+

1

2.

Z

3.

Z

4.

Z

5.

Z

6.

Z

7.

Z

1

1

1

2

1

1 7

5) < :

7 f m 0 fI ¤ f m F m G F m G ¤ 1 7 f m £ 1 f m D ¢ 3

fg3hf

fg3 0 fI ¤

F m GIH ¤

7 1 f m 0 fI 3

f ¤

7 f 3 ¢ ¤ 1 f 0 fI m g

4.7.6. Integrals containing Jν (z), Si (z) and ci (z) 1.

Z

-21 £ £ 0

" " 5 ! 7 3 3 G F ( _ 5 X 7 !

7( ( 7 ( 5

Y D F G F G 3 F ¢ 3

2.

Z

-21 £ 3 £ 0

" " 5 ! 7 3 F 3 G ( _ 5 7 !

5 7( 5 Y D F ¢ 3 G G F¢ 3 F

,9N N

5 F G

(

5 G 3 F IG H = 5 !

@CB

5 F G ( ,7 5 5 G 3 F 9G H (= 7 5 ! [email protected] B

4.7.7. Integrals containing products of Jν (z) 1.

Z

3.

4.

5.

6.

fT3

a g5 5_7 & N 1

m m

a a 5 5n & N g & a

Y

2.

65 m

Z

Z

7

7 m ( !

m

a 1 a a a a & 6 5

& a 5_7

5n 65

7 7 0fg3 D )(:7

1 a a & 5 5 1 6 a a ! 5_7 g 5 5 =m !

7 a 2- 1 fI g5 7 =m !

0fg3

5 5_7 g (87 7 D 65 (8 7 a 2- 1 fI 5_7 9m

7 m (

Z

a fI <3

m ! (& 65 5 7!

!

&

( [email protected]

a a 1 fI H 7 ( [email protected] B !

a a 1 0fI H =m

fT3

7X!9 g5 (* ! 7 D 65 (: 8 ( 7!& (:7X! a 2- 1 0fI 9m g5 5_7X!9 65 5nO! g5_7X!9 5_7X! a a 1 fI H = m

( m #$ 5 7 ( m #$ !

!

[email protected] B

!

. ,+ &-21 0fg3 & -21 / E / fg3

£ &-21 F 5 m G + a a 1 . a a F f f G

1

m= !

[email protected]

Z

Z

SEa fg3 a

/ E / fg3 3 ¢ SEa £ SEa . S ' 3 T3 Y F 5 m G + a a2SEa a 1 L $ L 3 8 3 S J J

,9N1

( [email protected]

a 1 &-

+

" . Y & " F 5 m G + a

7.

Z

8.

Z

(

7

1 3 F

Z

1

10.

Z

11.

Z

7 ( m 3 F 1

12.

a a a a2SEa a 1 F f f G = m ! ( (87 ! ( ')(:[email protected]

&-21 0fg3 &-21 / E 1 -; / E / fT3

'%& . 43 ¢ &-21 m G + a a 1 & -21 ' (87 7 G

L

3hf G

3 H f

D 0f O ¢ £ / f I KD ¢ L £ / f I H 3 £ / f I £ / f I ¤ 1 1

Y

F

,9N,

5:7! 5n7 ' 5 1 ( m ¤ O' 5 5 5n7 O ! O' 5n7 a

Z

13.

X- -

0f 0f

" '#" " m '%& &( '6565 5_7! 5_7 L 5g5 7 L 5 5_7 L ' ! J

J a a a 1 J a a 5_7 a 1 ( m N 565_ 7 5 5_ 7 565 5_!7 '65_7 ¤ 5 6 5 5 7 & L J

Y

Z

14.

0f

15.

1A- -

V-

- -

17.

18.

-

&(

&( 7 ( " ( J L 65_7X! 5 7! m

Z

16.

(& m ! L 65_J 7! 5_7X! 7 ( m

SZ

-

a a 1 a 5 7 #7 & ( ! N 65_7 5_7 g 5 _ 5 7 &- & a

#$

¤

a a 1 a 5_7 1 #7 ( #$ ¤ ! N g5 7 5 7 65 _ 5 7#7 ( #7,5

a a 1 a 5_7 1 #7 ( #$ L ! 65_J 7X ! 5 7 ! m N 65_7 5 7 6 = 5 5 7 #7 ( #7,5

Z m 5 m ( m £ f £ f O 3 £ f O

1 m £ £ £ £ 1 f O H f O <3 f O H1 f O Z m 5 m (

& 7 m £ f ] f9 3 £ £ f O 3 0f9 3 V £ f O 0 1 m £ £ £ £ Vf 3 1 f O H f O 3 f O H1 f O 0 ,9N 3

@CB

m

=m

=m

@ B

@CB

+

. 0f9 32

R+ &-21 & -21 F 43 ¢ m m Y ED F

Z

19.

1

20.

Z

21.

Z

22.

Z

1

1 7

&-21

m m F 3 G9H W = m ! ( [email protected] 7

1 fI <3 m fI 1 fI H ¤

7 ¢ 3 0f9 ¢ 0fI f fI fI <3hf9 0fI ¤ m 1 1

1 f

£ &-21 / G

7 f 3 D fI

1 0f

3 7 fg3 £ f2 Vf9 ¢ £ fI Vf9 3 m m 1 £ If H

1 f/ -21 Z

23.

f 3 G

5) L1

3 ¢

f /

¢

/

3 ¢

/

¢ £ If 1 £ fI <3 £ If H £ If ¤ 1

¢

£ N 1

WW

#$

1! ¤ 7 #7 # 7 &

4.8. The Bessel Function Yν (z) 4.8.1. Integrals containing Yν (z) and algebraic functions 1.

7 ( m

Z

F m G F m G

@ B

=m

=m

4.8.2. Integrals containing Yν (z) and Jν (z) 1.

Z

7

1 0 fg3

0fI 0 fI m 1 ,9N95

@ B

2.

Z

D

-

H

1 F ¢ 3

m m 5 m 5 m 5 G

= m

@ B

4.9. The Modified Bessel Function Iν (z) 4.9.1. Integrals containing Iν (z) and algebraic functions 1.

Z

-21 0fg3 -21 0fg3

2.

Z

3.

Z

4.

Z

F

5 I5 m & &( " ! G 5_7X! F G N & 5_7

#$

¤

0fg3 fg3

/ *F m G a 1 F G 1 a F m G 1

m fg3 F G

m -21 0fg3 -21 0fg3 V F G

=m

!

( [email protected] B

=m

@ B

=m

@ B

. fg3 £ - !- & -21 / 9f 4a + a & W 5 5 5n L 5n #$ ! ( [email protected] ! Y 5_J 7! 5 5 m = !

a L 1 N 5_7] 5 a J

Z . 0fg3 + - 0fg 3

6. £ + &- . / f + a 1 . -21 a F #" m G = m ( @ B ! 1

5.

Z

0fg3 4a 1

,9N;

+

7.

8.

9.

5) P1 ,

. - 0fg3 - + a 0fg3

m £ + a & . / f+ 1A- . -21 L G F # " -21

Z

Z

-21 0fg3 -!&4

Z

1

7

fg3

10.

Z

-21

11.

Z

-21 fT3 1

Z

-!&4 fT3 -21 1

13.

m F#" GIH

( @CB

( @CB

=m

0fg3

m D F#" m G F #" m 9G H 1

=m

=m

@CB

m F #" G

@ B

@CB

Z

!

-21 fT3 1 1 0fg3

MD / f F # " m G 3 / £

12.

=m

0fg3

m fg3 £ 1 F#" G =m !

m £ £ £ D / f F#" G 3 / f

=m

0fg3 m D

m F #" GIH

@ B

m F #" G 3 ¢ H

=m

@CB

4.9.2. Integrals containing Iν (z) and the exponential function 1.

Z

-21 0fg3 -21 + - . fg3 4

W & & " 5 1 5 ;5 m 5_7X! F G & N & 5_7 Ia 5 9a ! a 5 1

5 ! I5

! =m #$

,9N :

@ B

2.

Z

3.

Z

4.

Z

6.

7.

8.

9.

10.

@ B

=m

=m

@ B

. + - 0f63 4

5.

m . + - fg3 4 " ) f * ,

2

. + - 0f63 4 " ) f * ,

m 5_7

f , 3 m ) * *

. -21 + - 0f63 4

m m m W , 3 a 1 ) , * -21 ) W 7 Z . 3 4 m £ 3h9f K3 £ -21 + - 1 0f6

W

@ B

=m

Z

=m

=m

@CB

. -21 0fg3 -21 + - fg3 4

m m 7 W =m * , 3 a 1 ) , 2 - 1 ) W Z . 3 -21 + - 1 fg 3 4 m £ 3hf9 K3 £ -21 0fg =m

@ B

Z

. 1 / + - 0fg3

m D 3 F 7 h 3 f G " m / f H

@CB

@CB

Z

Z

. -21 / + - 0f63 " / f ,9N<

=m =m

@ B

@ B

+

5) P1 3

. -21 / + - 1 0f63

7 £ 3 / f / f K 3 £ m

Z

11.

. -21 fT3 -21 / + - 1 fg3

m

=m

@ B

Z

12.

3hf 3 ¢

=m

@CB

4.9.3. Integrals containing Iν (z) and trigonometric functions

Z

" ]" " 65 5_7X! 5_7X! a - 5_7 L a a 5_7 L J J W #$ a a 1 a 5 7

65 ! ( [email protected] B ! Y N =

5 7 a _ 5 7 a a 5 7

& -

Z 0f

2.

f " " 65 5 7! 5_7X! a - 5_7 L a a 5_7 L J

J W #$ a a 1 a 5_7

£ ! Y f £ N = 65 ! ( [email protected]CB 5_7 a - 5 7 a a 5_7 0f &

SZ 0f

3. -

0f W #$ 1 #7 £ m L f ! f £ J 5_7! N & 5_7# 7( #7<5 ¤ m

SZ £ m f £ f 4. -

m 0f f L 1 1 & #7 Y J 5_ 7! N 5 7 #7 ( V 7 ( #! 7,5 V7<5 ¤ 1.

f

,9N L

5.

SZ

-

-

¢ 3

SZ

6.

-

W a

-

Z

8.

Z

-

-

-

W a

-

f 0f " 5_7X! a 5_7 L - 5 7L J

J

Z

10.

Z

f f

#$

= m

@CB

0f £ £ f

1 #7 !

#7<5

N

#$ 1 & a 1 #7,5 ! 1 1 # 7,5 - # 7,5 a

= ( R7 @CB

" " " '& 65 5nO! 5_7X! a - a &L a a a &L J

a J a 5_ 7 a a #$ a £

& & ! 0f £ N f & 5_7 5 & a - a & a a a &

= g5 !

7 5 1 #$ L 1 J 5:7! m & N & 5n 7 7( ! 725 ¤

W #$ # 7 L 1 ! J 5 7! m N & 5_7# 7( V7<5

= m @CB

5 1 L J 5 7 ! m & N & 5_7 #7 (

9.

W #$ # 7 L 1 ¤ ! 5 7X! m N & 5_7V 7 ( # 7<5 J _

¢ 3 # - S 7.

-S

Y

,9N P

( @CB

+

5) P1 5

f

f " ]" 65 5 7! £ 0f a a 5_7 L f £ 5_7X! a - 5_7 L a & J a a 5_J 7 a a & #$

65 ! ( @CB ! Y N =

5_7 5 a a a a a - & & & &

Z

f

" " " '& 65 5nO! 5_7X! a - a &L a a a &L J J a & a a 5_7 a a & #$

! Y N = 65 ! ( [email protected] B & 5_7 5 & a - a & a a a &

Z

f

" ]" " 65 5_7X! 5_7X! a - 5_7 L a a 5_7 L J J

#$

a 1 a & a a 1 a 5_7 ! Y N = g5 ! ( [email protected] B 1 5 1 5_ 7 a - 5_7 a a 5_7

" ]" Z 5_7X! ( !

W #$ a (65 ()( - ! Y F G F G N & 5_ 7

a - 1

= ( ! @CB

Z

11.

12.

13.

14.

4.9.4. Integrals containing Iν (z) and the logarithmic function 1.

Z

-21

m 5 m ( Y

9a

N & 9 a a

9a L m & L # 5 ! 9J a a 1 JL 5_7! WJ W 9a #$ ! =m 5 ! 1 Ia 5_7 5_7

,N

@ B

m 5 m (

2.

Z

3.

Z

4.

Z

&

1

Z

6.

Z

7 1 f O <3

m 5 m (

f O <3

C

m !(:7

@ B

=m

=m

7 m 5 m ( f # f O 3 f 0f O 3 = m 7,5 7 ( m ¤

7( 7 ( f

m " " m &( 5 ! -21 m 5 m ( 5 ! 5 5 1 L 5_7! J

5 1 5 5 ! m #$ Y N 5 7 5 5 5 5_7 =m 5 ! 1 & &

5.

@ B

@ B

@ B

4.9.5. Integrals containing Iν (z) and inverse trigonometric functions 1.

1 -21

Z

0f

9a a 1 L L 5 !J 5_J 7! 9a L N & J

1

2.

Z

3.

Z

4.

Z

5.

Z

1

1

7 0f m

2

1 7

W #$ Ia 9a a 1 ! 9 a 5_7 9a 5_7 5_7

= 5 !

0fI ¤

m m f m F G 1 F G ¤ 7 0f m f 1 0 f

fg3 £

0fI

7 ( ,

m

m ¤

f-

0fI ¤

@CB

+

1

6.

Z

7.

Z

1

5) P1 :

m 1 0 f m D F G 3 ¢ H ¤ 7 1 f m fT3

fI 0 ¤

4.9.6. Integrals containing Iν (z) and special functions Z

4 3 £

" 5 5 5 5 ! 1 3 5 5 7 O 5n7 7 (1 L C 7258 ! & N 5 : ! J

( ! =

1.

-21

2.

£ ] fg3

f & O 1 2- 1 0f O 1 0f O 3 -!&4 0f O 4& 0f O

3.

Z

4.

Z

5.

Z

6.

Z

£ ] fg3

£ ]0fg3

fg3

SZ

Y

@CB

0f f

m * 3

-

/

f £ £ f

,!,

@CB

/ £f¤

m / £ f K3 * ¤

=m

/

J L _ 5 7X! m N

"

&(!& ( & & m 65 : 5 O!

/ E / fg3E

= m

£

!

( [email protected]

m

#$ 1 V7 ! ( ¤ a 1 5_7 5_7V 7 ( #7,5

7.

SZ

-

-

¢ 3 Z

8.

#-

-

/

-S

/

/

L J _ 5 7! m N

/

#$ 1 #7 ! ( ¤ a 1 5 7 5 7 #7 ( # 7,5

#7 ( L 1 J _ 5 7! m N a 5 7 5_! 7#7 ( #7,5 1

Z m 5 m ( 9. 2- 1

" " m & Ia L 5n ! 9a a L J 5_7X! J #$ 9a 9a ( Y N 9a a 9 a 5_7 ! a 5_7 5_7

1

Z1 F f ¢ 3 G / ¢ 3 / ¢ £ 10. -21

#$

= m

@CB

5 ! = n

@CB

W #$ 1! ¤ 1 N & 7 #7 &

4.9.7. Integrals containing products of Iν (z) 1.

2.

3.

Z

fg3 ]0fg3 4

7 7 F G a a 1 f O = m ! ( 7 @CB Z m V 0f63 f 0f O f O L1 f O <3 1 0f O L 0 f O = m @CB Z m m 0f O! # fQ3

f O L1 f O 3 1 0f O L 0f O = m @CB f a a 1 *

,3

+

4.

5.

6.

7.

8.

9.

10.

5) P1 <

V1 0f63 1 0 f O 3hf f O m 3 0 f O L1 f O 3 1 f O L f O 0 = m Z -21 V 0fg3 £ f 1 / £ f £ £ £ £ f 1 / f L1 / f <3 1 / f L / f = m Z £ £ 3

3 f 1 / f 1 / £ f -21 1 0fg £ £ £ £ f 1 1 / f L / f <3 / f L1 / f = m Z

Z

-21 0fg3 1 fg3 m / £ f L 1

Z

-!&4 0fg3 -21

Z

-21 0fg3 -21 V 1

Z

1 fg3

/ £ f <3 / £ f L1 / £ f

@CB

@CB

fg3 1

=m

0fg3

/ £ f <3 ¢ m £ / fg3

m 1 f <3 m

@CB

=m =m

@CB

@ B

] fg3 4

5 1L 5 1 m #$ F m G a 1 ! 5_7J ! 5 & L 1 N 5_7 5 & J

=m !

&0fg3

@ B

11.

12.

Z

] fg3 4

1

( 7 @CB

D £ f O £ f O L1 £ f O 3 1 £ f O L £ f O H ¤ 7 7 # 0f V1 0fI 3 V fI m fI 1 fI ¤ f

Z

,5

13.

14.

15.

7 1 0f m ¢ f 3 ¢ 0fI 3hf fI 1 fI 3hf

Z m 5 m ( m

£ f £ f O <3 1 £ f O 0 m £ £ £ £ f O L1 f O <3 1 f O L f O Z m 5 m ( #

& 7X m £ f ]0f £ £ f O 3 0f V £ f O 0 1 m £ £ £ Vf9

f O L 1 f O 3 1 f O L £ f O 0 1

Z

. ,+ -21 fg3 £ / E V

'& 5 m 1 L J

5_7X! 5 & L N J

16.

I ν (z )

Z

1 0fI ¤

=m

@ B

=m

@CB

/ fT3E

W #$ 1 5 1! = m @ B a 1 5_ 7 a & 1 a 5_7

" " m !& 5 Z 1L 5_7X! J 5

0fg3 / E / fT3E

&L W J

#$ 5 1 1 ! Y N = m @ B

a 1 5_ 7 5_ 7 1 5_7 5 &

Z1 f / ¢ / ¢ f / ¢ 3 / ¢ 3

W W #$ -21 1! ¤ £ N 1 7 #7 #7 &

17.

18.

19.

Z

- fT3 -

/ E - / E / fT3E 1 5 ! C7 ( ! F G -21 N

7( & ( & ( L m J

,!;

f 3E

- / g W #$ 1! a 1 1 5_7 5_7

=m

@CB

# #

5) N1

4.10. The Macdonald Function Kν (z) 4.10.1. Integrals containing Kν (z), Jν (z), Yν (z) and Iν (z) 1.

Z

2.

Z

3.

7

1 fg3 m 0 fI 3 1 0fI ¤

7

-

7 m D 3 1 H F m ( 3 ¢ G 7 1 f 1 f 3 m

Z

m 5 m (

m ¢

3 , )

= m

m = ?

@CB @CB

4.10.2. Integrals containing products of Kν (z) 1.

"

Z

5 m ! If

£ - V/ (:7X! f - 2- 1 T3 ¢ 0 fI £ T 3 ¢ f 1 fI 0 5 1 !

= m [email protected] 7 7 Z F G 3 = @ B &

%

2.

= @ B

3.

Z

4.

Z

&

3 7

5.

Z

3 7

6.

Z

7.

Z

= @ B

3 7

&

1

= @ B

7 7

= @ B ,:

= @ B

L

H

2 % & 1 7

= @ B

Z

8.

% & 1 3 7 Z 7 10. % 1& 7 7 Z X 7 11.

3 1

Z

9.

= @ B

= @ B = @ B

4.11. The Struve Functions Hν (z) and Lν (z) 4.11.1. Integrals containing Hν (z), Lν (z) and algebraic functions 1.

Z

-21 0fg3 -21

£ - f 4a a

Y

2.

Z

- 0fg3 -

3.

Z

4.

Z

L

0fg3

L

&

&N

fg3

0fg3 5 a 1 ! I5 a 1 ! a 1 5 & L 5 I5 5_7! J W W 7 5 a 1 I5 a 1 #$ ! 5 & 9 a a a 1 Ia a 1 5_7

=m 5 ! 5 ! !

L

H

m £ / f 1 - -21 *G 3 F -21

m 0fg 3 D F *G 3 ¢ H

L

H

5 7 O! F G 2- 1

0fg3 £

fg3 7] 5 5 a L 5 J 5 L 5 &L N & & 5 J J

4a 1 0fg3 Y

,!<

( [email protected] B

=m

@CB

=m

@CB

.

-!& + a 1

f9 4a a a 1 W #$ a !

& 5 5

m 5 ! ( [email protected] B = !

# #

5.

Z

-21 H

6.

Z

-21 0fg3 -21 H

7.

Z

8.

0fg3

D¢ 3

0fg3

5)

m F #" GIH

m

m F" G

-21 0fg3 1 H1 0fg3

¢ 3 F #" m G 3 m -!&4 0fg3 -21

H

1

Z

m m F#" G!

0fg3 3 m

11.

12.

Z

13.

-21 fT3 -21 L

fg3

-21 fT3 1 L1 fg3

¢ 3 F#" m G m

@CB

=m

m

=m

m m F#" G! 3

=m

m F #" G

@CB @CB

Z

-!&4 fT3 -21 L1

Z

@CB

=m

=m

fg3 m

-!&4 fT3 - L1 fg3

D £ F #" m G 3 / £ f F #" m G f

m

14.

@CB

Z

m F#" G £ &4

= m @CB

-!&4 0fg3 - H1 0fg3

DC3 £ F #" m G / £ f F# " m G f F #" m G9H m Z m ¢ D F #" G 3 H 10. 3

-21 L fg 9.

=m

Z

@CB

=m

@CB

m F #" G 3 £ &4

= m @CB

,L

m F #" 9G H

=m

@CB

L

H

4.11.2. Integrals containing Hν (z) and hyperbolic functions 1.

Z

( m

/ fT3

H

/ E

W '& !& 1 7 # m ! 5 L N & & a a J

#$

=m

!

( @CB

4.11.3. Integrals containing Hν (z), Lν (z) and trigonometric functions

Z

1.

- 2- 1

£

2.

Z

- 2- 1

0f f

0f

' 5 1 '65_7 6 " '#" " m ' & '& N ' 5 5 & '65 'g5 & L 'g5 5 & L & 6 J J

f

H

H

4.

Z

0f f

5:O!

L J 5 &L

W #$ a a & ( ! a a a & 5 &

= g5 ! K(

" " " !& 65 : 5 O! a - a &L a a a &L 5 &L J J ,P

J

L

W #$ ( ¤ ! & O'65_7

W #$ ' a 1 5_7 ! ( ¤ N & a 5 a ]6 & & ' 5_7

" " " !& 65 a - a &L a a a & J J

7 a 5_7 £ f £ N 0f & & a - a &

Y

Z

" '#" " m '%&(!& a & a & 5 L J L J

3.

H

7X! @CB

# #

Y

5.

SZ

- 2- 1

f

7 a

f £ £ N 0f &

&

H L

a

5_7 a a & ! - a & a a a &

= g5

" " '& ¢ 3 fI 5 & L m J

SZ H f

- -21 L " " '& fI 5 & m J L SZ H

- - -21 L " " !& ¢ 3 - S 5 & m J L Z H

- - -21 L " " !& 5 ! N m & & 5

&N

7.

8.

&N

&N

W #$

5 & ! (K

W #$ 1 V7#7 ! ¤

& 5 & #7( V7<5

W #$ 1 #7 #7 ¤ !

& 5 & #7 ( # 7,5

W 1 #7 #7

& #7 ( ! # 7,5

#$

= m

4.11.4. Integrals containing Hν (z), Lν (z) and the logarithmic function 1.

2.

7X! @CB

W #$ 1 #7V7 ! ¤ & 5 & #7( #7<5

6.

5) 5

@ B

&(!& !& &a a 1 L m L H -21

5 5:7! J a 5:J 7 L 5 & L L J

J

W W #$ 7 a a a a 1 1 ! Y N = m ! 5 ! ?( A7 @ B

5 n 5 a & & 7 &a a & &

7 Z m 5 m ( f 3 f O 0

H

= m [email protected] B Z

m 5 m (

, ,9N

#

3.

Z

" "! #$" #$$!

m 5 m ( 7 f O 3 f X L

@CB

=m

4.11.5. Integrals containing Hν (z), Lν (z) and inverse trigonometric functions

1 -21

f 0f

W !& 7 9 9a 5_7 L a a 1 9a 5_7 #$ m ! 5 5_7! J 5 & L 9a a 1 L & N & & 5 & 9a a & 9a a & J J

5 ! ( R7 @CB = Z1 7 C 3 0fI f = m [email protected] B 2. H f

m 1.

Z

H L

1 7

3.

Z

4.

Z

5.

Z

6.

Z

1

H L

1 7

1 f

C3

7 f m 0fI 3 ¢ 3 1 0f

¢ 3

L

fg3

H

C

0fI

C3

f

m

7

f

f

fI <3

fg3 m

7

f

= m

@CB

= m

@CB

@CB

= m

/ E L / fg3E

W #$ & '& 1 #7 ! m &( g5 5 ! N !

& & 6 5 5 & 65 5n

= m !

( @CB

4.12. The Kelvin Functions berν (z), beiν (z), kerν (z) and keiν (z) 4.12.1. Integrals containing berν (z), beiν (z), kerν (z), keiν (z) and algebraic functions 1.

Z

7

m (

m D F G 3 , ,

m GIH F

=m

@CB

# #

7 ( m

2.

Z

3.

Z

4.

Z

F m G F m G

7 ( m

7 D

7 ( m

5) 31

m G 3

F *

m G F

m G9H * F

m G F

@CB

=m

=m

=m

@CB @CB

4.13. The Airy Functions Ai (z) and Bi (z) 4.13.1. Integrals containing products of Ai (z) and Bi (z) 1.

2.

3.

4.

5.

6.

7.

Z

-21

Z

-21

7X

3

! 1;- L J 1a L - L J & J & 7 @VB 7 [email protected] = ! = L 9a 1 L 3 " 8 J I a L J - L J J = ! = @&#B [email protected] " 3 & 8 ! = @CB 9a L J 3 " 8 3 " 8

5_7 F G F G

@ & = = B [email protected]

¢ £ 4 a . - +

Z

-21

" & 3 8

Z

-21

Z

-21

Z

V-

!

3

7(* .

43 ¢ £ - + 4 a T F G = 7 = @VB [email protected]

#$

1L L 1 1 &]

7 J & 3 J & N 1 & ! 1

1 L L 1 1 3 J & J & N 1 & &]3

! 1

!

= = @VB 7 7[email protected]

Z

, , ,

#$

= = @ B 7[email protected]

& '

#$

1L L 1 1

8. X7 J & J & N 1 & 1 ! Z

L 1L 1 1 43 3

9. J & 3 J & N

1 ! &1 #$ Z

L 1 L 1 1 10. 3 3

3 J & N 1 & J & ! 1 Z

= = @VB C! @MB

#$

= = @VB [email protected]

= = @ B [email protected]

11.

Z

&

12.

Z

]& 3

13.

Z

14.

Z

15.

Z

16.

Z

V

#

= = @ B [email protected]

3

7

7 3 7X

= = @VB 7 [email protected]

7X 7

= = @ B [email protected]

&#3

= @ B

" 7 2 ;F G 3 7 F G

= @ B

7 G 7 F G 3 7X F

= @ B

4.14. The Legendre Polynomials Pn (z) 4.14.1. Integrals containing Pn (z) and algebraic functions 1.

1

Z

2.

Z

m 5_7 3hfI -21 j O'65_7
0f 3 -21 j

43 ¢ J 1 ' L % ' j F

, ,93

¢ 3hf G

= m

@CB

= m

@CB

# #

3.

4.

5.

1

7.

Z

8.

Z

11.

m (

! ' & j

m' ¤ 7 ( m

7 a '65_ 7 F m : ( 7 G 1

5 m ' & j a

1

( 7X! ' 6 ' 5 f - &- 0f9 ¢ -;&-!&4

m (87!

B

@ B

= m

= m

@ B

0f 3 -21 ¢ 3 - -;&-21 j

' 7 A. ( 7! F G f9 -21 ¢ 3hf9

-;&-21 j + -21 l ¢ 3 £ 9f O = m @CB

0f9 32

-21 ¢ 3 2

- ;- &-!&4 j a 1

' A. ( 7X! F G f 4a 1 ] ¢ h 3 f -;&-!&4 j + 4 a 1 l ¢ 3 £ f =m Z " ¢ O'65_7 ¢ 3hf9

-;&-21 j a 1 f O ¤ 3 2

-;&!- &4 j

1

Z

-21 Z1 -21

12.

7

-21 7 ( 7X! ' Z1

' & j

'65_7 f - &- f9 ¢ -;&-21

5 m

6.

10.

7(* m 5 m j

-21 Z1

Z

9.

7

1

Z

5) 5)

Z

7 (

7

¢ 3 j £

C

£ ¢ ¤

7 ¢ £ &4 _¤ C7 ( ! 3 j

j ¢ 0fg3 4

O'6m 5_7 ¢ m

*

, ,!5

¤

@CB

& '

13.

14.

15.

16.

0fg3 -21 ¢ - ;- &-21 # ! '! % '&

Z

j / ¢

f ¢ f O -;&-21 4 a 1 / ¢ f Z 0fg3 -21 ¢ -;&-!&4 j / ¢

'6m 5_7 ¢ f O -;&-21 / ¢ f

Z

-21 fT3 -21 j fg3

3 ¢ J 1' L % ' j ¢ 3 m

*

=m

@CB

@CB

=m

@CB

!

@CB

Z

-21 fT3 -!&4#j ¢ fg3

/ £ j ¢ m

, ) *

Z

j ¢ 3 0f63

m 4 3 ¢ ' % f -21 f / j £j 3 ,

&-21 ) ) & J L' 7 Z ¢ 3 fg3

20. 7( m ( ! j a 1 ' # ( 7! ' % & L ' J

, , ;

19.

=m

18.

17.

-21 fT3 -21 j ¢ fg3

" & m " ! ( ' ]'65_7 m 5 1L & N 7 5 1 ! ( = J

Z m m -21 j ¢ fg3 'g5 7 ) * ¢ , Z

@ B

=m

m ,

=m

m j a 1 ) ,

@CB

=m

!

'8>[email protected] B

=m

@CB

# #

Z

21.

43 ¢ Z

-21 j ) " ¢ ' (87! % a 1 1 J L' " 7( m (

5) 5) ,

3 0fg3 ,

m m m F G 1 * j ) * , 3 j a 1 ) * ,

" ¢ 3 0f63 ,

j a ! 1 ) 43 ¢ ' % j a

1 &L ' J

7,5 m ( ! Z 3 j ) 7( m ( ! ,

-21 ¢ 3 0fg " ' 3 ' % -21 ] f 3 £ a 1 j a

1 ) * & J L'

22.

23.

Z

-21 fT3 -!&4 j

24.

W

Z

-;&-!&#-R j F G

25.

=m

@CB

m , ) *

=m

m m (* ,

=m

@CB

@ B

0f63

¢ 3 m 43 ¢ / £ J 1' L % ' j

*

"' " m ' & V -R W F m G

=m

@CB

= m

@CB

4.14.2. Integrals containing Pn (z) and trigonometric functions 1.

Z

0f j

f " 65_7X! 1 L ' 3 ¢ ' % 5 7 L J a 5_7 L - _ J J

(' ]'65 1 a 1 Y N 5 7 & 1 - _

, ,9:

0f £ £ f 5_7 #$ ! a 5_7

=

( [email protected]

& '

" " 65nO! & L ' 0 f ¢ j a 1

43 ' % 2. a J a & L f - a &L

#$ J

( ' ] '65 5_7 J a £ & & ! Y 0f £ N = ([email protected] B & - a & a a & f &

'#" ' )(*'65_7! Z 1 0f G J L ' 5_7 j F 3. ' _ 5 7 a % f - - L - L J J

" # $ ( a _ 5 7 £ 1;- - 1 - ?')(:[email protected] ! Y 0f £ N =

* ( ' _ 5 7 a _ 5 7 & f - - - 1

(' '65 #$ m ! Z 1 & ¤ ¢ 4. f j 7 ( m &N

F G

&- & a ! (

SZ 0f j

5.

f ( ' ]'65 1 #7 ¤ 3 ¢ ' J 1% L ' m f £ £ N ! m f & 7 ( #7, 5

SZ 0f j

6.

f 3 ¢ m f £ £ N ( ' 7]( '65_V7<75 1 ! 7 ¤ m

m m f &

SZ 7 f j a 1

7.

f (' 'g5 #7 #$ & 1 ! 3 ¢ ' % F G m f £ £ N ¤

m V 7 ( # , 7 5 & f &

SZ 7 0f j a 1

8.

0f ( ' ]'65 & 1 7 m f £ £ N ¤ ! 7 ( # , 7 5

m

& f

Z

, , <

# #

9.

10.

11.

12.

13.

14.

15.

SZ

SZ

SZ

SZ

SZ

SZ

SZ

0f j ¢

f m m

f j F 0f m

7,5

7

¢

m

5) 5) ,

f £ £ f

(' '65_7 1 ( ¤ 7 ( #7,5 ! &N

G

f f

f j a

1 0f m m

F

( ' ]'65 1 1 ( £ ¤ £ &N

7 ( # <7 5 !

¢

f f

£ £

G

(' ]'65 & 1 ( ¤ 7( # 7<5 ! &N

7 f ¢ j ) 7,5 ,

0f ( 1- 1 ( m f £ £ N ! m f & 7( #7<5

f j F G

0f " ( 1;- 1 #7 ! £ m f £ £ J ' 1 % L ' £ O N m & 1 (*' V 7 ( #7,5 f

7 f j F G

0f ( 1;- 1 7 m f £ £ N ! m f & 7 ( #7,5

f j

0f '%& 1 L ' ( ' (' 1 #7 f £ m ' %J m f £ N & ( ' V7 ( #7,5 !

, ,9L

¤

#$

¤

¤

¤

& '

16.

SZ

j

-

3 ¢ J ' 1 % L ' ¢ 3 m

17.

18.

19.

20.

21.

22.

S Z "

j a 1

S Z "

7 j a 1 m ¢ 3

SZ

j ¢

V-

j F

SZ

-

¢

-S

7

m ¢ 3

&N

-S

G

7 ¢ 3 # - S m

SZ

7,5

"

j a 1 F

¢

-

¢

j )

7<5

7

7 ¢ 3 m , ,9P

( ' ] '65 1 1 ( #$ ¤ & N 7 ( # 7,5 !

G

7 ¢ 3 m

SZ

( ' ] '65 & 1 #7 #$ ¤ ! N & & #7( V7< 5

#$ (' '65 & 1 ! 7 ¤

7( #7, 5

( ' ]'65_7 ( #$ 1 ¤ ! 7 ( #7<5

&N

3 ¢ J & ' L% ' ¢ 3 # - S m

( ' ]'65 1 #7 #$ ¤ & N 7( #7, 5 !

-S

-S

( ' ] '65 & 1 ( #$ ¤ & N 7 ( # ,7 5 !

,

-S

( ;1 - 1 ( #$ ¤ & N 7 ( # 7,5 !

# #

23.

SZ

-

j F G

( ;1 - 1 V7 " #$ S N ! ¤ & 1 ( ' V 7 ( # 7<5

( #$ 7 7 ¢ 1;- 1 ! 7 ¤ S G

A j F m 3 - & N 7 ( #7,5

J ' 1 % L ' £ O ¢ 3 m

24.

25.

SZ

SZ

-

V-

j

£ ¢ 3 # - S

26.

Z

27.

Z

28.

Z

29.

Z

30.

Z

31.

Z

-

-

" "

#-

-

5) 5) ,

( ' ( ' 1 J ' % L m ' N ( ' V7 ( &

( ' 'g5 1 V7 #$ 1 ' L j 3 ¢ J ' % m & N 7 ( # 7< 5 !

(' '65 &L ' ¢ j a 1

43 J ' % m 5_7 & N

&-

j a 1 j a 1 j ¢ j F

( ' ]'65 3 ¢ J & ' L % ' N m & & V 7 (

&

#$

#$

¤

@CB

= m

#$ & # 7 !

& a

= m 1 V 7 #$ ! V7< 5

= m

( ' ]'65 & 1 #$ 7 !

m & N 7 ( #7, 5

(' '65_7 ( #$ 7 1 !

m & N 7 ( #7,5

(' ]'65 1 1 ! ( ¢ G 7 N m & 7 ( # 7<5

,93 N

1 # 7 ! #7<5

@CB

@CB @CB

= m

= m

= m

@CB @CB

32.

Z

& '

7<5

Z

33.

Z

34.

35.

Z

36.

Z

"

-

¢

j a 1 F

#-

j F 7

-

j F

-

j

¢

j )

G

( ' ] '65 & 1 ! ( 7 N m & 7 ( # 7,5

7 7<5 ,

#$

= m

( #$ ;1 - 1 ! ( 7 N = m m & 7 ( # 7,5

( 1 #7 " 1 ' L 1 £ J ' % m O N G

' #7 ( #7,! 5 & 1 (*

= m

#$ ( 7 1;- 1 ! 7 G

= m m & N 7 ( #7<5

(' V(' V7 #$ 1L' 1 £ J ' % m N & ( O' #7( #7,5 !

= m

@CB

@CB

#$

@CB

@CB

@CB

4.14.3. Integrals containing Pn (z) and the logarithmic function

; $ # $ f Z

1.

fg3 &-21

-

2.

Z

¡=

m 5 m (

( L 5 m J ' G j F m G

F m T j ¢

7

','g5 7! 3.

Z

m 5 m (

'

' % £ 3 ¤ 1 J L'

'65_7! % . ¢ ¢ j + -21 lC-!&4 f O <3 ¤ 1L'& a 1 J

j / ¢ 3 E

7 '65_7X!AO')(87! ¢ 3 '65_7X! % j + -1 lC- . ¢ 3 £ f O ¤ a 1L'& 1 J

,931

# #

4.

Z

m 5 m (

5) 5) 5

' & ( 7 ( !

7 ( '65_7X!A7 O'65 7!

¢ 3 '65_7X! % j + -1 lC-21 . ¢ 3 £ f O ¤ a 1L'& 1 J

4.14.4. Integrals containing Pn (z), Jν (z), Iν (z) and Kν (z)

1

Z

1.

2.

3.

4.

-21 Z1 f -21 Z1 -

FOf

' ' % '6m _ 5 7X! ¤

¢ 3 T

G j

/ ¢ 3 / ¢

j

"' ' ' ! O'6m 5_7X! ¤ %

T G j F m G &4 - a 1 F

m

1

m ¢ 32

- a &4 F 7 ( GTj a 1

" 3 ¢ ' m % F G fI Z

Z

6.

'65 3 ¢ J ' % & L f 1 a £ / f

&4

0/ E j F m 3 ¢ G / ] f a 1 / £ f

4.14.5. Integrals containing products of Pn (z) Z

7 j j F m G '65_7 0f9 a 1 3hf - ¤

Z

7 j a 1 j a 1 F m G '65

1 1

@CB

= m

=m

£ / E j a 1 F " m 3 ¢ G

2.

Z

5.

1.

@ B

= m

f9 a 3hf - &-21 ¤ ,93,

=m

@CB @ B

& '

3.

7 j ¢ 3 £ j F ¢ 3 m G '65_7 0f a 1 3hf -; ¤

Z

1

4.

1 7

j ¢ 3 £ j F ¢ 3 m G 3

Z

5.

Z

Z

10.

@CB

f KD j F ¢ GIH

(' 'g5 7 a 5_7 #$ " " 5_7X! 1 1 N 7 #7

5 7 ( _ 5 _ 5 7 _ 5 7 a - 7 a _ - L L ! J

J

= ( [email protected] " 5:7! ¢ f

D j F GIH

0f a 5n7 L - 5:7 L J

(' O' 5n7 a J 5n7 #$ £ 1 1 Y 0f £ N 77 = ?( [email protected] B

0f - 5n7 a 5n7 ! (

0f j

f (' 'g5 7 1 1 7 ¤ m f £ £ N ! 7 V 7 ( # , 7 5

m f &

¢ G9H

0f j D F f ( ' ]'65_7 1 1 ( m f £ £ N ¤ ! 7 # 7 ( # < 7 5

m & f

( ' ]'65_7 #7 $ 7 1 1 0 m ¢ 3 - S N & 7#7( V7< 5 ! ¤ - j

9.

7 1 7 j j F m G m 3 ¢!¤

6.

8.

=m f

Z

7.

SZ

SZ

SZ

,93 3

# #

SZ

11.

12.

Z

13.

Z

14.

Z

1

-

-

#-

Dj F

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5) !;

GIH

(' '65_7 1 1 ( #$ ¤ ! N 7 #7( # 7,5 &

( ' ]'65_7 #$ 7 7 1 1 j m N & 7V7 ( #7< 5 ! = m @CB

(' '65_7 1 1 ( #$ 7 ¢ !

D j F 9G H

m N & 7 #7 ( # 7,5

= m @CB 7 ¢ 3 m

j ST j 0

5_7X!A7 O'Q5_7! N

(

1 (

-S

5_7( ' ]'65_7 1 1 1 1 7 #$ ¤ ! & 5 1 (*' & 5n ' 1 # 7#7 #7

4.15. The Chebyshev Polynomials Tn (z) 4.15.1. Integrals containing Tn (z) and algebraic functions 1.

1 C7 ( 5

Z

4 3 ¢ ]J 1 ' L % ' ! f - &-21 f ¢ -;&-21

= m

@CB

1 C7 ( ! " ' & 3 ¢ J'6& 5_L ' 7X! % f - &-21 f ¢ -;&-!&4

a

5 m

1 = m @CB

2.

Z

3.

Z

4.

! " '& m

0f 3 -21 ¢ 3 -;&-21

3 ¢ ¢ h 3 f9

-;&-21 j F Z -21 ¢ 3 -;&-!&4 / ¢

O '6m 5_7 ¢ f O -;&-21 ,9395

¢ 3hf G

¢

/ f

@ B

=m

=m

@ B

) '

5.

6.

7.

8.

Z

-21 0fg3 -21 ¢ -;&-21 / ¢

¢ f O -;&-21 j / ¢ f

Z

-21 0fg3 -21 fg3

3 ¢ j ¢ 3 m j ¢ 3 m = '8>?7 ! m , , ) & -21 ) m ( ! " Z " m ¢ ¢ 0f 3

j ) , =m 75 m ( ! a 1

0fg3 -21 fg3

& 5_7X! 43 ¢ m 5 &L &N

J

Z

@CB

@ B

( ' ]' _ 5 7 W 1 5 & !

=m

( 7 @CB

!

0fg3 -21 a fg3

1 ( ' ]'65_7 5 #$ '65_7! m & 5 & L 43 ¢ W &

& 5nJ ! & N

& 5n !

=m !

Z

@CB

9.

=m

10.

0fg3 -21 " ¢

) & 5_7! m 5 &L &N

J

" m ( ! Z a " ¢ , 7 5 ( !

1 ) m & 5_7X! m 5 &L &N

J

Z

11.

fg3 ,

(' ' ] _ 5 7 1 5 & (

!

fg3 ,

( ' ]'65_7] 5_7 1 5 & ! (

,9 3;

=m

=m

!

!

( @CB

( 7 [email protected] B

( 7 @CB

# #

5) !; ,

4.15.2. Integrals containing Tn (z) and trigonometric functions 1.

Z

0f

f " 43 ¢ - 5_7 L J

Y N

2.

£ g5 7 ! f a 5_7 L 0f £ J ( ' ' a 5_7 #$ 1 ! & 1 - 5 7 a 5_7

=

( [email protected]

0f a

f 1 " ]" '65_7! g5n ! £ 3 ¢ 0f a a & L f £ - a &L J (' J '65_ 7 5_7 a #$ & ! Y N = ( & - a & a a & &

( ' ]' #7 #$ m ! Z & ! ( ¢ f 7( m N &

F G

1 &- & a

Z m ! a F ¢ G

< 7 5

1 ( ' ]'65_7#7 & ! ( 7 ( m ! N m & 1 &- & a

'#" ' )(*'65_7! Z 0f F G a - 5 7L f - - 5_7 L J J

" #$

( £ 1;- - a 1 - 5_7 ! Y 0f £ N = ( 0f & 7 (*' - - 5_7 a - 5_7

SZ 0f

f 3 ¢ m f £ £ N 7( ( ' ]' ##7<7 5 ! m f &

Z

3.

4.

@CB

#$

¤

¤

5.

6.

,93 :

[email protected]

¤

) '

7.

8.

9.

10.

11.

12.

13.

SZ

7

f a

f 1 £ 3 ¢ m f £ £ f

SZ

SZ

0f

f m

7

SZ

SZ

SZ

f ¢ 0f

f F 0f 7

0f a 0f 1 m

m

SZ

¢ m

( ' ]'65_7 #7 #$ O'65_7 1 ! ¤ m N & & # 7 ( # 7,5

m

f £ £ f

m

f f

( ' '65_7 #7 7 £ £ & N 7 ( V7<5 ! ¤

m

f f

( ' ] ' #7 £ £ & N 7 ( V 7
f f

( ' ] ' #7 ( £ £ & N 7 ( # 7,! 5 ¤

G

m

f a F ¢ 7,5

1 0f m f £ £ m f 7 ¢ f ) 7<5 0f m f £ £ m f

(' ' V7 7 N& 7 ( V 7<5 ! ¤

,93<

( ¤

G

( ' ] '65_7#7 ( 7 ( V7<5 ! ¤ &N

,

( ;1 - #7 ( ¤

7 ( #7,5 ! &N

# #

14.

SZ

15.

16.

17.

18.

19.

20.

21.

22.

SZ

SZ

SZ

G

( 1 - 1 #7 " '' '*>[email protected] ! m N & 7 (*' V 7 ( # 7,5 =

G

( 1- #7 7 m f £ f £ & N 7 ( #7,5 ! ¤

3 ¢ m m

f 0f

' m

V-

-

a 1

7 ¢

m

S Z "

SZ

¢

-

( ' 1 (*' #7 7 ¤ ! & N 7( V7<5

7

a m

1

= '*> @CB

¤

(' ] '65_7 1 #7 #$ ¤ ! N & & #7 ( # 7<5

( ' ]' # 7 7

3 - S & N 7 ( #7,5 ! ¤

(' 'g5 7 #7 7 ¢ 3 - S & N 7 ( #7<5 ! ¤

( ' ]' #7 ( #$ 7 ¢ ¤ ! S m 3 - & N 7( V7<5

,93 L

3 ¢ £ ¢ ¢ 3 m

SZ

f £ £ f

( ' 1 (*' 1 #7 f £ m f £ N & 7 * ( ' V7 ( #7,5 ! ' ( ' ' V7

( X 7 ! ¢ S m 3 V- & N 7( #7,5 !

S Z "

f 0f

SZ

F f £ f £ f A F 7 0 f m

f 0f m

5) !; ,

-S

) '

23.

24.

25.

26.

SZ

-

SZ

F "

7,5

SZ

-

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¢

a 1 F

G

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28.

)

SZ

SZ

-

-

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Z

30.

Z

31.

Z

7,5

7

,

-S

( 1 - V7 ( #$ ¤ & N 7 ( V 7<5 !

R ¢

-

(' '65_7 #7 ( & N 7 ( # 7<5 ! ¤

( 1 #7 " #$ ]' " ' 1 m N & 7(* ' # 7 ( # 7,! 5 ¤ l

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A F m 3 - N & 7 ( #7<5

-

F G

#-

( ' ]' #7 ( & N 7( # 7,! 5 ¤

G

7 ¢ 3 m

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-S

7 ¢ 3 V - S m

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27.

7 ¢ 3 m

R ¢

' ( 7 ! m

7 N 7 m & J

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( ' ]' #7 7 ( m #7,5 ! m L

7 N 7(( ' ' VV7<7 5 ! 7 m &

,93 P

( ' 7 (*O'

( ' 1 V7 #$ 1 *

#7 ( # 7,5 ! ¤

= m @ B @ B

= m

= m

@ B

# #

32.

33.

Z

Z

34.

Z

35.

Z

36.

Z

37.

Z

38.

Z

39.

40.

-

"

a 1

a 1

5) !; ,

3 ¢ O'6 5_7X! N ( ' ]'6 5_7#7 ! m 5 7 & & - & a

= m

( ' ]'65_7 1 #7 3 ¢ £ ¢ N ! m & & # 7 ( # 7<5

" ( ' ]'65_7#7 7 7 m & N 7( #7,5 !

a 1

(' ' V7 ( #$ 7 ! ¢ - m & N 7 ( #7<5

7 ¢ -

) 7,5 ,

( 1 - V7 ! ( 7 N m & 7 ( V7<5 (' ' # 7 (

7 F ¢ G - m & N 7 ( V7

" ¢ 7<5 a 1 F G

#$

#$

7 N (7 ' ( '6 5_# 7 7<#5 7 ! ( m &

= m

-

@ B

= m

@ B

= m

@ B

@CB

= m

@ B

= m

= m

F G

( ']" ' 1 #7 " #$ 1 ¢ ! = '*> ! m m N & 7( ' V7 ( #7<5 l

#$ ( V 7 7 Z 1;- 7 ! 7 G

= m - F m N & 7 ( #7,5

Z

@CB

@CB

,!5N

@CB

@ B

+ * '

Z

41.

-

']" m

( ' 1 ( ' 1 #7 N & 7(* ' #7( V7<5 !

= '*>

#$

!

@CB

m

4.15.3. Integrals containing Tn (z) and special functions Z

0 f a 1

1

1.

2.

1

Z

3.

1

Z

7! m m '65_ D V;- &-21 F G 3 # a 1 F IG H m = m

3 fI 1

-21 3hfI -21 j a 1 / E F#" m G £ f ¢ h

@CB [email protected]

m = h

-21 3hfI -21 j a / E a 1 F#" m G £ f a 1 ¢ 3hfI 1

= hm

[email protected]

4.16. The Chebyshev Polynomials Un (z) 4.16.1. Integrals containing Un (z) and algebraic functions

j ¢ 3 £ f9

0f9 32

-21 3 ¢ P

1.

Z

2.

Z

3.

4.

0f 3 -21 3 -;&-21 F G

43 ¢ 3hf9

-;&-21 j a ¢ 3 m ,

1 ) *

=m

=m

!

m

@ B

[email protected]

&L ' 1 C7 ( ! ' & 3 ¢ J6 '

5_7X! % f - &-!& f9 ¢ -;&-21

5 m

= m @CB C7 ( ! Z1 ' 3 ¢ ' % J & '6L 5n ! f - &!- & f9 ¢ -;&-!&4

a

5 m ' &

1 = m @CB Z

,!5)

# #

5.

Z

-21 0fg3 -21 fg3

5) :1 ,

43 ¢ j ¢ 3 m , )

=m

@CB

4.16.2. Integrals containing Un (z) and trigonometric functions 1.

Z

Y

0f f

f £ £ 0f

43 ¢

( ' ]'65_7 a J

1 N & 1 - 5_7 a

" 65_7! 5 7L a 5 7L J 5 7 #$

! = ( [email protected] B _ 5 7

0f a

f 1 " '65_7X! g5:O! m 0f £ f £

' % a a &L - a &L J ( ' '65n 5_7 a #$ J

& ! Y N = ( @CB

a a a & & - & &

Z 3. f F ¢ G

£ ¢ m ! N (' 'g5 7 #7 ! ( ¤ 7 ( m &

&- & a

Z m ! a F ¢ G

4. < 7 5

1 £ ¢ 7 ( m ! N (' 'g5: #a 7 ! ( ¤ m &

&- & '#" (*'65 7!

Z 0f F G

5. a - 5_7 L

f - - 5_7 L ( - a J1 - 5_7 " J#$

£ ; 1 0 f ?')(:[email protected] B ! Y = £ N

( ' _ 5 7 a 5 7 & f - - -

SZ 0f 6.

f 43 ¢ m f £ £ N (' 7 ( '65_#7,7 #5 7 ! ¤ m f &

,!5, 2.

Z

+ * '

7.

8.

SZ

SZ

0f f

£ m f f

7

f f

&43 ¢ 9.

SZ

7

10.

f 0f

11.

£ 12.

SZ

13.

SZ

7,5

7

f f

m

F m

( ' ]'65n #7 #$ £ 6 ' 5_7 1 ! ¤ £ m N & & 7 ( # 7,5 #

a 1

m f f

f 0f £

¢

f m f

f 0f

£

SZ

m

0f 0f

SZ

a 1

( ' ]'65_7 #7 7 #$ £ O '65_7 1 ! ¤ £ m N & & 7( #7< 5 #

m

F G

f f

¢

f 0f

( ' ]'65n #7 7 #$ £ 6 ' 5_7 1 ! ¤ £ m N & & 7( V7< 5 #

( ' ]'65n V7 ( #$ £ 6 ' 5_7 1 ! ¤ £ m N & & #7( V 7<5

" #$

( £ ! ' 1;- 1 V7 ! £ m N & ( ' #7 ( # 7,5

G

( ' ]'65_7 #7 £ '65_7 1 £ m N & & #7( V 7<5

¢

a 1 F G

(' ]'65n #7 1 f £ £ '65_7 N

m # 7 ( V 7<5 & f &

,!53

f f

¤

!

( #$

( #$

¤

!

¤

# #

14.

15.

SZ

f ¢ 0f £ m

SZ

f 0f

16.

17.

SZ

S Z "

a 1

f f

f f

18.

19.

20.

21.

-

S Z "

SZ

SZ

-

F

¢

£ '& £ m

¢ ¢

m 3

-S -S

" £ ¢ 7( m

a 1

¢

-

£ £

O'65_7 ¢ 3 m

,

#$ ( '65_7 ;1 - 1 #7 ! ( ¤

& #7 ( # ,7 5 m N &

7<5

)

£ 43 ¢

SZ

7

( ' V( 1 (*' 1 V7 N & ( O')(:7 V7 ( # 7,5 !

' ( ' ] 6 ' _ 5 7 # 7 ( 7! ¢ S !

m 3 - & N 7 ( #7<5

m

-

5) :1 ,

¤

¤

(' ] '65n 1 #7 #$ ¤ ! N & & #7 ( # 7<5

( ' ]'65_7 #7 7 #$ #7< 1 5 ! ¤ N V 7 ( & &

( ' ' 5* 7 7 #$ ( 1 ! ¤ N 7,( 72 5

&

&

"( ¢ 7 ( m

(' '65n 1 #7 ( #$ ¤ ! N & & #7 ( # 7,5

G

"( £ ¢ 7 ( m ,!5 5

( ' ] '65_7 1 #7 ( #$ ¤ ! N & & #7 ( # 7,5

+ * '

22.

23.

24.

25.

SZ

SZ

SZ

SZ

Z

27.

Z

28.

Z

29.

Z

Z

"

725

26.

30.

-

¢

-

-

-

-

a 1

-

a 1

¢

G

"( £ ¢ 7,( m

( ' ' 58 1 7 ( #$ ¤ ! N & & ]7,( 75

7

) <7 5 ,

( #$ 1;- 1 #7 ! ( ¤ '65_7 ¢ 3 - S N

m & & V7 ( #7<5

G

F " #$ ( ' 1- 1 #7 ! ¤ ! ¢ 3 - S N m & ( ' V7 ( #7<5

' ¢ 3 m

" "

a 1 F

a 1

( ' (')( 1 1 #7 N & ( O')(:7V7 ( V7<5 !

( ' ]'65_7V7 ( 7! ' = m m & N 7 ( #7,5 !

(' 'g5: #7 # 6 ' 5 7 ! ! 3 ¢ 5_7 N m &

&- & a

= m (' '65_7 #7 7 #$ O'65_7 1 ! = m m N & & V7 ( #7< 5

(' 'g5: 1 #7 #$ 6 ' _ 5 7 £ 3 ¢ ! m N & & V7 ( # 7<5

= m ( ' ' 58 7 7 #$ 1 ! ]' 5n7X! N = m m & & 7( 7 5

,!5; -S

#$

¤

@ B

@CB

@ B

@CB

@ B

# #

Z

31.

Z

32.

-

Z

Z

35.

Z

36.

"

7<5

-

-

-

¢ F

34.

Z

33.

¢

a 1 F

5) :1 3

( ' '65n 1 #7 ( '65_7 N ! m & & # 7 ( # 7,5

@CB

= m ( ' ]'65_7 V7 ( #$ '65_7 1 !

G

m N & & #7( V 7<5

= m

¢

@CB

G

( ' '65n 1 #7 ( ]'65_7X! N ! m & & # 7 ( # 7,5

7 ¢ ) 7,5 ,

( 1;- 1 #7 ! ( '65_7 N m & & V7 ( # 7<5

( ' # ! 1;- 1 7 ! F G

m N & ( ' #7 ( #7,5

#$

#$

= m

#$

@ B

= m

@ B

" #$

= m (' V(')( V7 #$ ' 1 1 !

m N & ( ')(:7 #7 ( #7,5

= m

@CB

@CB

4.16.3. Integrals containing Un (z) and Kν (z) 1.

Z

0f

m m D

G9H G 3 a F F ;- &-21

1

4.16.4. Integrals containing products of Un (z) 1.

SZ

0f £ f ( ' ]'65: 1 #7 £ Y N &

& # 7 ( #7,5

,!5:

!

m 7

@ B

f £ £ '65_7! m f (' '65n #7V7 7 3 N & #7 ( #7,5 ! ¤

#$

= m

2.

3.

SZ

Z

-

, '

'65_7! 0

m (' ]'65: #7 7 #$ ( ' '65n #7V7 7 1 ! £ ¤ Y N & & 7( #7, 5 3 N & #7( #7,5 ! #

¢ 3

-S

0

( ' ]'65n V7 7 '65_7X! £ N 1 ! m & & # 7( #7, 5

-

#$

3

(' 'g5: #7 #7 7 N & #7( V7<5 !

m =

@CB

4.17. The Hermite Polynomials Hn (z) 4.17.1. Integrals containing Hn (z) and algebraic functions 1.

-21 0fg3 -21 fg3

43 ¢ O' '% ! % f 4a -21 N & &

Z

W W (' ! 1 9a 9 a a 1

#$

=m

@CB

-21 0fg3 -21 a fg3

1 W W ( ' 5 1 I5 1 #$

6 ' _ 5 X 7 ! 7 7 % £ 43 ¢ ' % f 4a F G N ! & & & 9a a 1 Ia 5 7 = m ! ( 7 @CB Z 3. 3 1 0f63

1 0fg m 3 ¢ O'6')5_(877! ! % % m £ 1 m 3 f9

, ) &-21 ) '8, >?7 [email protected] B m = ! '! % m Z m ¢ 4. 3 -21 fg 3

43 ' % ) 1 0fg , = m [email protected] B 2.

Z

,!5<

# #

5.

6.

7.

8.

Z

-21 0fg3 -21 fg3

43 ¢ ' '% ! % m , )

'65_7! % m ¢ a 0f63

43 '65_7X! % 1 ) 1 '65_7X! % m Z ¢ a fg3

43 '65_7X! % 1 1 Z

5) !< ,

m

0fg3 4a 1

Z

)

,

m ,

@CB

=m

@ B

=m

=m

@ B

0f63

W #$ ( ' ] 5n O ' ! 5 n

!

% 43 ¢ £ - -21 / 9f 4 a 4& ' % ! 5 L N

1 5

J

=m

! Z 4 a

1 0fg3 a 3

9.

1 0fg W #$ ( ' ] 5 5 L

6 ' _ 5 7 ! % 3 ¢ £ - -21 O/ f9 4a ' % ! J 5 ! N & 5

= m ! ( '65_7X! % m Z m ¢ 10. 3

43 '65_7X! % 1 ) , -21 a 1 fg =m Z ! 4 & 11. 3 3

-21 fT

fg m 3 ¢ ' '% ! % / £ m ) , =

( [email protected]

@CB

4.17.2. Integrals containing Hn (z) and the exponential function 1.

Z

2 3hf9 O &-!&4 -

W W

,!5L

@CB

@CB

£ &-21 f9 :F 3 7 G - W W ' >[email protected] B m= ! ! 8

Z

2.

W W

2 3hf9

&-21 -

£ fI 4 F 7 G - W W a

1 = m !

Z

3.

&-21 -

W

, '

3 ¢ f -; F m G

= '*>?7

!

@ B

m

@ B

4.17.3. Integrals containing Hn (z) and trigonometric functions 1.

2.

3.

4.

5.

Z

0f a 1

'%& 3 ¢ 7 ( m ! F G m

Z

SZ

SZ

f

43V m ! F 7 G N 7( m & &

7

0f

f £ m f f

0f f £

a ¤

f a

1 f 3 ¢ £ a 1 m m

SZ

( ' #7 N &- & !

( ' #7 £ ( ! ' 7 £ m F G N 7 ( #7,! 5 ¤

f f

G

F

m f £ f £ ,!5P

(' V7 & #$ ¤ ! 1 &- & a

£ £ F G &N &

( ' 1 #7 #$ ¤ ! & V7 ( #7,5

(' 1 (*' 1 V7 ! ' ¤ m N 7 ( # 7,5 ( "

!

# #

6.

SZ

0f f

£ 7.

8.

SZ

-

S Z "

a 1

a 1 a 1 F G

(' V(')( 1 1 V7 m f £ ! ' & ¤ f £ m N 7 ( #7,5 ( "

! ' (' V7 ( ! 7 ¤ ¢ S

m F G 3 - N 7( #7,! 5

9.

10.

-

SZ

11.

Z

12.

Z

V-

G

F

' ! ¢ 3 m

13.

-

-S

a 1

#$ 1 #7 ! ¤

#7,5

( ' 1 (*' 1 #7 ( " #$ ¤ ! N

7 ( # <7 5

(' V(')( 1 1 V7 ( " #$ ¤ ! N 7 ( #7, 5

' (' #7 ( ! 7 m F G N 7( #7,! 5 = m @CB

( ' 1 #7 #$ ! m & N & & #7 ( #7,5

' (' 1 (*' 1 #7 !

F G m N

7 ( # 7,5

43 ¢ a 1 £ a 1 F G

Z

a 1 a 1 F G

'%& ! ¢ 3 V- S m

"

-

( ' F G m & N & #7 ( &

43 ¢ £ a 1 ¢ 3 # - S

SZ

5) !< 3

, ;9N

= m ( "

!

= m

@CB

#$

@CB

Z

14.

-

, '

a 1 a 1 F G

( ' ( ' ( '& ! N 7 ( # 7<5 m

#7 1 1 " ( !

#$

= m

@CB

4.17.4. Integrals containing Hn (z), erf (z) and erfc (z) 1.

2.

3.

4.

Z

Z

Z

0f

0f

Z

#-

W

( X7 ! ' '! % . ¢ j + 1 lC-;&-21

3 m , m ) &L ' J

= m ( 7X! ' '65_7! % . ¢ '65_7X! % j + 1RlC-;&-21 a 1

3 m , m ) = m 0f a 1

'65_7X! % . ¢ ¢ j + &4 lC-;&-21

3 m , = m 43 m ) L' J

'#" ' % m . m 5n ¢ 43 m 5n j + 1 lC-;&-21 ) m 5n f a 1

@CB

@CB

@CB ¤ ,

4.17.5. Integrals containing Hn (z) and Kν (z) 1.

2.

3.

']" ' m ¢ £ ' & 43 m 0f

; - &-21 ) 3 ,

Z

Z

Z

1 0 f

1 0 f a 1

']" ' 43 ¢ £ ' & -;&-!&4 3 m , m ) ' '& 3 ¢ £ ¢ '%& m , ;

= m

@CB

= m @CB m -;&-21 ) 3 , = m @CB

# #

5) !< :

4.17.6. Integrals containing products of Hn (z) 1.

2.

3.

4.

5.

6.

7.

S Ea 1 / E a 1 / fT3

( & ' 5_7X! % % '65_7X! % % ( ! 5n'6 5_7!A 5n'65:O! f SE a 0f = m Z a a 1 / fT / E 3

1 W #$ ( ' £ R a 1 2 f9 N ! =m & 1

Z 0fg 3 -21 SEa 1 / E / fT 3

43 £ SEa 5_75n! % %'6O5_'T7 (:7X! % % f 1 0f

= m SEa Z 3 -21 S / E / fg 3

-21 0fg 43 £ SEa £ 3 ¢ £ 3 ¢ SEa f = m Z 3 -21 / E / fg 3

-21 0fg W #$ ( ' 7 £ R F G N ! =m 1

1 #7

Z 0fg 3 -21 a 1 / E / fg 3

W #$ (' 7 £ R a 1 F G F G f N ! =m 7 & 1

SZ 0f /

/

f ( ' ]'65 V 7 '& 7 £ m 1 1 ! f £ N

V 7 m F G f 1 1 & ( # 7,5

Z

, ; ,

@ B

@CB

@ B

@CB

@CB

@ B

#$

¤

8.

9.

10.

11.

12.

13.

14.

SZ

, '

f a /

1 f £ R a & F G m m SZ - / / ' ¢ 3 #- S m 7

S Z "

Z

a 1 / ¢ 3 - S

a 1 £ R a

a

1 /

(' '65 & f £ £ N V 7 f & & (

( ' ] '65

F G N V 7 1 1 &

7

#$ 1 1 # 7 ! ¤ ( # 7<5

/

( ' ] '65 V7 #$ & 1 ! ¤

m F G N & & # 7 ( # 7,5

/

( ' ]'65 #7 #$ ' 7 1 1 ! m F G N 1 1 & #7 ( #7<5

Z " / a a

1 /

1 ( ' ]6 ' 5 & 1 V7 #$ £ R a F G N #7,! 5

m # 7 ( & &

Z a 1

£ &-21 43 ¢ 3 ¢ ' % '6 5_7X! % FD -

Z

/

#$ 1 V7 ! ¤

#7,5

@ B

= m

= m

@CB

'65_7 ¤ H G -

/ £ a 1

' 5_7 ¤ £ &- '6J & 5_L ' 7 43 ¢ 3 ¢ ' % '6 5 7 ! % F D 6 H G , ;93

# #

5) L1

'#" ! % ( '! ( 1 L ' J ¤ ST F G

( * ' %

1 ( J L ( V(' W Z 1 £ _ i E S a / &N

#&-21 7 #7 ¤ S

! ( ](' #$ W Z £ ¢ £ ¢ 1!7 ¤ " * 3 3 & N

V-

S

1 ( 1 ( '

W Z 7 1 L "

a 1 £ 4 J $& % ! ¤ -

Z

15.

16.

17.

18.

W

4.18. The Laguerre Polynomials Lλn (z) 4.18.1. Integrals containing Lλ n (z) and algebraic functions 1.

7X! ' 4a -21 0fg3 -21 c fg3 4 ( P5_ ' % f 2 1 W ( ' #$ Y N 5 7 9a ! 9a a = @CB & & 1

Z 3 c c fg 3 4

c 0fg 1 IF m G c a 1 '65 P5_7X! c a 1 m ( [email protected] B , = '65 P5 & L ) J

' Z % m fg 3 4 43 ¢ '65_7X! % a 1 ) , ¤ Z m & (P5_7X! ' 5n ! 4 a 0fg 3 1 c fg 3

& ' % 5 L J

#$ ( ' ] 5n

( [email protected] ! Y N m = !

5_7 5

' Z % m 3 3 ¢ O'65_7X! % * a 1 ) * , -21 0fg = m @CB Z

2.

3.

4.

5.

, ;!5

# . '/

4.18.2. Integrals containing Lλ n (z) and trigonometric functions 1.

SZ

m

m (' 1 #7 £ ( P5_7X! ' ¤ ! Y f £ ' % N _ 5 7 # 7 ( # < 7 5

& & f

" 65_7!A(P5_7X! ' Z 0f c

' % a 5_7

f - 5_7 J ( ' L a J 5_7 L #$ £ 1 ¤ ! Y 0f £ N

_ 5 7 a 5 7 # 5 7 f & & - ' m

SZ # ( ! 0f G ' %m c F

f ( ' ( Q(*' V7 #$ f £ 1 ¤ Y f £ N 7( #7<5 ( ! 1

SZ (' 1 ]7 #$ (5)7X! ' ¢ S 3 - ' % m & N & 5)7 7I( ! 75 ¤ - c

SZ G

- c F (' V( 6( ' #7 #$ ' ( ' % ! ¢ 3 - S N #7,5 1 ( ¤ m 7 (

! 1

#$ ( ' 1 #7 ( P5_7X! ' Z @CB ! #7,5 ' % = m - c

N

m _ 5 7 # 7 ( & &

( ' ' ( 6(*' 1 #7 #$ ( ! ' Z ' % m N 7( V7<5 ( - c F G

1

! = m @CB Z -21 m 5 m ( c

m (P5_7! ' ! N P5_( 7' 5 ! m 5_7 = m ( 7 @CB

1 ! ' % 5 1 L & & J

, ; ;

0f f

c

2.

3.

4.

5.

6.

7.

8.

# #

5) L1 3

4.18.3. Integrals containing Lλ n (z) and erfc (z) Z

1.

7 . 0f / E #'65 7 ! m j + 1RlC-;&-21 F ¢ 3 m G

= m

@CB

4.18.4. Integrals containing products of Lλ n (z) 1.

2.

3.

4.

5.

Z

cI0fg3

Y

Z

Sc ]0fg3 4

d ¢! ¢ f c a a 1 ESc a a a 1 0f O =

c 0fg3

5 '! % n %' %

SZ

Z

0f f

( [email protected]

c 43]0fg3 4

& & m /' ! %

SZ

W W #$ ( ' (P5n'65_7! 65n'65_7! ! 565n ! 5 7 c a a & 1N c a _

=

c 43

c

m

m

( [email protected]

f f W

£ £

#$ (' P5n'65_7 1 V 7 5_7! ¤ ! Y D ' % ' H N c a 1 c 5 7 # 5 7 #7 ( V 7<5

-

c 43

c

5_7! ' ¢ (P ' % ! m 3 -

-S

W #$ (' # 5n'65_7 1 #7 ¤ ! N a 5_7# 5 7 #7 ( # ,7 5 c 1 c

c

W ' ( ' # 5n'65_7 1 #7 ( 5 7 ! ' ! ! % m N c a 1 c _ 5 7# 5 7 #7 ( # ,7 5

c 3

, ;9:

#$

= m

@CB

0 ' /

" /" £ d ¢ F 7 G F 7 G 2 c - Sc c

% ' %! S #7 $ ( V( ' 5 1 1! Y 0 d ¢ 4ST0d ¢ N = ( 7 X @ B

( * ( ' # 5 7 & 1 1

Z

6.

4.19. The Gegenbauer Polynomials Cnλ (z) 4.19.1. Integrals containing Cnλ (z) and algebraic functions 1.

2.

Z

-21 0fg3 -21 c 0fg3

43 ¢ ( &' ! % ' j + l c -21 . ¢ 3 m ¤ , ) Z (9! ' & . ¢ m ¤ ¢ c a 0fg 3

3 '65_7X! % f9 j + 1Rl c -21 ) 3 , 1

4.19.2. Integrals containing Cnλ (z) and trigonometric functions 1.

Z

0f c

f " 65_7X!A(9! ' 43 ¢ ' % a 5_7 L - 5_7 J L J

(' # 5:' a 1 Y N & 1 - 5 7

2.

Z

0f c f a " 3 ¢ ' % J Y N

f £ £ 0f 5_7 #$ ! a 5_7

1

65n !A(9! ' & ££ 0f a a & L f - a &L J ( ' 5:'65_ 7 _ 5 7 a & #$

! & & - a & a a &

=

( [email protected]

=

([email protected] B

, ; <

# #

3.

4.

5.

6.

7.

8.

9.

SZ

SZ

7

0f c

f 3 ¢ # ' (&% ! ' m m

f c f a 1 43 ¢ (&' ! % '%& m m

SZ

7

5) P1 ,

( ' P5n' V7 £ £ & N 7 ( #7,5 ! ¤

f f

f £ f £

(' # 5:'g5 7 # 7 7 ( V7<5 ! ¤ &N

0f c

0f a 1 ( ' P5n'65_7 #7 7 !' & 1 ! £ m f £ £ '6&5_ X 7 ! % N

m P 5 V 7 ( # , 7 5 & f 1

SZ 0f c F ¢ G

f ( ' P5n' V7 ( 1 ! £ m f £ £ '&!! % ' N

m P 5 # 7 ( #7<5 & f 1

SZ 7 f c a F ¢ G

< 7 5

1 f ( ' P5n'65_7 #7 ( 1 ! £ m f £ £ O '695 ! '%7& ! % N

m P 5 # 7 ( # 7<5 & f 1

SZ 7 0f ¢ c ) 7<5 ,

f ( 1;- 1 #7 ! ( £ m f £ £ ' &% ! ' N m & 5 1 # 7 ( #7,5 f

SZ 0f c F G

f ( 1;- 1 #7 " £ m f £ £ ( ' &% ! ' £ O N ( 6( ' V7 ( #! 7,5 & 7 f m

, ;9L

#$

¤

#$

#$

¤

¤

#$

¤

¤

0 ' /

10.

11.

12.

13.

SZ

-

c

S Z "

43 ¢ ( ' 9% ! ' ¢ 3 V - S m

c a 1

£ 43 ¢ ( &' ! % %' & ¢ 3 m

SZ

V-

S Z "

14.

15.

16.

SZ

SZ

-

V-

V-

c

'&! ! % ' ¢ 3 m

c a

1

¢ 3

SZ

(' # n 5 '65_7 1 V 7 #$ ¤ ! N & & #7 ( #7<5

-S

(' .65h' & 7 & N 7K( 7 5 ! ¤

-S

c ¢

¢

( ;1 - N & 7 (6 ( ' V 7

1 # 7 " #$ ¤ ! ( #7<5

' 9% ! ' ¢ 3 m

c F

7 7 #$ 1 ! ¤ 7 5

( ' # n 5 '65_7 1 V7 7 #$ &! ' & ¤ ! '65_7! % m N & P 5 1 # 7 ( V7< 5

c F G

( ' 9% ! ' £ O ¢ 3 # - S m

( ' g5h' N & g5 7E( 1

-S

-S

( #$ ¤

( ' ] 5:' N & 5 #7 ( 1

1 # 7 #7,5 !

( ' P5n' N & P5 V 7 ( 1

1 V 7 ( #7

G

O'9!! % ' ¢ 3 # - S m , ;9P

#$

¤

# #

17.

18.

SZ

SZ

19.

20.

Z

Z

22.

Z

23.

Z

24.

Z

"

7,5

Z

21.

-

c a F ¢ G

1 ( ' # : 5 'g5 7 1 #7 ( $# !' & ¢ '6& 5_ ¤ ! S 7X! % m 3 #- N & P 5 1 # 7( # ,7 5

7

¢

-

c

-

c a

1

"

-

"

-

5) P1 ,

c ) <7 5 ,

( 1;- ' &% ! ' ¢ 3 - S N m & P5 1 V7 (

( ' P5n' #7 ( 9 ! ' 43 ¢ ' % m & N 7 ( #7<5 !

1 V 7 ( #$ V7

= m @CB

43 ¢ ' (9 ! '5 & 7 N ( ' P5n '6 5_a 7 # 7 ! % m &

<& - &

c a

1

@ B

= m

( ' # 5n'65_7 1 V7 #$ £ 43 ¢ ( &' ! '%% & N ! = m @ B m & & V7 ( #7<5

( ' P5n' #7 7 #$ &! ' 1 c O'! % m N & P5 #7( V7<5 ! = m @CB

1

(' # 5n'65 7 #7 7 #$ &! ' & #7, 1 5 ! c a

6 ' _ 5 X 7 ! % N m P 5 # 7 ( &

1 1

= m @CB

c F G

( ;1 - ( ' 9% ! ' £ O N

( ' V 7 m 7 ( 6 & ,9: N

1 # 7 " ! ( #7<5

#$

= m

@ B

25.

Z

Z

26.

Z

27.

0 ' / c ¢

-

c F

-

¢

-

¢

( ' ] 5:' ' 9% ! ' N m & 5 1 #7 (

( ' P5n' '9!! % ' N m & 5 1 #7 (

G

c )

7,5

7

(

1 # 7 #7,5 !

#$

1

= m @CB #7 ( #$ ! #7,5

=

m

@CB

,

( ' &% ! ' N m & P 5

;1 1 V7

1 V 7 ! ( ( V<7 5

#$

= m

@CB

4.19.3. Integrals containing products of Cnλ (z) 1.

Z

f KD c F ¢ 9G H

( ' P5n' " " 5_7X!A&! ' 5 1 #7,5 ' % ! - 5_7 L a 5_7 L N P J

J

2.

Z

3.

SZ

a 1 # 7,5 ( #$

- #7,5 ! a

= ( [email protected]

f D c F ¢ G9H

0f " 5_7X!A&! ' £ f ' % ! - 5_7 L a 5_7 L f £ J J (' # P 5n' a 1 V 7<5 ( #$ ( [email protected] B ! Y N =

5 # < 7 5 V < 7 5 a - 1

0f c m f £ £ m f f ( ' '65n # #7 7 #$ 9! ¤ Y D ' % ' H N # 5 V7 ( 1 #7,5 ! 1

,9:1

# #

4.

5.

6.

SZ

0f D c F

f m m

SZ

SZ

7.

Z

8.

Z

-

#-

-

f f

G9H

( ' ] '65n # #P 5 1 #7(

1 # 7 ( #$ ¤

#7

( ' ]P5n' 5 1 ] #7 (

#7 7 #$ 1 # 7<5 ! ¤

£ 9! ' £ D ' % H N

c

"( 7( m

D c F

-

¢

5) ,9N1

¢

D c F

¢

9! '

D ' % H N

G9H

"( 7( m

c

7 D ' 9% ! ' H N m

( ' ] '65n # 1 V7 ( #$ ! ¤ 5 1 ] #7 ( #7<5

$ # ( ' ]'65n # 1 #7 7 ! 5 1 ] #7 ( #7<5

= m @CB

9! ' D ' % H N

GIH

7 D ' &% ! ' H N m

( ' ] '65n # 5 1 #7 (

1 # 7 ( #7,! 5

#$

@ B

= m

4.20. The Jacobi Polynomials Pn(ρ, σ) (z) 4.20.1. Integrals containing Pn(ρ, σ) (z) and algebraic functions 1.

7X! ' -21 0fg3 -21 j + l . ¢ 0f63 5_ ' % ( ' 5 g5n'65_7

W Y f 4a -21 N & 5_7 9a 9a a 1 ! ( =

Z

,9:,

@CB

2.

&' 3 465 78

0fg3 4a 1

7 Y F !£

Z

5 7! ' . j + l ¢ 0fg3 £ - -21 ' % £ G f9 4 a 4& N ( ' 5 g5n'65_7 5n = &

5_7 5 ! (

( [email protected]

4.20.2. Integrals containing Pn(ρ, σ) (z) and trigonometric functions 1.

2.

3.

SZ

0f j + l .

f £ fI fI fI . 0f j + l

f £ m f £ £ f

SZ

SZ

-

. j + l

Z

5.

Z

-

#-

( ' 5 g _ 5 7! ' 5n'65_7 ' %m N & 5 7V7 ( #7,5 _

1 # 7 #$ ¤

1

!

('

#$ 5 7 ! ' 5 g5:'g5 7 1 #7 ¤ ' % m N & 5_7#7( #7,5

! 1

(' 5 g5:'g5 7 #7 #$ _ 5 7 ! ' 1 . j + l ' % m N & 5 7 #7( #7,5

1

! = m @CB

¢ 3

4.

( ' 5 65n'65_7 #7 _ 5 7X! ' 1 ' %m N & _ 5 7#7( m V7<5 m 7 ¤ !

. j + l ¢

-S

(' 5 g5n'65_7 V 7 1 ' 5_% 7X! ' N m & 5_7V7 ( #7<5 (

!

#$

= m

@ B

4.20.3. Integrals containing Pn(ρ, σ) (z) and Jν (z) 1.

1 ¢ 3 ¢ f / ¢ 3 j -21 £ a a . g5n'65_7X! ' % + & & f - -21 Z

,9: 3

. + l

a

a a 1 / £ f

=

( [email protected]

# #

5) ,9N1 5

4.20.4. Integrals containing products of Pn(ρ, σ) (z) 1.

SZ

0f j + l . 3 j + l .

f 43 ¢ £ m f £ £ 5_7X! ' g5_7X! ' ' % ! m f (' a a 1 a 5_7 5 g5n'65_7 1 V 7 Y N ¤ 5_7 g5_7 5 g5_7#7( #7<5 ! 7

SZ . . V- j + l 43 j + l

43 ¢ ¢ 3 - S 5_7' ! ' ! 65_7! ' % m

2.

Y SZ

5 7 5 g5n'65_7 a _ 5 g 5_7 #7 ( #7<5

1

V7 7

V-

¢

¤

#$ 1 # 7 ¤

(

m

#$

!

. G j + l F ¢ G

g 5 7X! ' 65_7! ' 43 ¢ ¢ 3 - S _ ' %! m (' a a a 5_7 5 g5n'65 7 1 Y N

_ 5 7 g _ 5 7 5

g _ 5 7 # 7 ( V7<5

! Z . . 3 ¢ 5 ' 7! ! 65_7! 4. #- j + l 3 j + l

% m (' a a a 5_7 5 g5n'65_7 #7 #$ 1 1 Y N = 5 g5 7 #7 ( #7,5 7 5_7 g 5_7

! Z . ¢ . ¢ 5. GTj + l F G

#- j + l F3 ( ' a a a 5 7

1 5 g5:'g5 7 43 ¢ 5_' 7X! ! g5_7X! N % m 5 g5_7#7( #7,5 5 7 g 5_7

! = 3.

. j + l ]F 3

(' a a 1 N

5_7 65_7

1 # 7

(

,9:95

m

@CB

#$

@CB

K

4.21. The Complete Elliptic Integral K (z) 4.21.1. Integrals containing K (z) and algebraic functions 1.

C7( ! " C 7 ( m ! & 1

Z

1

2.

Z

3.

Z

4.

Z

5.

6.

Z

7.

Z

9.

K

K

K

m

0fg3

f

@ B

C7 ( m ! B

( m ! 5 1 L & N 5 1 5! 1

J

= ! C7 ( m ! @ B

fg3

m

m

( m! B

( m! B

m

m ¢ £ 2 K 0f 3

f9 3 f * 3 & Z m m 3 -21 0fg3 1 K 0fg , 1) Z m 1) 3 -21 K 0f6 3

-21 0fg ,

W W #$

1 1 & f 4 a -21 N & 7 9 a I ! a a 1

& ( m ! B !

0fg3

¢ K / 3 E

-21 0fg3 -21

K f

1 ¢ -21 f

Z

8.

¢ 32

-21

K

#$ ! 1 ¢ 5 3hfI -21 N 1 5 m 1! 1L J

C 7 ( m ! = !

,9:;

m

( m! B

( m! B ( m! B

# #

10.

Z

4a 1 0fg3

K

12.

Z

Z

1

K

-21

K

#$ ( 7 1 1 5n W N& 7 5 !

m ¢ m fg3

3 * 3 m 0f

¢ F #" G

fg3

!

14.

Z

( m! B

& ( m ! B

fg3

( m m £ m m 3 , ,

) 1) Z m 3

-!&4 fT3 -21 K fg 1 ) , & ( m

1 0fg3 -21

m F" G

13.

n 5 O! 5 L J

( m ! B

fg3 £ - - &4

f9 4a &4

Y

11.

5) ,

K

15.

Z

3 -21 1 £

16.

Z

3 -!&4 -21 £

17.

Z

K

£ 3

K

" 7,5 m ( !

£ 3

K

7

!

!

B

B

7 ¤ ¢ D F G F GIH

- F G ¤

m ( ! 7,5 m ( !

m m ¢ #" * ,9: :

5

m! B

K

4.21.2. Integrals containing K (z), the exponential, hyperbolic and trigonometric functions 1.

2.

1 -21

Z

¢ 3 E

K /

1 -21

Z

0f;/ E 0f;/ E

/ ¢ 3E

5 5 1L m J _

5 7 ! N & &

3.

1 -21

Z

f / E 0f / E

K

/ ¢ 3E

! 5 1 J

F

1

4.

Z

5.

Z

6.

Z

7.

Z

8.

9.

1

-21

1

1

1

Z

-21

Z

K

/ ¢ 3

0f;/ E

0f / E

1

0f;/ E

K

/ ¢ 3

/ ¢ 3

0f;/ E

K

K

=

K

m ! 5 1 L N 5 1 ! 5 1 J

1 5 1! 5 7 5_7

W #$

=

W

N & 5 ! 5

1 1 1 L

m m G 1 F G ¤

#$

K 0f

Y

=

@CB

F m G F m G ¤ 1

/ ¢ 3E

£ 'g5 1 J ' % ! L N &

( 7 [email protected] B

F m G ¤

D F m G 3 F m G F m G 3 m 1 m 0f / E K / ¢ 3 F G ¤

£

@CB

m

1 F G9H ¤

&- 43f9

a 1 a 1 '6 5_7O'6 5_7 ,9:<

!

a 1

m

C7( m ! B

# #

Z

10.

Z

11.

" " 5_7X! 0f K

a 5_7 L - 5_7 L J J

a V 7<5 #$ 1 1 1 ( 7 Y N 7 #7< 5 V7<5 ! a = ! & -

f K

0f £ " " 5_7! 0f £ f - 5 7L a 5_7 L J J 1 1 a 1 V7<5 #$ ( 7 Y N 7 #7< 5 V7<5 ! a = ! & -

f K m m 0f £ Y f £ N 7 1 ( 1 # 1 7,! 5 7 ( f &

1 1 #$ 1 ¢ 3 - S N ! K

m & 7( # 7,5

7(

5) , 3

@ B

SZ

12.

SZ

13.

-

Z

14.

-

K

1 1 1 ! N m & 7 ( # 7<5

1

Z

2.

Z

1

¢ 32

-!&4

(

K

7 (

(

K

m

!

£ 3 ,9: L

B

7 (

3 ¤

B

#$

4.21.3. Integrals containing K (z) and the logarithmic function 1.

@ B

£¤

B

3.

4.

5.

6.

1

Z

¢

f

-!&4

¢ f

K

K

1 -21

Z

1

Z

8.

Z

F f;/

N &

Ff/

¢

7

7<5 m

" 7<5 m 1

Z

C7<5

E 0/

=

m!

f 0 C7( m !

@ B

@CB

/ ¢ 3

m 5_7 , 3 m )+*

K

E

F f;/

Y 1

7<5 m

Z

"

1

/ ¢ 3

m 5_7 , = C7,5 m ! @CB m )+* m 5 1L ¢ ¢ f G K / 3 E

J _ 5 7 ! #$ ( 7 7,5 1 1 5 1 5 1 ! m B

& 5 7 5_

7 ( m

!

m f G K / ¢ 3E £ 3Pf9

<3 & 3f9 O 7<5 m B m 5 1L ¢ ¢ f G K / 3 E

J 5_7 ! 7 #7 5 1 5 1 #$

( 7 7,5 m B & 5 7 5_7 ( m ! !

m 5 7,5 m ¢ ¢ m f G K / 3 E

7<5 m B

£ 0f ¢ 1

7.

10.

m 5_7! K 5 &L 7 #7 5_7 5_7 J

( 7 Y N 5 5 ! & & & ! ( m =

Z1 ¢ 3 f K / ¢ 3E £ ¢ 3hfI K / f< 3 !- &4 h 1 -21

Z

Y

9.

N & F f;/

FOf;/

¢ f G K / ¢ 3 E

F f ¢ f G

,9: P

7<5 m

B

# #

11.

1 -21

Z

,7 5 m 7( m

1 # 7 5 N & _ & 5 7

¢ 3E

K /

1

Z

13.

Z

14.

Z

7<5 m 7 ( m

1 7

1 -21

1

15.

Z

16.

Z

-21

Z

17.

18.

19.

Z

Z

7,5 7 (

K

,7 5 7 ( 7(% 7 (

K 0f

m 5 m (

K

m 5 m (

&

K

m 5 m (

5_7

#$

:3 £ m G

f

( 7

7( m

7 ( m

!

E fI 0

B

B

¤

L 1 1 9J a 1 L N & 7 9 a 1 J 7 (

= !

m FOf

m 5_7 m

#$

!

@ B

¢ 3hf 3 ¢ G

7 ( m B m #$ m J L N 1 1 !

9 a 1 L & 7 9a 1 5_7 J 7( m B !

f-

K

Df

0f O

K

f ] £ f9

m 5 m (

Y f9 ¢

/ ¢ 3E

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m 5 1L J 5_7!

1 5 1

5_7 m

!

Y 12.

¢ K / 3 E

5) , 3

D

¢ h 3 f 3 ¢H 7 ( m B

¢ ¢ 7 ( m f 3 9f ¢ 7 ( m B 0f O 3 m 5 7 K F 5_7 G

f ] £ f9 3 F f f ¢ G 3 ¢ H 7,5 m B

, <9N

Z

_ 5 7 Df FOf

20.

1

Z

21.

m 5 m (

,7 5 m 7( 7( m 7 (

1 -21 F f /

K

Z

1

Z

m D 3

, 2D ¢ 3

1

Z

25.

¢ f G

K

K

/ E

B

= &C7 ( m ! H

E 0fI

7<5 m

m 5_7X! 5 &L J

( 7 7,5 m !

@CB

¢ f GIH

B

7 ( <7 5 m £ 3 ,

) 7<5 m

/ ¢ 3

¢ 3hf f FOf-

32

32

f ¢ ¢H

-!&4 F f /

K

¢ f G K / ¢ 3E

22.

7 #7#7 5 7 5_7 Y N ] 5 5 (

m & & & !

Z1 23. 2- 1 F f;/ ¢ f G K / ¢ 3 £ F ¢ ¢ f G 7,5 7,5 m

24.

5_7 G

KF

f ¢G 3

B

7,5 m

B

" " 5_7! ! 5 J L ( [email protected] =

4.21.4. Integrals containing K (z) and inverse trigonometric functions 1.

1

Z

3 E

f;/ E K / ¢

m £ f <3 f 0

&

2.

1

Z

/ E

K

/ ¢ 3E F G ¤ , <

7 ( m

B

# #

1

3.

Z

4.

Z

"

1

/ ¢ 3E

7 ( m

7 7 7 E F G 3 F G ¤

f;/ E K / ¢ 3 E

m 5 1L J 5 7 !

7#7 5 1 5 1 #$ = N & _

& 5 7 5 7 ! m

" Z1 ¢ 7 ( m f;/ E K / 3E

Y

5.

K 0/

5) , 5

( 7

f

!

7 ( m

7 ( m 7 7,5 7 ( m ¢ 3 m 7 ( m f;/ E K / 3E

7 ( m 1 1 9a 1 m 9a 1 L 2- 1 K f

7 5_! 7 J L N & 7 5_ 7 ( J

m

6.

7.

1

Z

1

Z

8.

9.

Z

@ B

B

B

B

1

Z

K

1

1

10.

Z

11.

Z

12.

Z

1

1

£¤

K 0f K

7 £ f9

<3 f9

0

&

K

D 3

!

K 0f

7 ( m

B

7 & - F G ¤

7 £¤

H ) , < ,

¢ 3hf 3 7,5 7 ( m 3 ¢

, 7 ( m

B

13.

Z

14.

15.

"

1

m

_ 5 7 m

1 -21

K

9 a 1 L J L J

( 7 ,7 5 m !

KF m _ 5 7 G

1 1 Ia 1 ( m ! Y N 7 5_7 5_ 7 &

Z1 m m 5_7 K F m 5 7 G

Z

7 £ 43f9

<3

7<5

3 E

0f / E K / ¢ #7 5 5 5 1 1! m J 1 L N 1 5_7X! &

& 5_ 7 5 7

(

( m

1

0f;/ E

K

/ ¢ 3E m D £

¢ f

7

EF

!

1

Z

18.

Z

19.

20.

1

/ E

0f / E

1

Z

K

B

7,5 m

B

B

m

7<5 m

7 7 / ¢ 3 E D F G - F G 3 £ &4 H ¤

K

/ ¢ 3E m D0 ¢ f9

1

EF

m 5_7! 5 &L 7 #7V7 5 7 5_7 m J

( 7 Y N ! & 5 & 5 &

-21 f / E K / ¢ 3

7 ( 7( m 7,5 7 ( m £ , 3 ) ,

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1 -21

Z

7<5 m G 3 H

17.

m

#$

Z

& 3Pf9

16.

B

f / E K / ¢ 3 E

, <93

m

7<5 m G 3 H

7<5 m

7( m

7 ( m

B

!

B

B

# #

1

Z

21.

-!&4

f;/ E

K

/ ¢ 3 F

5) , ;

¢ 3hf f

1

22.

Z

23.

Z

1

-21

-!&4

/

0/ E

K

K

3 ¢ £ £ ¤

/ ¢ 3

/ ¢ 3

:3 £ ¤

fT3 ¢ G 7 ( m B

4.21.5. Integrals containing K (z) and Li2 (z) 1.

2.

m 5 7! ¢ K / 3 E 5 & L 7#7 #7 5_7 5_7 J m Y N 5 & 5 & ! ( 7 !

Z1 -!&4 0 f K / ¢ 3E £ £ 0fg3 ¢ K 0 / f 1 -21 0 f

Z

1

Z

-!&4 4 3f Y £ E 1

4.

Z

5.

Z

-!&4 1

K

K

-!&4 4 3

m _ 5 7 , m ) *

K

B

f <3 7 ( m B

/ f ¢

/ ¢ 3E

m 5 7 , 3 m ) *

7( m

E /

3.

3 £

7,5 m

B

/ ¢ 3E £ P3 ¤ / ¢ 3E £ / 2 F G 3 £ ¤

K

4.21.6. Integrals containing K (z), shi (z) and Si (z) 1.

1 -21

Z

0f;/ E

K

f;/ E

/ ¢ 3E

5 1L m J _

5 7X ! & N

1 5 1 5 1!

& & 5 7

5 7 ,

W #$

=

( 7 [email protected] B

2.

1

Z

¢ f;/ E K / 3 E

K

D f F m G 3 £ F m G F m G hf F m GIH ¤ 1 1

4.21.7. Integrals containing K (z) and erf (z) 1.

1 -21 0 f;/ E

Z

/ ¢ 3 E

5 5 m 5 1L ( m 1 1 1 ! J 5_7X! N 5_7 5 7 & & &

¢ 3

0f;/ E K / K

#$

=

2.

W 1 -21

Z

m 5 1L

7 5 1 5 1 m ! J 5_7! & N & & 5_7 5 7

#$

=

( 7 [email protected] B

( 7 [email protected] B

4.21.8. Integrals containing K (z), S(z) and C(z) 1.

1 -21 f;/ E

Z

/ ¢ 3

5 & 7 m L J 5 N * & J L K

2.

1 -21 0f / E

Z

/ ¢ 3

5 " m J 1 L N

5 & & J L K

W

& 5 & 5 & ! ( & 5 5

#$

=

( @ B

=

( 7 @ B

1 5 1 5 1! ( 1 5 & 5 &

W #$

4.21.9. Integrals containing K (z) and γ(ν, z) 1.

1 -21 f K / m 5 5 5 J

Z

!

¢ 3

5 5 ( m N & 5_7 5 5 1 ! 5 5 1 & 1L

, < ;

= 5 !

@CB

+

2.

1 -21

Z

f K / m 5 ! 5 5 1L & J

5) , N

¢ 3

7 5 5 m N & 5_7 5 5 1 5 ! 5 1

= 5 !

@CB

4.21.10. Integrals containing K (z), Jν (z) and Iν (z) 1.

1 -21

0 f;/ E f;/ E

Z

K

Y 0 f;/ E f;/ E

K

/

1 f;/ E

K

/

1 0 f / E

K

1

2.

Z

3.

Z

4.

Z

5.

Z

1

1

1 -21

f

1

Z

7.

Z

8.

Z

0 f;/ E

f;/ E

N & 5

5

a

/ ¢ 3 m H fI <3hf / ¢ 3 m f L2- 1 fI 3

K

K

m 5 L !& 5 7!

5 J a 1 L _ J W #$ 5 ! = 5 !

1 5 a 1 5_7

L fI ¤ H 0fI

/ ¢ 3

/ ¢ 3 m

Y 6.

/ ¢ 3

K

7 / ¢ 3 m

7 / ¢ 3E m £

fI 0 ¤

L

5 ! m 5 5 G F _ 5 7 ! 1L 5 J 5 m #$ ! = 5 ! 5 1 5 5 1

£ ¤ fI £ fI

7 ¢ £ If f £ If 3 ¢ ¤ /

f;/ E K 3E

m 1 V1 0f;/ E

¤ -21 fI 0

1 1

5 1 5 7 5 &N & _

H

@ B

, <9:

fT3hf

£ fI 0 ¤

@ B

9.

1

Z

f;/ E 1

Z

10.

V f;/ E

1 7

Z

11.

1 7

Z

12.

/ ¢ 3 E

3 C 7 3 7 3 7 m m V1 0f;/ E

1

1

Z

14.

/ ¢ 3E

C3

£ fI <3 f

7 / ¢ 3E 7 m 3

K

1 0f;/ E V1 f / E

K

K

7

£ fI

m

7 / ¢ 3E 7 m '¢ ¡ f

£ fI £ £ ¢ £ ¤ f9 3 If

£ fI £ £ ¢ f9 f

7 m

K

£ fI

7 7 3 m 7 m

K

#

7 / ¢ 3E 7 m

K

1 0 f / E

Z

13.

K

£ fI

£ fI

£ fI 3

£ fI 3 ¢ ¢

#f9

7 / ¢ 3E 7 m ¢ ¡ f

£ fI 3 ¢ ¢

1.

1 -21

H L

f;/ E 0f;/ E

1 H f / E

K

1 L 0 f / E

K

Z

3.

Z

£ fI ¤

£ fI ¤

£ If ¤

£ fI £

3 #f fI

Y 2.

£ fI

4.21.11. Integrals containing K (z), Hν (z) and Lν (z) Z

£ fI 3

£ fI

£ fI ¤

&N

K

/ ¢ 3

/ ¢ 3E

7 5 & 5

a 1

& 5

¢ m 3

5 a 1L

/ ¢ 3 m 0fI 3 ¢ ¤ , < <

'&

L 5J 5_7 L J 5 & L J J

W 5 a 1 #$ ! = 5 ! 7 5_7 5 5_

0fI ¤

¤

( [email protected] B

+

1

4.

Z

5.

Z

1

H

0f;/ E

K

L

0f;/ E

K

/ ¢ 3 m f 3

5) , !,

¢ ¤ 9 3hf fI ]f 1 0fI

/ ¢ 3 m & ¢ f9 O 0fI 3 #f 1 0fI 3hf9 3 ¤

4.21.12. Integrals containing K (z) and Lλ n (z) 1.

1 -21 c 0f

Z

K

/ ¢ 3

!&( 5 7! ' N P5_(7 ' 5 ! m 5

1 1 ' % 5 1 L & & J

=

@ B

4.21.13. Integrals containing products of K (z) 1.

1

Z

7 (

7

K

" 3 7 F 7 G F 7 G N £

1

2.

Z

3.

Z

7 (

1 -21

K f

K

7 N

/ E

K

/ ¢ 3

#$ 1 1 1 1 V7 & & & & ! 7 V7 #$ ¤ & & & & F G N 7 !

#$ 1 1 1 1 1 1 ¤

1 # 7V 7# 7 #7 V7 7 ! ! 1 1

5 1 L N & 7 5 J

5.

6.

/ ¢ 3 m 0 fI 3 43fI 0 Z1 ¤ ¢ K / E K / 3 E

1

Z

1

Z

K 0f;/

E

K

7 ( ¢ 3hf9

7<5 m m m ¢ f9

fI <3 43fI , <9L

¢ 3E K 0f;/ E K /

4.

!

m #$ ! 1 5 1 7 ( m B

7 ( m

7 ( m

B

B

1

7.

Z

8.

Z

9.

Z

K 0/

1

E

K 0/

K

E

K

"

1

7<5 m

/ ¢ 3

7 ¤

/ ¢ 3

O7 7 ¤

KF

m 7,5 m G

K

#

( #$ ! m 1 1 5 N & 7 5 ! 5 1 1 1 J L

7 Z1 ¢ 3 7<5 K )+* 7,5 , K /

1

11.

12.

Z

7<5

1

K

K

"

)+*

7<5

Z

13.

K

) * K

,

K

7,5 ,

K

1 1 & &

!

7,5 m

B

¤

/ ¢ 3 ¢

¤

G

" 3 7X F 7 G

£ 4

7

F G N

7,5

G

¤

/ ¢ 3

Z

/ ¢ 3

10.

E

1 1 V 7 & & 7

!

#$

3

N

& & # 7 #$ ¤ 7 11 11 !

4.22. The Complete Elliptic Integral E (z) 4.22.1. Integrals containing E (z) and algebraic functions 1.

C7( ! " C 7 ( m ! & E

! ¢ 3hfI -21 N 5 7 ` ! m 5_7X! 5 1 L

1^ J

= ! C7 ( m ! @ B 1

Z

, <9P

# #

2.

3.

1

Z

7 (

Z

E 0f

£ ¢ ¢ m D f2 9f 3hf

-21 0fg3 -21

5) , ,

E

Vf9 3 ¢

W W ( 1 1 & f 4 a 2- 1 N & 7 9a I a ! a 1

m

!

4.

5.

6.

8.

#$

( m! B

m m ¢ E 0fg3

m * 3 & ( ! m m B ! Z m m m ¢ E 0f63

m * 3 & ( ! m m B ! Z

Z

Z

m

2

E

7.

C7 ( m ! B

fH

fg3

0f63

& f 3 ¢

1 0fg3 1

E

m m £ ¢ f ] ]f * 3 & ( ! m m B ! m m 0fg3

3hf9

1 ) , ( ! m m m m B 3 , , !

m £ f9

)

& )

m m 7 P3hf 1 -21 0fg3 1 E 0f63

, ) ( ! m m m m m f

, 3 , ! & ) )

Z

,9L N

B

#

E

m 3 f9

1 ) 9. h -21 0fg3 -21 E fg3

, m m m ( ! m m f

, 3 , ! & ) ) " m ( ! C" Z 10. 3

7 ( m ( ! E 0fg W W #$ 1 & &f 4a -21 N ! Ia a

7 9 a & 1

( m !

m Z

7 7 ( m (

!

m fg3 ] ( m m ! Z m 3 ] ( m 12. 7 ( m ( ! E fg m ! m ( ! Z 13. 7 ( m ( ! E fg 3

( m ! £ f9

m <3hf9 m 3 m

1 m ! m ( ! Z " 14. 3

7 ( m ( ! E fg ( m £ £ 3 f9

m m m , m

) , 1) m ! m ( ! " Z " 3

15. 7 ( m ( ! E fg # ( ! £ £ 3_f m m m , m

) , 1) m ! 11.

Z

!

E

B

B

,9L1

& ( m ! B

& ( m ! B

m

m & & ( m ! B

3

m

3

m

m

,

& ) ( m! B

m

,

& ( m ) ! B

# #

16.

Z

4a 1 0fg3

( 1 & N 7

Y 17.

Z

1

Z

-21

E

E

1

5

7 m £f m *

18.

19.

Z

20.

Z

21.

Z

E

5nO! fg3 £ - - &4

f9 4a &4 5 L J

# $ 5n ( 7 ( !

m m B W ! ! !

¢ F #" m G ] f 3 ¢ ¢ 3 m * & ( ! m m B !

fg3

5) , ,

fg3

+ m 7 * m

m ¢ F # " G * 3 m

( m! B

1 0fg3 -21 E fg3

m ¢ 3 m m 3 ¢ 3 £ f

m 3 m m 7X ) , 1 ) , ,

) & ) , & ( m m ! B ! fg3

m m m m m , ) , 3 & ) , & ( ! m m B ! m & 5n ! 3

! E fg & 5 L J

#$ 1 & 5n W ( 7 ( m! B

m

7 ] 5 ! ! !

m m 5n ! ¢ m -21

3

3 , ! E fg ) m 3 F " G m ! ( m ! B

-21 fT3 -!&4 E ¢ 3 m ) , 1 )

& m (

7 (

Z

7 (

!

m (

Y 22.

m (

&N

,9L,

Z

23.

Z

24.

Z

25.

Z

26.

1

Z

27.

" 7 ( m ( !

#

E

m fg3 * ( m & ( m m ! m ( ! " ¢ m -21 m 3

7 ( m ( ! E fg ) 3 , Y ¢ f

m , 3hf m , 3 m m ,

)

) & ) & ( m m ! " m ( ! " 3

7 ( m ( ! E fg ¢ 3 m -21 9 ¢ 3hf

m ) , , 1) ( m m m m m f , 3 & , ! ) ) " m ( ! 5 7<5 E

m ( ! 7,5 m ( ! * &5 m m ! ! -21 ¢ f E / ¢ 3 5 7! 5 1 L ( 5_7 m J

C7 Y N ! ! & 5 1 5 & =

E

!

B

!

B

!

m m !

B

B

( m ! @CB

4.22.2. Integrals containing E (z), the exponential, hyperbolic and trigonometric functions 1.

2.

1 -21

Z

1 -21

Z

E

/ ¢ 3

0 f / E E / ¢ 0f / E m 5_7! 5 5_7X! J 5

5_7 ! 5_7! 5 1 L 5 & L N & 5 1 5!

J J

m

3

5 1 5 &

7! N &

! & 5_7 5n

1L

,9L 3

W #$

=

&

=

@CB

( 7 [email protected] B

# #

3.

1 -21

Z

f;/ E 0f;/ E

E

/ ¢ 3

! 5_7X! 5 1 L J

¢ m D F 0 f / E E / 3

N

1

Z

5.

Z

6.

Z

7.

Z

8.

Z

9.

Z

10.

1 7

1 7

1

1 7

0f / E

1 7

Z

f;/ E

0f;/ E

E

f;/ E

E

/ ¢ 3

f;/ E

E

G

m D F m G

/ ¢ 3

Z

Y

0f E

( 1 1 N & 7 # 7,5

m

1 F GIH ¤

D F m G 3

m

1 F G9H ¤

3 £ - &-!& 43f9

J

')( 1 L '65 1 L

' % ! J

')( 1 '65 1 '65 1 N& 'g5 7 O'65 7 m = 7 ( m ! " " 5 7!

- 5_7 L a 5 7L J J

a 1 V7<5 #$

( 7 7 ( ! =

# < 7 5 a ! -

Y

11.

@ B

m

1 F G9H ¤

E f

D # F mG # F mGIH ¤ 1

D F m G F m G 3 m 1

/ ¢ 3

m

=

m D # F m G 3 # F m GIH ¤ 1

/ ¢ 3

/ ¢ 3

E

£

E

W 5 7 #$

_ ! & 1 5 1 5 &

m m 1 F G 3 V1 F G9H ¤

4.

5) , , ,

@CB

,9L95

@ B

"

Z

12.

7 (

Z

" " - 5_7 L J

0f E

1 & a 1 5 N & 7 5_7 a -

f E

0f 5 7! 0f a 5_7 L f J

0f 7,(

f " " 5:7! - 5n7 L a 5:7 L J

J

Z

14.

E

15.

Z

E 0f

16.

17.

18.

SZ

SZ

E

" " 5 7! a 5 - 5_7 L J

J

= ( 7 7 (

7 ! 5_7 #$

!

7L

f £ £ 0f

f 0f

E

3 £ -;&- f

E-Q1 & N 5_7

m ( Y N 7 &

,9L;

#$ 1 & a 1 5n7 ! N

& 7 - 5n7 a 5:7

= ( 7 ! 7( @ B ' E-Q1 a 1 J L 5_J 7 L L a 1 J a 1 7 ( m [email protected] B '6 5_7 m = !

f £ £ f 1 1 1 7 ( ( #7, 5 ! =

( 1 1 1 #$ ¢ S m 3 - & N 7( V7< 5 ! - E

7 ( = " 1 & ¢ 1 S m 3 - & N 7( # 7,! 5 7 ( E

7 ( =

@ B

( a 5_7 #$ £ 1 1 1 £ N & 7 5 7 a 5 ! 7 -

= ( 7 ! 7( @ B

Y

SZ

Y 13.

#

m

@CB

@CB

@CB

#$

# #

Z

19.

E

-

( 1 1 1! N m & 7 ( # 7< 5

5) , , 3 #$

m

!

7 (

B

4.22.3. Integrals containing E (z) and the logarithmic function

1

1.

Z

2.

Z

¢ 32

-!&4

1 -21

¢ f

E

3 ¤

/ ¢ 3

m 5_7X! 5_7X! N 7 #7 5_5 7 5n5

& 5 ! 5 & L &

J

=

E

1

3.

Z

4.

Z

5.

Z

6.

7.

1

-!&4

¢ f

E

-!&4

¢ 3hf

E

/ ¢ 3 £ 0f ¢ 1

/ ¢ 3 £

E fI <3

E

!

( m

( 7

!

&C7<5 m !

@ B

m 5 _ 7 , 3 ,

m )* = C7,5 m ! @CB = &C7 ( m !

,

@CB

m 5_7X! 5 1 L ¢ f G E / ¢

F f / 3 E 5 7! J 5 7!

1 5 1 5 & #$ 1

( 7 7,5 m B Y N m

_ 5 7 n 5 ( !

& & !

Z1 ¢ ¢

F f;/ f G E / 3E

9 f9 ¢ 43f9 O 3 9 ¢ 3 £ f9

43f9

<3 ]f9 43f9

0 m

1 & 7<5 m B 1 -21

1

Z

7

7<5 m

F f;/

£ f9 3 £ m )

¢ f G

E

¢ f £ f9

,9L :

/ ¢ 3 E

7,5 7<5 m ,

7<5 m

B

8.

9.

"

1

Z

7<5 m

1 -21

Z

Y 10.

7<5 m 7 ( m

1

11.

Z

12.

Z

1 7

7<5 7 (

7,5 m 7 ( m

7<5 m

7( m

7 ( m

1 # 7 5 N & _ & 5 7

¢ 3E

E /

/ ¢ 3

E

1 5 &

5n m

!

K 0fI

#$

( 7

9 ¢ 3 £ f9 O E fI 0

!

1 7

Z

14.

Z

15.

16.

1 -21

1

Z

1

Z

7

7,5 7 (

B

E

B

B

3 ¤

/ ¢ 3 m 0f9 3 ¢

K fI

E fI 0

13.

E

7<5 m 7 ( m 9f 9 f9 3 ¢ m 1

Z

E

¢ f G E / ¢ 3E

F ¢ f 3 ¢ G

m m 5 7! 5 1 L ¢ 3E

E / # 5_7X! J 5_7X!

F f;/

#

7 ( m

B

/ ¢ 3 £ ¤

,7 5 7 ( 7( 7 (

( L m #$ 1 1 ! E f

IJ a 1 L N & 7 9 a 1 5 7 J 7 ( m @ B = !

,7 5 7(

7( 7 ( E f

D f f- f9 £ ¢ 3hf 3 £ H 7 (

m 7<5 7 ( F ¢ 3 ¢ h 3 f G

7 ( m ( 7 ( E 0f

m 7 ( ,9L<

m m

B

B

# #

7<5 7 ( 7<5 m 7 ( 7 ( 1

17.

Z

18.

Z

-21

m 5 m (

E

m

5) , , 3

F m

m

¢ f 3 ¢G

7<5 m B

7<5 m G

EF

m

! 5_7

( L 1 1

9a J 1 L N & 7 I a 1 J

m

19.

20.

Z

Z

m 5 m (

7X D f

&

E

m 5 m (

7 ( m

7 ( m

B

0f O 0f9 £ ¢ h 3 f 3 £H 7( m m !

B

!

7

0f O Vf 3 ¢ 9f # ¢ h 3 f 3 H 7( m m B !

"

Z

#$

m 5 m (

Y D ¢ ]f ] #f9

21.

E

E

L m #$ 1 & !

Ia J 1 L N & 7 Ia 1 5_7 J

m !

22.

Z

m 5 m ( 7 ( 7

E

m

,9L L

7 ( m

B

7( m

B

1 0f

m

!

23.

Z

1

24.

Z

25.

Z

Z

26.

m 5 m (

7 (

7 (

7

7,5 7(

E

¢ m 5 m (

7 D f FOf ' &

F ¢ 3 m

E

¢ 3hf G

7( m !

7 7 7 F G ¤

E

1 -21 F f;/

Z

27.

¢ f G E /

m 5_7! 5_7! N 5 ! 5 & L J

¢ 3E

7V7#7 & 5

=m !

1

Z

28.

-!&4 F f;/

)

_ 5 7 & 5

(

¢ f G E / ¢ 3

¢ f 3 7,5 ,7 5 m 3 ¢

,

B

¢ D # f ] Vf9 3

B

FOf ' f ¢ G 7<5 ¢ ¢ m m #f f V f 3 H ! 7

5 7 G

_

EF

f ¢ G 0f 3 £ f ¢ £ H , 7 5 m m ! m 5 m ( E F 5_7 G

¢

#

5

n ! ( m 7 ,7 5 m !

B

@CB

=m

!

7,5 m

@CB

4.22.4. Integrals containing E (z) and inverse trigonometric functions 1.

1 -21

Z

Y

f;/ E E / ¢ 3 E

5 1L 5 &L J # _ 5 7X! J 5_7X!

1 1 5 1 5 & N & 5_7 5n m & !

m

,9L P

#$

( 7 !

7( m

B

# #

2.

3.

1

Z

1

Z

f;/ E E / ¢ 3 E

&0f9 3 ¢ f9

9 £ f9 ¢ f9

<3 ] f9 f9

0 m

1 & 7 ( m

4.

5.

Z

6.

Z

1 7

7 7 7 f;/ E E / ¢ 3E 7 F G F G

1 7

Z

7 ( m

8.

1

Z

7

7 ( m

f;/ E E / ¢ 3 E

B

7 # 7 5 N & 5_7 &

1 5 &

5n m

!

f;/ E E / ¢ 3E

5 1L 5 &L J ] _ 5 7! J 5_7!

m

#$

( 7

m ,

F¢ 3

( 1 9 a 1 L 9. E f

J L N & 7 J

Z1 ¢ £ £ ¤ 7 10. E

1 -21

B

¤

f

,7 5 7 ( £ 3hf9 3 £ ¢ h 3 f 3 £ f9 m ) " Z1 ¢ f / E E / 3 E m 7 ( m Z

7 ( m

7 7 7 / E E / ¢ 3E 7 F G 3 F G ¤

"

1

m

0 f;/ E E / ¢ 3 E

Y 7.

5) , , 5

,9P N

7( m

7 ( m

!

¢ 3hf G

7 ( m 1 9a 1 ! m

5_ 7 5_7

7 ( m !

B

B

B

B

11.

Z

12.

Z

13.

Z

1

1

2

E

14.

( 1 N & 7 5

"

1

Y Z

1 ¢ -21 f

£ ¤

m

E F 7<5 m G

1 9a 1 7 5_7 ( m

!

7 ( m E 0f

1 & 9a 1 9 a 1 L J L N & 7 5_ 7 5_ 7 m J

! 7 Z1 7 ( m E 0f m f 1

7 (

1

7 (

1

7

Z

17.

Z

18.

Z

E

E

7<5 m

EF

7 ( m

7 ( m

!

19.

1 -21

F f m

m

7<5 m G

0 f;/ E E / ¢ 3 E

Y

m

B

B

¢ f G

7<5 m

B

5 1L 5 &L # J 5_7X! J 5 7!

1 # 7 5 1 5 N & 5_7 n 5 ( & !

7 3 £ £ ¤

Z

B

7 7 D F G 3 ¢ F G9H ¤

16.

9 a 1 L J L J

7,5 m !

15.

E

7 7 7 F G F G ¤

E

#

,9P1

&

m

#$

!

7,5 m

B

# #

1

Z

0f;/ E E / ¢ 3 E

m m 7 5_ 7X! 9 £ 9f ¢ E F m _ 5 7 G 3 m

20.

1

21.

Z

22.

Z

1 7

/

1 -21

Z

23.

E

/ ¢ 3

f;/ E E / m 5n7X! 5 ! 5 J

1

Z

f;/ E

E

, ¢ 3 )

1

Z

25.

-!&4

-!&4

/

E

¢ 1

m 5 7 GIH

7<5 m

B

¢ 3 E

7 77 5n7 5* 5n7X! N O 5 & 5 m &L & !

?( 7 ,

7 ( m !

B

/ ¢ 3

¢ 3hf 7<5 7( m

,

/ ¢ 3 , ¢ 3

m K F m 5 7 G 3 m 5_7 7<5 m B m

3 7X7 " D F 7 G L¢ £ F GIH ¤

0f;/ E E / ¢ 3E m 0f9 Y D K F m 5_7 G 3 E F m m

24.

5) , , ;

7( m

B

£ ¤

4.22.5. Integrals containing E (z) and Li2 (z) 1.

1 3 E

-21 0 f E / ¢ m 5_7X! 5_7X! N 7 # 7 #7 5 5 7 5

& 5 ! 5 & L &

J

=

Z

,9P,

5

n ! m ( 7 C7 ( m ! !

@ B

#

E

4.22.6. Integrals containing E (z), shi (z) and Si (z) 1.

1 -21

Z

0f;/ E

E

f;/ E

/ ¢ 3E

m 5_7X! 5 1 L 5 7! J 5 7! & N

2.

1

Z

0 f / E E /

W

1 5 1 5 &!

& & 5_7 5n

#$

=

( 7 [email protected] B

¢ 3E

D f9 F m G 3 £ f F m G F m G ¢ f9

F m GIH ¤ m 1 1

4.22.7. Integrals containing E (z) and erf (z) 1.

1 -21 0 f;/ E E / ¢ 3E

m 5_7! 5 1 L 1 ] 5 7! J 5 7! & N & &

W Z1 -21 0f;/ E E / ¢ 3E

m 5_7! 5 1 L ] 5_7! J 5_7! & N & &

Z

2.

5 1 5 & 5_7 n 5 ( m ! #$

7 5 1 5 & 5_7 5n m ! #$

=

( 7 @CB

=

( 7 [email protected] B

4.22.8. Integrals containing E (z), S(z) and C(z) 1.

1 3 E

-21 f;/ E E / ¢ 5 ! 5 & L 7 m J 5 O ! 5

* J L Z1 -21 0f;/ E E / ¢ 3E

5_7X! 5 1 L m " 5 ! J 5 & L J Z

2.

&N

W

& 5 & 5 ! ( & 5 5

=

1 5 1 5 ! ( & N 1 5 & 5 ,9P 3

#$

( @CB

W

#$

=

( 7 @CB

# #

5) , , P

4.22.9. Integrals containing E (z) and γ(ν, z) 1.

1 3 E

-21 f E / ¢ 5 5 5_7 ( m m 5 ! 5 !

5 5 1 L & N & 5_7 5 5 1 5 ! 5 & 5 5 7! J

= 5 !

Z

2.

1 -21

Z

f E / ¢ 3 E

7 5 5 _ 5 7 m m 5 ! 5 !

5 5 1 L & N & 5_7 5 5 1 5 ! 5 & 5 5 7!

J

= 5 !

@CB

@CB

4.22.10. Integrals containing E (z), Jν (z) and Iν (z) 1.

1 -21

0 f;/ f;/ m ! & 5 5 Z

1 0f;/ E

2.

Z

3.

Z

4.

Z

5.

Z

1 7

1 f;/ E

1 f;/ E

1 7

E

1 0 f / E

/ ¢ 3 m f

E

E E / ¢ 3

E W #$ 5

5 ! 5 L 5 5_7 ! 7! 5 J 5_ 7 L 5_7! N & 5 a 1 5 a & 5 7 J

= 5 O! @CB

E

/ ¢ 3 m

H

0fI 3

H

/ ¢ 3 m

L

,9P95

¤ 1 fI 0

¤ 1 0fI

/ ¢ 3 m f L 0fI 3 E

H

¤ 1 0fI

L

¤ 1 0fI

6.

1 -21

Z

/ ¢ 3

5 ! 5n 5 7! 5 J 5 1 5 Y N 5_ 7 5 & &

f

1 -21

Z

f;/ E 0f;/ E E / f;/ E f;/ E 565 ! 5 5 g5 5_7X! J 5 a a a J 5_7 Y N 65_ 7 1 5_7 6 5 5

1

9.

Z

1

E

1 f;/ E

E

1 7

10.

Z

11.

Z

12.

Z

13.

f / E

1

0f;/ E 1 1 7

Z

/ ¢ 3

E

E

V1 0f / E

E

7

m f

&

= 5 !

@ B

#$ m ! a a &

= 565 !

5_7

@CB

£ fI #f9 £ If 0 ¤

£ fI <3

m (& 5_X7 !

£ fI Vf £ If ¤

7 / ¢ 3 m £ If £ f9 3 ¢ ¤

/ ¢ 3

E

#$

m

£ £ m If 3 f

/ ¢ 3

m F G

7

/ ¢ 3

1 0f;/ E

1 V f / E

5 ! 5 1 L 5_7!

5 5 7! 5 1 5 5

a L " ]" " 65_7!

a a 1 L

5 a 5 a 7 5 a a 1

5

Z

E

¢ 3

8.

E

7.

7 £ m f

7

m f

,9P;

£ fI 3

7 / ¢ 3 m

£ fI 3

£ fI #f9

£ fI 3 #f9

£ fI 3 £ f9 3 ¢ ¤

£ fI ¤

£ fI ¤

+

5) , ,

4.22.11. Integrals containing E (z), Hν (z) and Lν (z) 1.

1 -21

Z

H L

f / E 0f / E

Y

1 H f;/ E

E

1 L 0 f;/ E

E

2.

Z

3.

Z

4.

Z

5.

1 7

H

0f;/ E

1 7 L 0 f;/ E

Z

&

/ ¢

/ ¢ 3 E

5 5_7X! 5 5nO! 5 7 5 J a N 5

& &

3 m ¢ E

5 a 1 m a 1 J 7 5L G F _ 5 L J &L

W #$ 1 5 a & ! = 5 ! 5 5_7 5 5n

3 fI 0 ¤

( [email protected] B

/ ¢ 3 m fI <3 ¢ ¤

E

E

/ ¢ 3 f 0fI 3 1 0 fI m 1 0 fI H fI <3

fI H fI 0 ¤ 1

/ ¢ 3 f 0 fI 3 1 fI 0 m 3 1 fI L fI <3 0 fI L1 fI 0 ¤

4.22.12. Integrals containing E (z) and Lλ n (z) 1.

1 -21 c 0f E / ¢ 3E

( ' 5_7 m !9(P 5_7X! ' N ' % 5_7X! 5 1 L & & 5_7 5 1 ! 5 & J

Z

4.22.13. Integrals containing products of E (z) and K (z) 1.

1

Z

7 (

K

E

N

( 1 1 1 1 1 1 #$ ¤

1 # 7V7 #7 #7 7 !

,9P :

=

@CB

2.

1 -21

Z

/ ¢ 3

! 1 1 5_7! 5 1 L N & 7 5 J

K f;/

E

E

1 -21

K

/ ¢ 3

m

_ 5 7 ! 1 5

&

#$

7( m B ! ( #$ ! m 1 1 5 E 0f / E

N & 7 5 5 ! 1 1 1 J L 7 ( B m

Z

E

3.

!

4.

1

Z

1

Z

K

/ ¢ 3 E 0f;/ E

f 7 ( m ,77 5( m 0 fI 3 3PfI m m

/ ¢ 3

7 7 ( m 7<7 5( m 0fI 3 43fI m m m m

Z1 ¤ ¢ 3 7 6. K / E E / Z1 ¢ 7 ¤ 7. K / E E / 3

7 ¤ Z1 ¢ 7 7X

K / E E / 3

8. ! " Z1 ¢ 5 3

9. 7 ( m E f / E K / 1L

#$ J m Y N 7 1 & 5 ! 5 & 1 1

7 Z1 ¢ 3 7,75( m 10. 7 ( m E 0f;/ E K / m m 5.

K 0f;/

E

E

7( m

B

7( m

B

7 ( m

7 ( m

,9P<

!

B

B

+

11.

12.

"

5) , , 3

! ¢ K / 3

5 1L J

( #$ m & 1

7<5 m Y N 7 5 ! 5 ! & 1 1

7 Z1 m ¢ 3 f ¤ / G F E K <7 5 m 7<5 m m " ! 5_7X! Z1 m ¢ G / E 3

7<5 m K F 7,5 m

5 1L 5 &L J J

5_7 ( #$ m 1 1

7,5 m Y N 7 5 5 ! ! & 1 &

7 Z1 m ¢ 7<5 m K F 7,5 m G E / 3

3 0f9 3 ¢ fT3 AfI 3 3AfI m

m m 7<5 m

1

Z

m E F 7<5 m G

<7 5 m

13.

1

15.

Z

16.

Z

17.

Z

18.

Z

19.

Z

K

1 ¢ /

K

1

1

7

7<5

7<5

/ ¢

K

7,5

) *

,

/ ¢ 3

E

)+*

,

E

7,5

)+*

K

/ ¢ 3

1 ¢ -21 f 1

Y

K

E

G3

£ ¤

G

£ ¤

7<5

/ ¢ 3 £ 7,5

) *

( 1 N & 7

,

B

/ ¢ 3

B

] ¤

G

£

,

# ¤

G

! m 5 E F 7<5 m G

1 J L 1 ( m #$

7<5 m

5 1 ! 5 1 !

,9P L

/ ¢ 3

E

B

14.

B

E

4.22.14. Integrals containing products of E (z) 1.

2.

1

Z

E

7 (

1 -21

Z

E f;/

E

1

Z

1

4.

Z

5.

Z

6.

Z

7.

Z

( 1 V ( 1 1 1 1 1 #$ ¤

1 V7 #7 # 7 7 !

1

/ ¢ 3

( _ 5 7 m #$ ! _ 5 7X! 1 1 !

5 1 L 5 & L N & 7 5 1 5 & 7 ( J J

m !

/ ¢ 3

m 5 7 m ( m (:7 m 3 m m

E 0f;/

E

1

E

8.

3.

N

/ E

E 0/

2

E

E /

"

E

E E

E

E

/ ¢ 3

7 ( m 7<5 m 0 fI 3 4 3fI 7 ( m

7 ( m

B

/ ¢ 3

E f;/

E

E

¤

/ ¢ 3

¤ 7 7X ¤

! 5_7X!

5 1L 5 &L J

J

5_7 #$ m! 1 & 7 ( Y N 7 5 5 ! & 1 &

7 7<5 m Z1 7 ( m E f;/ E E / ¢ 3 m D0 f9 ¢ 7 ( m 3 £ f 7 ( 1

B

E

/ ¢ 3

,9P P

m

B

H

m

B

# #

9.

5) ,931

1 ¢ m f 1 E / ¢ 3 E F 7,5 m G

5 m 5_7 7 m 7 3 g f 3 m AfI 3 4 3 AfI m m 7 ( m B

Z

7 7<5 m 1

10.

Z

11.

Z

12.

Z

1 ¢ 1

1

m 7,5 m G

EF

E

¢ 1

7,5

) *

E

E

,

E

7<5

) *

/ ¢ 3 m f 0f9 3 ¢ f 7 ( m B

,

/ ¢ 3 £

E

¢ ¤

G

/ ¢ 3 £ £

G

]

¤

4.23. The Complete Elliptic Integral D (z) 4.23.1. Integrals containing D (z) and elementary functions 1.

Z

-21 0fg3 -21

D

W W & 1 & f 4 a 2- 1 N & 9 a 9 ! a a 1

2.

Z

3.

Z

4.

Z

D

D

fg3

0fg3

£ m F 3

!

fg3

F £ 3

3 f G h

( m

( m

( m

3hf G

1 0fg3 1 D fg3

m m m , 3 ) , 1)

#$

B

B

B

3 N N

m

, & ) ( m

B

5.

Z

-21 fQ3 1

D

0f 3

m £

1)

D

m

, 3

) ( m

6.

Z

-21 fQ3 -21

D

fQ3

£

1)

m

, 3

) ( m

7.

8.

9.

4a 1 0fg3

Z

Z

1

Z

D

-21

D

£ - !- & &4

f9 4a &4

fg3 W #$ ] 5n ( 7 Y N 1 & 5 !

! &

m m fg3 F #" G 3 *

D

fg3

Z

Y 11.

12.

£ -

f 0fg3 m m m 1) , 3 ) , & ) ,

-21 fT3 1

Z

D

-!&4 fT3 -21

SZ

V-

B

m

,

B

n 5 ! 5 L J

( m B ¢ 3 m ( m

B

( m

B

0fg3

£ -!&4 £ m 3 m , ,

) 1) m ¢ 1 S D

3 #- & N 7 (

D

,

-21

10.

f 1 ¢ ¢ 3 m *

m

( m

( m

B

3 N1

B

1 & #$ # 7,! 5 3 ¢

7 ( B

# #

Z

13.

D

-

( ( 1 1 m N & (

5) ,931 , 1 !

m

#$

3 ¢

!

7 (

m L -21 D

9 a J 1 L J m! #$ 7( 1 & Y N Ia 5_7 m m ! & 1

5m m ( Z m

D

0f 7( m m ! Z m 5 m (

D

¢ 3 ¢ 3hf 7<5 7 ( m )

, 7( m m ! 9a 1 m 9a Z1 -21 D f J 1 L N 1 & 5_ 7 5_! 7

& 7 ( J L m m 5 m (

Z

14.

15.

16.

B

B

B

B

B

B

B

B

17.

!

1

Z

18.

m F f

D f

f

¢ 3hf 3 ¢ G

7 ( m

4.23.2. Integrals containing products of D (z), K (z) and E (z) 1.

1 -21

Z

K

/ ¢ 3

D f

/ E

#$ ! 1 & ! m 5 N

1L & 5 1 5 1 J 7 ( m

!

2.

1

Z

K

/ ¢ 3

D f;/

E m D ¢ h 3 f f 3 N,

7<5 m 7 ( m H

7 ( m

1

3.

Z

4.

Z

K

/ ¢ 3

1 -21

E

E

D 0/

/ ¢ 3

D f

£¤

! 5 7 ! 1 &

5 1 L 5 & L N & 5 J J

_ 5 7 ! 1 5

5.

1

Z

/ ¢ 3

E

E

D f;/

/ E

f m (:7 ,77 5( m m , m )

m & !

#$

7( m

7 ( m

1

6.

Z

7.

Z

1

E

/ ¢ 3

E

/ ¢ 3

D 0/

B

B

E ¤

D 0/

E

¤

4.24. The Generalized Hypergeometric Function p Fq ((ap ); (bq ); z) 4.24.1. Integrals containing p Fq ((ap ); (bq ); z) and algebraic functions 1.

2.

-21 0fg3 -21 N a m ! ! ! m ( ! ` 1^ W m ! #$ Y a N a ! 9a I! a a m

1

Z -21 0fg3 -21 N a m ! ! ! m ( 1^ #$ ! m ! Y a N a ! 5

1 1 1 3 N 3

Z

!

f 4 a 2- 1

`

=m

!

( m

" " ! m 5 1L J

( m !

B

!

@ B

# #

5) ,!5) ,

4.24.2. Integrals containing p Fq ((ap ); (bq ); z) and trigonometric functions 1.

2.

3.

4.

m ! ! ! N ^ `

m ! 1 V7 m f £ £ a N a ! ! # 7 ( # < 7 5

m

f

! Z

m ! m £ N ! `

^ ' !' ' ( & m ! m a N a m !95:' ]'65 1 ! m ' % ! ' 1 1 !95n' O'65_

7 m ! Z

£ ¢ ! ! N `

^ m !' " ']" m !95n' ]'65 & ' % 3 O ! ' a 1 N a 1 !95n' O'65n ! m ! Z f ! ! N `

0f ^ " 5_7! ££ f 0f - 5_7 L a 5_7 L J ! a 5_7 #$ J

Y a N a ! m 5_1 7 a ! 5_7 = ( 1 -

! Z m ! m ! N ^ `

! " ' ' m !95 a 1 m ( m ! ' m ' N !95 '6 5_! 7 !' 5_7 L J

m ! Z

! ! f N `

^ m ! a 1 5_7 #$ " " 5_7X! ! a 1 N a ! - 5_7 a 5 7 a 5_7 L - 5_7 L J

J

= ( SZ

0f f

¤

¤

¤

5.

6.

[email protected] B

¤

3 N95

[email protected]

7.

SZ

-

m ! N ! ! `

^ "( 7( a N a m

1

#

.

. W + + a 1 #7 ! \ U $ . .

+ + a 1 #7 ( #7<5

W m R

.

R. a ( " + 5_7#7 \ U #$ 7( + 1 ! m 5_7 a 1 N a + . a 1 + . 5_7 & ( & 5

= m @CB SZ m ! ! ! 8. V- N `

^ "( m ! 1 #7 #$ 7( a N a ! ¤ m

! #7( V7<5

SZ m ! ! ! N 9. - `

^ ( "( m ! V7 & #$ 7 ( ( 7X ! 5_ 7 a N a ! ¤ m

! &<- & a

m ! Z ! ! `

10. - N ^ W

R.

R. a #7 \ U #$ 7 + + 1 ! m a 1 N a + . + . a 1 #7( V7<5

m #$

. a

. 5_7 #7 W \

U + + 1 ! = m @CB a N .

.

m 5_7 a 1 + a 1 + 5_7 &<- & a

11.

Z

-

m ! m ! 7 ! ! N ^ `

m a N a ! 7( ! 3 N;

#$ 1 V7 !

V7<5

m =

@CB

# #

Z

12.

-

5) ,!5) 3

m ! ! ! N ^ `

m ! #7 & 7 a N a ! m 5_7 ! & - & a

#$

@CB

= m

4.24.3. Integrals containing p Fq ((ap ); (bq ); z) and the logarithmic function 1.

Z

-21

m 5 m (

m ! N ^ ! ! `

m ! ! f

5 1 L a N a ! 5 J

m ! 1 5 7

=m

@CB

4.24.4. Integrals containing p Fq ((ap ); (bq ); z), K (z) and E (z) 1.

m ! m ! ! `

m ! m ! a N a

5 1 L ! 5 1 ! 5 J

! Z1 m m 2- 1 E / ¢ 3 N ^ ! ! `

m ! 5_7 ! 5_7X! ! a N a

5 1 L 5 & L ! 5 1 5 J J

1 -21

Z

K

/ ¢ 3

N ^

1

=

&

=

2.

m

@ B

@CB

4.24.5. Integrals containing products of p Fq ((ap ); (bq ); z) 1.

2.

-21 0fg3 -21 N ! L N ! m ( ! `

1 1J 1 1^ f 4a -21 N 5 !5 m L 1 1J

Z

-21 0fg3 -21 N ! L N ! ( m ( ! `

1 1J 1 1^ W W 4f 4a -21 N ! 1 Ia 9a a 1

=m

@CB

Z

3 N :

#$

=m

@ B

3.

-21 0fg3 -21 N ! L N ! ( m ( ! `

1 J 1 ^ W W #$ ( f 4a -21 N ! =m

a 1 9a 1 9a a 1

Z a a2S -21 0fg 3 &-21 N 1 K3 N 1 3 fg3 4

. 3 ¢ SEa ; F 5 m G + a a2SEa &-21

S . ' Y L $L ¢ 383 S ¢ 3 )3 F 5 m G + a

& J J £ f2 = m Y a a a a2SEa &-21 #$ ( Z . 1 E S a ! a 1 0fg 3 + -21 N 1 3 ] fg 3 4 1 N '65 (*' & 1

7 ' 7 3 ¢ F G R ¢ 3 S $ L F G J &Y S? a a 1 + 1A- .

. / f / / f + -21

Y m . + -21 / f / 3 / f = #$ ( V( ( #$ (' V(' V(')( ( 1 Z 1 ! !

- S -;&- N

N

1 ( V( (: 7 1 (*' ( O')(: 7 1 43 ¢ SE a # % ! ! % ' % '! ! % D C £ £ F 7 GIH = m ! ( Z 2 1 fg 3 3 N a 1 ^ !&! ! ` ¡ N 1 f 5 m ( !& L ( m ! ! J Z

4.

5.

@CB

@CB

6.

7.

3 N<

@ B

'[email protected] B

7 B

Chapter 5

Finite Sums 5.1. The Psi Function ψ(z) 5.1.1. Sums containing ψ(k + a) 1.

2.

3.

1 fI fI

¢ <3 <¤ fT3 ¢ ¢ 3 0fg3 ¢ 0&3 0f 2 ¢ 3 0f 0 ¤ 7 £ m (8 3

f £ f fI m 5n ' fg3 3 £ ¢ 3hf f- ¢ 0 ¤

' fI ¢ ¢L£ f ¢ m (:7! 7 3 fI 0fg3 ¢ '6 5_ 7 ¢L£ f & 3 5. & fI £ £ £ R 4fg3 4.

6.

7.

m (87! m

$ " ¢ £ ¢ 3 1

3hfI <3 f2 £ fg3

3

f- ¢ ¢ X £ ¢ f- ¤

# f9

7 £ ¢ R ] 0 fT3 4

f9 fI 7 ¢ 4 3 0fg3 ¢

f9 R f- ¢ ¤

C

£ 3 £ -; £ "' 6 ' 3 5n

7 #7 ( 7 ' 5 ! L = '8>[email protected] B N 1 J 6 £ f 3 £ If £ f 0fI 3 £ ;- £ 0f 0f$ ¤ - T 1 3 N P

+

¢ '$ L 3 J 1 m !" ¢ 9. $&% !" 10. $&% O m !" ! 11. " fI ( 7 5_7X! K3 ¢ m

F £ 3 '7 G

8.

7

m

C

;

7 F¢ 3 ' G

= '8>[email protected]

¢

m 5 7 ! ' ' % m f ¢ <3 ¢ ¤

5_7X! ' ' % O 3 ¢

¢ ¤

m m 5 7! ' m m 5_7! ' ! ' 0 fg3 ¢ D ¢ ! ' H fI 3 ( m (:7X!A m 5 7! ' !' f f ¢ h 3 f 0f ¢ < 3 ¢ ¤

12.

13.

14.

15.

16.

3

m (:7 ¢ !" m ! " fI ( m _ 5 7! fg3K3 ! '%& 3 ( m 5_7! m ! ' m ')( 9 $ %

¢ fg3 ¢ 0

¢ 0fg3 3 ¢ 0f$ ¤

!" 5 7X! fI $ _ m !' & '65_7X! % 0 f ¢ 2 3Pfg3 <3 ¢ h 3 fI ¢ ¤ 3hfI 3 4 3fg3

£ ' £ If $ L 3 I f J 7 £ &-21 F m 5_ G ( 7! % O')( $ : ')( $ ! % 1 ']" 1 L ' ' J D£

£ 3

C3

D£

7 '

£

m F G

m 5_7 F GIH ¤

7 F G 3 £ H

£ ')( $ ! % ¢ £ ')( $ ! % &-21 ¢ 3 31N

C

¤

= '*>[email protected]

17.

' O ' -21 m (:7! " $9% fI J $ L J $ L " ' 5 7 m ! ' D £ If <3 f 3 F m _ G 1 L ' m !' J

F

m 5:'g5 7 ¤ GIH

5.1.2. Sums containing products of ψ(k + a) 1.

2.

3.

£ £ g ¢ fI ¢ h fI E f 3 3 fI E fI ¢ £ 3 £ If fI fI E fI ¤ 3

!" m ! " fI O m (:7 m ( (87! m ( (:7 O fg3 ! '%& 3 m ! ' m ( (87! D m ( (:7 ¢ f- ;

f 3 & I ¢ £ 3hfI & 0 fI

¢ ¢ 0 fg3 3 ¢ O ¢

fg3K3 ¢ ' ¢ H ¤

¢ 3 £ fI 0fI £ g f 3 ¢ E 0fI £ fg3 ¢ fI ¢ 3 £ 3 £ fI E If 7 7 ¤ fI & If fI <3 fI

5.1.3. Sums containing ψ (k + a, z) 1.

¢ ' 4 3 J $L 1

2.

( 7 ! " ' _ 5 7 $L $ J 1

F 7 G 3 ')(:7! % D C £ £ 1 J L'

F 7 IG H = '*>[email protected]

F 7 G ' % 3 &L' J

' 7 ( ' V7#7 #7 N & ] ! 7 &L ' &

31 J

= '*>[email protected]

+

3.

0 fI 3 7 E 0fI 3 If fI I f 7 7 £ fI T f 3 ¢ £ ¢ 3hfI If 0 0fI 7 ¢ £ £ 3 f £ fI fI 0 3

; ,

fI ¤

5.2. The Incomplete Gamma Functions γ(ν, z) and Γ(ν, z) 5.2.1. Sums containing γ(nk + ν, z) 1.

2.

3.

4.

5.

6.

7.

8.

")" " 7 #7 1 5 $ (:7 & (:7 N ]! L J

7 #7 " ! 7 L ¤ 3 5n' N J 5n'65_ ( 1;- #7 ( 7 ! " !¢ 7<5h ( X7 ! ' ( ! ' ¤ 3 ' % -&N

" &$ % & ' % " '65_7 ¤ ' 1 ! ' & 1 N 1 J 5:'g!5 7 L J $ L43& - & 7 ¢ ' ! " $ 5 7! $ 4 3 L & J 'g5 7 7 D T3 ¢ )3 ¢! & <3 ' ! % -21 - -21 0& H ¤ ' " m (:7X! " 7 (*' m ¤ ¢ ' & ! ' L m ( '! " ! " $ 4 3 L N J 5_7 m (* J ( 7X! " ' ¢) ¢ S ¤ S 5 ! $ % J $ L & 9l 3 3 - SE- a -21 0& 7 ' ¢! ¢) ¢) 7IGIH ¤ $ L & ;G 3 F 1 D F J ' % ' $ 5n'65_7! % $ 5_7! % 43& - ¢) & '65_7 $ L J ' % ! 'g5: ¤ a

6 ' n 5

! % - 1 N 1 J O'65 ! L 3 31!,

γ (ν, z )

Γ(ν, z )

( ' m #7 ' m !" ! " 3P& - !¢ & & N 1 " ` $ L J ^ ! ( ' m V7 ( " ¤ ( m !' 3 ! ' - & N 1 ^ m ( (*! '65_7 ` m 5 ! ( ' m 5 !" ¢! 2 a S 5 ! 5 5n'65_7 L 10. J $ L $ % 3P& - & 3P& Jm ( (*' m #7 C7( m ! ' 3 5:'! % S V- & N 1 ^ m ( ' ( " ` ¤ ! ' n 5 ' ! " ¢ £) & 5_7! " ! " $ 4 3 L 11. J " '8>[email protected] 5_7')! (: 7X! 5_% ' 7% ! ' ']" 3P& & -21 0& = 9.

5.2.2. Sums containing products of γ(ν

k, z)

' $ L 1 & - £ 3 1 3 & J * W #$ '& ' '65 1 '65_7 5n' £ ! ¤ ! ' & 5n'! & N a 1 5n' 5:'g5 7 5n'6 5 7 1

a 1 O'65_7 £ ¢ 2. & 3 3& nJ $ L W #$ '& 8 '& '%& '65 & '65n 5n'65_7 3 ¤ £ £ ! 5n'65_7! N ! '& & a & 5n' 5n'65n 5n'65n &

5.2.3. Sums containing Γ(ν k, z) ' 1 & - 77( ( 5n ' L ¤ $ L 4 3 & T3 1. ! J J ' ¢ 1 ¢ ¢ '8>[email protected] 2. J $ L 3 ; T3 & 3 &-21 3 -; V- & -;-2 1 & = C7 ( ! " ' % a '65_7 ¤ ' C 7 ( * ( ' ! ! 3. & " 3& T3 ' V- J 5:'g!5 7 L J $ L ')(:7X! % ¢ ' 7 ! " 1 & 3 ! ' - & -21 & $ 4 3 L = '8>[email protected] 4. J 1.

313

+

; 31

5.3. The Bessel Function Jν (z) 5.3.1. Sums containing Jν

nk (z)

1.

( &$'% ! " £ & - a 0& * -; 1

2.

( &$'% ! " 43 £ & - 3 ¢ & * &- -21

3.

!" ' £) n 5 ' 6 _ 5 X 7 ! $ L a a " 0& 1 F G J ;

4.

!" ¢ ' £) ' 5_7! " $ 4 3 L ; 5n6 J

FO$

' ¤ G

' ¤ G

F

a 0 & ¤

a

& W " " 5 '65 1 ( #$ n ¤ ! ! 1 N 5 ] 5n

7

6 ' _ 5 1

5.

"

( L ( L J J (*'65_7X! ( ! ' J $ L " a 0 & '

'

¤ -; &

5.3.2. Sums containing products of Jν 1.

2.

3.

¢ 4 3 ; a & 1 ( 7! ' a

nk (z)

7

£ ¢

& <3 - a 1 0& ¢ a a 0 & 3h a a 0& ¤ 1

5 ( ! " ' L 1 ¤ $ L 5n'65_7X! " a 0 & J ! 1 N 5 7 ! 5n'65_7 J

a 1 0 & a a 1 0& £ -

a 1 £ )£ ¢ '65_7 3 J $ L ;- &-21 Q ;- &-21 0 & ' & 5 a n 3 ¤

(')( 1 L F G 1 -;&-21 & J 315

5.4. The Modified Bessel Function Iν (z) 5.4.1. Sums containing Iν

"

1.

2.

3.

4.

nk (z)

'

( L L ' J J (*'6 5_7X! " a 0& ( ! ' -; 0& ¤ $ L

J 7 ( $&'% ! " £ & - ¢ ¤ 0 & 43 - <1

£) ¤ ; a 0& a 1 & <3 a a 1 0& 1 !" ¢ ' £) ¤ $ 4 3 L 5:'g5 7! " a 0& ; a 1 F G a 0& J

5.4.2. Sums containing products of Jν 1.

' J $ L

-; a 0& (

nk (z)

and Iν

nk (z)

a &

#$ ( 7X! ' ( ! ' ¤ 5_X7 ! F G -; N & 5_7 - a 1 - 5_7

' 2. J $L -; a 0& - & #$ ' 5 1 ( ¤ ( 7 ! ( 5 7 ! ! ' F ,G -; N ! 5 7 1 a 1 5_7 ( a 1 ( _

' G 3. -; a 1 Q -21 & " J ']$ " L F ¤ A& a 1 1 -; A& 3 A& a 1 1 -; 3 A& ¢ ' G &- -21 Q -21 0& 4. 4" 3 ']" J $ L F ¤ A& a 1 &-21 A& 3 A& a 1 &-21 3 A& 3 ' & 8 ¢ ' ¢ 5. 43 J $L -21 & &- a 1 0& 3 '65 ! O'65 ! £ £ ¤ Y D a / a / H 1 1

31!;

+

6.

7.

8.

9.

10.

11.

12.

; 5) ,

3 '%& 8 ¢ ' 43 J $ L 2- 1 & -; a 1 0& '65 ! £ ' O'g5 ! Y D /

1 -;

1

£ ¤

-; / H

3 '%& 8 ' ¢ J $ L - -21 0& -; a 1 & 43 '65 ! O'65 ! £ Y D a / ' 1

a 1 / £ H ¤

3 '%& ¢ ' 4 3 J $L a 1 & -; a 1 0& O'65 ! £ Y D

-;&-21 /

8 '65 !

3 '%& ¢ ' 4 3 J $ L 2- 1 & ;- &-21 0& '65 ! £ Y D

-;& -21 / E3

8 '65 !

-;&-21

£ ¤

/ H

-;&-21

£ ¤

/ H

-;&-21

£ ¤

/ H

Y

3 '%& 8 J $L 1 - 0& &- a 1 & O'65 ! £ '65 ! /

D -;&-21

Y

3 '%& 8 J $ L - -21 0& & - -21 0& '65 ! ' 5 ! £ E3 6 £ ¤ / D

-;&-21

;- &-21 / H

'

'

3 ' & ¢ ' ¢ 43 J $ L a 1 & &- -21 0& 43 '65 ! O'65 £ Y D a / 3 1 31:

8 !

a 1 / £ H ¤

13.

14.

' J $ L 1 Y D ' J $L 1 Y D

3 '%& 8 £ K3 O'65 !

0 & &

&- -21 '65 !

1 -; / 0& -;&-21 & '65 ! a / 1

! " ¢ ' ' 5_7! " V a & $ L 4 3 - 5:6 J

4.

5.

6.

£ ¤ 1 -; / H

5 J L ! N 5_7 1 !5n'65_7 ¤ 1

65 (*'65_7X! ' a ' ¢ 6 _ 5 7 ! n 5 6 ' _ 5 X 7 A ! ( ) ( ! G $ L 0 & a & 3 F ' J a a a a a 5 7 1 ¤ Y N 6 _ 5 7 n 5 6 ' _ 5 7

g5 5_! 7

& ' J $ L - 0& - 0& a - a 1 a - 5_7 ( ! ' ( ! ' a 6 ¤

! 5_7! 5_7X! F G

N & )(*'65_ 7 (*'65_7 65 ( 'g5 7

a 1 ¢ £ £! ¢ '65_7 3 3 J $ L -;&-21 Q -;&-21 0& ' & 5 a 3 ¤

( ' ( 1 L F G 1 -;&-21 & J

" ' ( 7 ! G $ L V

3 V

D F F SEa 1 $ GIH $ - S -21 $ J 1 3 ¢ £ 1A- - &-21 = O' '% ! % @ S 3 3 ¢ S = '[email protected] B l " ( 7! M! '%& M! " M! "'& M! 43 ¢ '%& M]! M]! ¤

3.

nk (z)

2.

3 '%& 8 ¢ 3 £ K3 O'65 ! a / £ H ¤ 1

5.4.3. Sums containing products of Iν 1.

31!<

+

; ;

5.5. The Macdonald Function Kν (z) 5.5.1. Sums containing Kν 1.

2.

3.

'65 ')( $&% 7

nk (z)

$ !% $ ! % a 1 & 7 a 1 ) 3 * 3 a , 1 ) ")" 7 $&% ( ' (! $ ! % ' % " V- ¤ &- -21 & 1 ( m (*' L)" J ')( $ ! % C7 ( m (* '! " F G a 1 & " '! F 7 G 5n Om '5n ! - - m m 5nO' 3 '65_7X! % m 5:'! £ &-21 O 1 - N

O

7

* 3

,

= @CB

2- 1 0& m 5n 'g5 7 #7 O m 5nO'65_7O'6! 5n L ¤ J " "']" ' L ¢ E S a - S -;&-21 0& ¤

( ' )S - a 1 0& 3 5 1L' L" J

F G a & a & ¤

4.

5.

( ' J J $ L 1 ( J

' J $ L

5.5.2. Sums containing Kν 1.

¢ ' 43 J $L

nk (z)

and special functions

a -; & a 0 & ( L J

F3 G N & E - 5_7 a 5 a 5 7L J

( a E Q 1 L 3 J a a F3 G N & E- a & a a & L J

) ' ( ! ¢ 3 5_7X! F ,G -; N & - a 1

#$

7 1

#$

(

31L

& &

- 5_7 5_7

#$

¤

2.

3.

2.

Lν (z )

¢ ' G <1 Q 2- 1 0& 43 J $ L F " ']" 3 ¢ a Q3 Q 1 & -21 Q3 Q Q a 1 &-21 Q 0 ¤ ¢ 0& 3 0& a 4 3 & -21 0& 3 a 1 & & 0 - ¤ ¢ 3 a & &- a 1 & <3 a a 1 0& &- 0&

5.5.3. Sums containing products of Kν 1.

Hν (z )

nk (z)

)£ ¢ a 1 Q a 1 0& 5 a $ 5:'g5 7! % ¢ / $ 5_ 7X! % ')( $ ! % F G 1 a 1 & = @CB $ 5_7 ¢ ')( $ ! % $ 5n'65_7X! % a 1 Q a 1 & 4 3 43 ¢ ' % F 5 G a 1 a & ¤ 1

5.6. The Struve Functions Hν (z) and Lν (z) 5.6.1. Sums containing Hk+ν (z) and Lk+ν (z) 1.

2.

3.

( ' ! " '! " &$ % F G H -21 & 3 ¢ " a F ,G D £ £ 1

( ' ! ' ! " " $&% F G L -21 0& " a F G D £ £ 1

' F]3 G H a & $ L 0 I f J " '& ( m 5 &L ' J 5:'g5 & L J

31P

¢ a F , G 3h a F , GIH ¤ 1

&4

¢ a F G a F GIH ¤ 1

&4

( m 5:'g5

N & 5:'g5 &

W & V7 ( #$ ¤ ! ( m 5 & &

+

4.

' G L a & J $ L 0fI F]3 " '&

( m 5 &L' 5 '65 & #7 ( m n ! J :

N 5 'g5 & L & 5:'g5 & ( m 5 J

; <

W

& &

#$

¤

5.7. The Legendre Polynomials Pn (z) 5.7.1. Sums containing Pm 1.

2.

3.

nk (z)

( ' ( ' (')( #$ 1! ¤ O'! % E (:7 G &N

j

1;- & ' % ! F 1 (*' V( O')(:7

£) ¢ '65_7 (:7 j a 1 & <3 j & ¤ j & 7,5 4 ' ¢ £ j & $ j F 7,5: 4 5 G ¤ $ L J $ ')( $ ! £ 7 '8>[email protected] ')( $ ! % & j & F G £ & = 1 " ( 7 ! ')( $ ! % m ! " FO63 3 ¢ G j & 7 ! j + !- &4 lM1A- -; . F 3 # V 3 ¢ G ¤

m ' & -21 ¢ ' ¢ G j 0& F $ L 3 3 3

J ' 1L' ('! " J ' % F ¢ 3 (87 G -; £ - C7 (*'! " F (:7 3 ¢ G ¤ ' m !" 7( G j & J $ L $&% F ( ' #( ' #$ 19- <- ]7,( a ¤ m ' ! % ' F 727,5: ( G N & 7,( m ( ' 7( m ( ' 7

!
4.

5.

6.

7.

8.

3,9N

& '

)" " ' 9. J $ L ')( $ ! % 1 (*' L " ) 7 J

" ' F ¢ 3h 1 J L' m !" ')( $ ! % $ 5n'65_7! % ( 10. m !' ' ! O'Q 5_ 7!A % 11.

12.

13.

14.

15.

16.

8 ( 7 8 ( (8 5 # (:7 , j & #$ ( 1;- 1 ¢ ¤

3 G ;- & N 7 #7 7 (8 5 (:7 !

m ! " j & #$ ( ' ( ' ( m 5_7 & ! a

( m !' F G & N

1 ¤ 1 (*' V7 ( m ( '

m !" !" £! ¢ ')( $ ! % $ 5n'65_7X! % ( m ! " ( ! " j & ( ' ( ' ] ( m (

' ! m ( ! ' ! ! ' ( ! F 7<5 G N ¤ 7( m (*' V7 ( ! (*1 ' a % ' m ' &

¢ ' ¢ 3h G j 0& $ L 3 4 3 F

J ¢ 3h 3 ¢ j + 9lC-;&-21 . F £ 3 £ 3 ¢ 3 ¢ G ¤

¢ ' m !" j 0& $ L

3 J 1L" J

( ' ( ' &- - #$ ! '

; 1 m W ¤ - ¢ 3h

N & ( ( ' V 7 ( ( ' m m W 1 1 1 J L '

! -Q1 ' m !" ¢ - j & $ L h 3

J 1 J L" (' 1 (*' 1- - #7( a " ¤ m ! ' ¢ 3h

-; N 7 ( * ( ' # 7 ( * ( ' V 7 (

m m & 1L' ! J

$ 5_7 O'6' 5 % 7 ¤ j 0 & ')( $ ! % '65 & )L "

J

1L'& ¢ ' ¢ J 1 L)" J ' % ¢ 3h

¤ j 0& $ L 4 3 '65

J &L " J

3,

+

17.

18.

$ 5_7!&'! " 7 ( ' ' % 0& ¤ $ ! % & (*' L " & 5n' L " j & J

J

m !" - j & 1 J L " ( ' ( ' &<- - #$ ! '

; 1 m ' % - ¢ 3h

N W ¤ & 1 ( m (*' V7 ( m ( ' 1 ! W

-Q1 ' m !" ¢ - j J $ L 1 L " 3h O & J

(' 1 (*' 1- - #7( a " ¤ m ! ' ¢ 3h -; N 7 ( * ( ' # 7 ( * ( ' V 7 (

m m & 1L' ! J $ 5_ 7 ')( $ ! % $ (:7X!A $ 5 ! 'g5 & L " j & J

5_7 (' (*' V7 " #$ O 6 ' 1 ' % ')(87!A '65 ! N ¤ !

* ( &

1 ' ( ' $ 58'! % 1 L " ' 43 ¢ J $L ¢ $9% ( ' J 58' j & 1 & J L"J L" £ ¢ I j 0& ¤ ' _ 5 7 $ 1 " L j 0& 43 ¢ J $L $ (:7X!A $ 5 ! '6J 5

& )L "

J

(' V(' V7 #$ & 1 W ¤ L ' ! ' % ')(:J 7X!A '65 ! ¢ 3h O N 1- &

1 (*' (*' m !" !" 1 L " ' J

j 0& 43 ¢ J $ L ¢ '65 & L)" & ( m L" & ( L"

J ] m ! ' ! ' 1 L J ' & J

( ' ( ' & ( m ( 1 W ¢ ¤ J 7( m (* ' V7 ( ! (*1;' - ' % & ( m L ' & ( L ' 3h & N

J

J

$ 5 O'6' 5 % a 1 ¤ j a 0 & ')( $ ! % '65 L " 1 J

¢ 43 ')( ¢ ' 43 J $ L

19.

20.

21.

22.

23.

24.

; <

3, ,

25.

26.

27.

28.

29.

& '

' m !" ¢ - j a & $ L h 3

1 J & J L" (' V( 1 (*' 1- - #7 ( a " ¤ m ! ' a 1 ¢ 3h

-; N 7 ( * ( ' V 7 ( ( ' V 7 (

m m & &L ' ! J

¢ ' m !" j a 1 0& $ L 3

J & L " J

( ' ( ' - - #$ ! '

& m ' % 1A- ¢ 3h N W ¤ & 7 ( m (*' & ( m ( ' & ! W

-Q1 ' ! " m ¢ $ L 3h - j a 1 & J &L " J

(' V(')( 1 1- - #7 ( a " ¤ m ! ' a 1 ¢ 3h -; N 7 ( * ( ' V 7 ( ( ' V 7 (

m m & &L ' ! J

& L '%& &L " ¢ ' £) j a 1 & J ' % ; ¢ 3h ¤ $ L 4 3 F G '6J 5

J L" J

7 $ 5 ')( $ ! % $ 5_7X!A $ 5 ! '65 j a & L" 1 J

( ' ( ( ' V7 " #$

6 ' 5 1 ' % '65_7X!A '6

¤ ! a 5 O! 1 & N

( 1 ( ' & (*'

30.

31.

&L " ¢ ' $ 5 $ 5_7X!A $ 5 ! '6J 5

j a & $ L 4 3 J L" 1 & L ' &J

( ' V(' V7 1 W #$ ! ' % '6J 5_ 7X!A '65 O! ¢ 3h N ¤ ( 1 (*' & 1(*- ' &

! ! " " m & ¢ ' J L " ( j a 0& $ L 4 3 '65 ( J L" mL" L" 1 J J

] m ! ' ! ' & L ' & J

( ' ( ' ( m ( 1 W ¢ ¤ J

7( m (* ' V7 ( ! (*1;' - ' % ( m L ' ( L ' ; 3h & N

J

J

3,93

+

32.

33.

34.

35.

; <

( ')( 1 ¢ $ &L £ 3 ¢ J ( '65

* J J £ SEa 1 ' ( ' % ! % F 7 G ]F 3 3 7 S

L" j & 1 L " & -

S G SEa ¢ h 3 & SE- a 1 S & 1 = ( ' ( ' 43 ¢ £ 3 ¢ J $&% 1 L" j & F G ' %! ¤ &-

* ( ' ! ( " £ 3 ¢ m J 1 L" j 0& J & L ' a 1 0& ¤ $&% 1 ( m (*' L " &-

m 5 1 J L' J

U0\ U0\ & O')( $9 % $ 5_7 F3 O'65_7 G j 0& ] j l F E(:7 G ¤ &!- & ('! " ( R a 1 L " & 3 ¢ £! ¢ $9% ( J 'g5 1 L " j &- 0& J

( ' ( ' ( a 1 R & W ! £ J O& 'L ! '% 3 ¢ N ( ' ( 1 V( 'g5 <1 - &

( ' ! ( a " R &L " & 3 ¢ £! a 0& J' ( j ( 9 $ % 1 L " &- 1 J

( ' ( ' ( a & & L R & W ! £ a 1 J O '65_' 7X& ! % ; 3 ¢ N , (')( & V(')( 1 &

¢ ' ( ! " ¢ ! 3 G j S - & 43 J $L m " F m 5 ! ! ' F 3 ¢ G S

m ' ( #7( m ( 7 (8 5n (:7 1 Y N 7( m ( ! (*' V7 = > &

5_7X! " ¢ ' ¢ G j a2S 0& 3 43 J $ L (*'65 & L " FO$ J

1L' S 3 ¢ G - - ( J ( 1 L ' F J

Y j S + lC- S -;&-21 . V V 3 ¢ 3 43 3

'[email protected]

36.

#$

¤

37.

#$

¤

38.

39.

3,!5

'[email protected] B

¤

& '

5.7.2. Sums containing Pn (z) and special functions 1.

7 ' ' '#" " (8 $L O]! " & - j 0& O]! '#" j &-21 & J

= '8>[email protected]

2.

O '65 7 '65_7 £ $ 5_7 L& & - &- j a 1 & ' '#" 1A- j & ¤ J

3.

F Q3

¢ G 3

¢ j & £ j + 1RlC-;&!- &4 . F V 3 3 # 3 ¢ G

. ¢ ¢ ¢ F 3h 3 G -21 D 3 j + a lC-;&- F # 3 3 # 1

5.7.3. Sums containing products of Pm 1.

2.

3.

4.

3 ¢ G9H ¤

nk (z)

)£ ¢

j 2 & 0 'g5 7 ! j 0& 3 £ j & j a 0& j a & 0 ¤ 7 (* 1 1 $ 5_7 1L' ¢ ¤ ')( $ ! % $ 5n'65_7X! % j 0& J ' % ! ¢ h

3 4 3 $ 5_7! '65 1 L " 7 (* ' ¢ J 5_7X! % * j 0& ' % ! j & ¤ 4 3 ) ' ( ! n 5 6 ' ( ' $ % $ & J L" 7(8 7,5 ¢ ¢ J 1 L" 4 3 $9% j * , j * , ) ) 7 (8 7,5 £ 43 ¢ a 1 ¢ J & ' L % ' j a j a ,

) *

) * , ( ' ]'65 #$ L ' ¢ ¤ J ' % ! 3h

N & & 7 (8

! 3, ;

+

5.

¢ 43

; < 5

7 (8 7<5 a j ,

1) * ) * 7 (8 j a & * , j a & ) ) L' ¢ 7VJ ' % ! 3h &4 N &

&L " J $9% j a 1 £ 43 ¢ a 1 ¢ J ' L % '

,

7,5 * , ( ' ]'65 #$ ¤ 7* (

!

5.7.4. Sums containing Pm (ϕ(k, z)) 1.

2.

3.

¢ ' j 43 J $ L S ¢ '$ L j S ¢ 4 3 J 1 ¢ ' $ L j SEa 3 J

= '[email protected]

& ¡ & 3 £ & S F 7 G S

S l 3 ¢

= '8>

@CB

&

' 5 6 £ F 3 G S J $ 5n' 1 L ! % " £ & a a 1 Q ¤ S a

4.

5.

6.

7.

8.

1 L ( ¢ ' j S / 3 ¢ S S J

$ S l 4 3 L J 1L( J

& L ( ¢ ' j SEa 1 / 3 ¢ S SEa 1 J

$ S l 4 3 L

2 1 J &L ( 1 J

&L ( 3 43 ¢ S J % ¢ ' $ L j S / ¢ 3P& S J 1 L ( 4 3

J 1 ( J L ( 7! " ' j SEa 1 / ¢ 3V& S $ L < 7 5 $ J 1 ¢ ' S j S:F ¢ $ G 43 ¢ S S l $ 4 3 L J 1 3,9:

= '8>

@CB

@ B

= '8>

@CB

% & L ( J 5_7X! % S l 3 ¢ = '8>

@CB

S l

= '8>

1L( 3 J % £ & S

= '8>

@CB

& '

¢ 4 3 1 10. 3 1 9.

11.

12.

13.

14.

15.

16.

17.

7 ' J $ L & S j S +) * $ 5 , F G S S l 3 S

= '8>

¢ ' $ L & SEa 1 j SEa 5 , / F G S S l $

1 )+* J ' > 3h SEa 1 = * '8> ¢ ' $ L 3h& S j S F $$ 5(8 G 3 ¢ S ? S 3h S 3 = l J 1 7 1 ( ¢ ' S J L ! % £ & S j G G $ L S S 4 3 F 3 = '8> F $ S l

J 1 ¢ ' SEa j $ L

E S a 4 3 F 1 1 $ G F G S S l J 1 & L ( 3 J 5_7! % £ S SEa 1 = '8> ¢ S ? 1 ( ¢ ' S ¢ J L ! % #& S G j $ $ 3 S 3 4 3 L S : # F "

l J 1 = '*> ( ¢ ' $ & j SEa 1 F " ¢ $ G 43 ¢ S _ S l $ L 4 3 n 5 $ J 1 & L ( 3 J 5_7X! % V& S = '8> $ ¢ ' 43 ¢ S _ $ 4 3 L & S j S * $ 5 , ) J 1 1 3 J

@CB

@CB

@B

@CB

@ B

@CB

@ B

= '8>

@CB

= '*>

@CB

S l L( % 3& S

$ ¢ ' & SEa 1 j SEa 5 , $ $ L

4 3 2 1

1) * J 1 3 ¢ S _ S 3 J & L % ( 43& S l 3, <

+

18.

19.

; < ;

¢ ' $ L S j S F $ 5 G 3 ¢ S F 7 G - S S 3 $ S l J 1 1L( 3 J % S = '8> ¢ ' S S j S ) 4 3 J $ L & 1 ¢ _i S ;- J 1 L(*( '"! ' 43 %

@CB

$ 5 3h S J 1 L % ( $ $ 5 ] ! , S -; ')( ! " (]! ")" ¤ 5n'! % ')( 5 a $ 1L" J

5.7.5. Sums containing Pk (ϕ(k, z)) 1.

2.

¢ ' $ L f f O &- -21 & j F $ 5 4 3 $ 5 J 1 ' 3 & -21 ' ( m 5 m

$ G ( m j F ( m G ¤

¢ ' ¢ f ¢

$ L 4 3 f &- 2- 1 J ( ]! ' $ m 5n 5n 1 L ' Y j $ 5_7!A $ 5 5_7! , J ' % ' m _ 5 7! ¤ m m )

5.7.6. Sums containing products of Pm (ϕ(k, z)) 1.

2.

3.

¢ ' j Sg $ 4 3 & L J

&

=

¡

¢ ' S j S:F#" ¢ $ " $ 4 3 L J 1 _ 3

G j S:F#" ¢ $ 3 " $ G $ T ¢ S S 3 J 1 L ! ( 3#& S = '*> % l ¢ ' S j ES a 1 F#" ¢ $ " $ GTj SEa 1 #F " ¢ $ 3 " $ G $ L 4 3 J 1 _ 3 ¢ S S 3 J & L ! ( 3#& S = '*> % l 3,9L

'[email protected]

@CB

@CB

) ' * '

5.8. The Chebyshev Polynomials Tn (z) and Un (z) 5.8.1. Sums containing Tm+nk (z) 1.

2.

3.

4.

5.

6.

'

7,5 4 ¢ £ 0& F 7<5n 4 5 G ¤ m !" F 7(8G & 1L" J

(' ( ' 1;- - 1 m ! ' F 7<7 5(8 G N & 1 ( m ( ' V7 ( m ( ' 1L' J

7 7 0& 0& ¤ 7 3 O & M (:7! £ 0 3 ¢ £ 0& 3h

$J L ' J $L

O' 7 O' ')( $ L & £ &-21 ' L ¤ J J ' ! 7 7 ')( $ ! % '65 $ ! % 0& ]O'! % ]' % ! ¤

' m !" 43 ¢ J $L - 0& 1 J L" ( ' ( ' ! ' 1 1;- - m ¢

m (*' #7( m (*'

- 3h N & ( 1 1 J L'

' m !" ¢ - & $ L h 3

8. J 1 J L " ( ' ( ' 1 1;- - m ! ' ¢ 3h -; N

( ( * ' # 7 ( m (*' m & 1 1L' J

7 &-21 0& 9. a 1 0& O'65_7 ')( $ L a 1 & £ a 1 ¤ 10. J 7.

3,9P

1

&<- - !

, a - 1

#$

¤

&-21 0 & = '*>[email protected]

& - - #$ W ¤ 1 W

! -Q1 & - - #$ " ¤ 1! (

= '8>[email protected]

+

11.

12.

13.

14.

15.

16.

17.

18.

19.

7 O' 5n'65_7 ) ' ( $ $ J 3 R

; L1

' '& L a & O '65 7 ¤

1

¢ a 0&

1 7 M : ( 7X! £ ¢ a 1 0& 3h & 0 ¤ 7 ' ' & ¤ ')( $ ! % $ 5n'65_7! % a 1 & O'65_7X! % $ 5_7 ' ¢ ¢ ')( $ ! % $ 5n'65_7X! % a 1 & '! % 3h ¤ 43 ' m !" ¢ - a 0& $ h 3 O

L 1 J & J L" #$ ( ' (')( 1 1;- - & - - " ¤ m ! ' a 1 ¢ 3h -; N ' & 1 ( m (*' #7( m (* 1! ( &L ' J

m !" ¢ ')( $ ! % $ 5_7X! % F G a 1 0& 3 ( ' (*' - - #$ ! 7 * ( '

1 & m O'! % 1A- W ¤ , N & 7 ( m ( ' & ( m (*' & W )

! -Q1 C7,5 m 5n'! " ¢ ')( $ ! % $ 5n'65_7X! % C7 ( m ( '! " a 1 & 4 3 ] 5n7 '!A ! 0& ¤ m m ' a 1 3 ¢ ' $L C7 ( m ! ( " '! & ] ' %! ' & m m " &-

J ' m ! ' ( 7! ' ¤ C7 ( * ( ' ! m

'

l

( ' 8 ( 7 & J $& & % L)" & j + -I & lC- & . F G &-!& ( L ' 7 ¤ & & l J & % 3 3 N

&

20.

21.

22.

23.

) ' * '

(*' " ( O! ' . 43 ¢ $&% OJ ') 1 (* $ L 5_ X7 ! &- a 1 & O '65_7Qj + 1 lC-;&-21 ¢ 3 # O ¤

¤ £ S J $ L SEa &- & & & O'65_7 £ $ 5_7 L & - &- a 1 0& J O' ']" " (:7 $ L O]! " &- 0& J

O'65_7 ' '#" 0& ¤ ' ! ']" 1A- &-21 &

= '8>[email protected]

5.8.2. Sums containing products of Tm+nk (z) 1.

2.

')( 1 L " ¢ J

43 ')( $ ! % $ 5:'! % J ]' 7 ! % ¢ 43 a 43 ¢

( ' L " 0 &

& * 7

( 1 L ' 1 L ' j 0& 3h j &-21 & 0

J J

7(8 7<5 a , 1 ) * , 1) * '65_7X! 7(8 a 1 '65 a & ) * , a & ) * '65_7X!A'65n ! ¢ ¢ 3h

3 ]O'Q5 ! a & F 7 ¢ 3 N 7(8

= '8>[email protected] B

7,5 , G

(')(* '65_7 ¤ 1 1 ! 7(8

5.8.3. Sums containing Tn (ϕ(k, z)) 1.

2.

3.

¢ ' ¡ $ 4 3 L T S & J ' ¢ 3 J $L SEa & S £ &-21 ;3P& a ¢ ' $ L S S:F ¢ $ G 3 ¢ S _ S 3 £ S -21 4 3 l J 1 3 31

= '[email protected]

S

! " a ' S Q ¤ $ 5n = '8>

>[email protected]

+

4.

5.

6.

7.

8.

9.

¢ ' / 43 J $ L S ¢ ' 43 J $ L -21

1

1

1

10.

11.

12.

13.

¢ S £ S SEa _

SEa 1 / 4 3 1 S l 3 43 ¢

5.8.4. Sums containing Um+nk (z) 1.

= '8>

>[email protected]

¢ S £ = '*> @CB

' S $ L S F $ G ? S l 3 £ S 2- 1 S = '8> >[email protected]B ' SEa _ $ L

E S a F 1 1 $ G £ ¢ S l 3 £ S SEa 1 = '*> @CB ' 43 ¢ J $ L S S:F" ¢ $ G 43 ¢ S _ S l 3 £ S -21 S = '*> >[email protected] ( & ' $ ¢ 43 J $L $ 5n SEa 1 F#" ¢ $ G 3 ¢ S ? S l 3 #& S = '*> @CB ' = '8> @CB 43 ¢ J $L & S S ) * $ 5n , _ S l 3h S 1 ' ¢ E S a

5 1 43 J $ L &

SEa 1 )+* $ , 1 43 ¢ SEa 1 _ £ ¢ A/ S 3h SEa 1 = '8> @CB l ' $ 43 ¢ J $ L & S S ) * $ 5n , 3 ¢ S ? S l 3 3P& S 1 = '*> @CB $ ' 43 ¢ J $ L -21 & SEa 1 SEa 1 ) * $ 5 , 43 ¢ S ? S l 1 3 £ ¢ X3P& S = '8> @ B

¢ 4 3 J 1 ¢ 4 3 J 1

3 ¢ S ? £ S 2- 1 S S l

; L1 5

" ' 5 7 $ L & J $

7<5n 4 5 ' 7<5 ¤ F 7<5n 5 G '65 7 3 3,

2.

3.

4.

5.

6.

7.

8.

) ' * '

¢ ¢ O '65: £ ¢ ¢ ' ( $ L & R 3h& ¤ 4 3 ) J ']" $ 5_7 ¢ ')( $ ! % $ 5n'65n ! % & '65_7X! % ¢ 3h& ¤ 4 3 " m !" ')( $ ! % $ 5n ! % ¢ 3h& - 0& ( ' ( (*' '#" 1 &- - - - '65_ m 7X!! ' % F 7<7 5(8 G N 7( m (*' & ( m (*' & , - &

! a 1 m !" ')( $ ! % $ 5n ! % F 7(* G 0& ( ' ( ' ( '#" 1 &- - - - '65_ m 7X!! ' % F 7<7 5(8 G N 7( m (*' & ( m (*' & , - &

! a 1 $ 5_7 ')( $ ! % $ 5n'65:O! % $ 5 7!A $ 5 ! 0& " 7 (8 3

; & 2 1 a & L ' '65 ! 7(* 1 ) * J

! ! " " m ¢ ')( $ ! % $ 5:'g5:O! % ( m ! " ( ! " & ( ' ( ' ( ( ']" m ( 1 '65_ 7! % m( ! ' ! ! ' ( ! ¢ & N m ' ' & 7( m (*' #7( (*' !

1a ¢ ¢ £ ¤ 3 43 0& #$

¤

#$

¤

¤ ,

#$

¤

¢ £! ¢ 4 3 0 & 3 O7 3 ¢ ¢ a 0& 3 # ' 5h ! % - j + -!&4 lC-21 . ¢ 3 £

¤

a

&L ' J

' 5 7 ! $ ')( $ ! % $ 5n'65_7! % 0& '! % ¤ 10. 9.

3 3 3

11.

12.

13.

14.

15.

16.

17.

18.

19.

')(

+

m !" $ ! % $ 5_7X! % F 7 (8 G & ( ' m !' £ ¢ O'! % & 4 3h

-; N & 7 (

; L1 5

( ' &- - - - #$ 1 * ¤ m (*' & ( m (*' & ! ( "

¢ ' m !" $ L 4 3

& J &L " J

( ' ( ' ( # ! ' 1 1;- - &- W - $ ¤ m ¢ - 3h N & ( m (* ' #7( m (*' 1 W &L ' 1 J

'%&

! -Q1 ' $ 5_7 ')( $ ! % $ 5n'65nO! % a 1 0& O'65 7! % ¤ ' ¢ $ 5_7 ¢ ')( $ ! % $ 5n'65n ! % a 1 & O'65_7X! % 3h

¤ 4 3 m !" ')( $ ! % $ 5n ! % F 7 (8 G a 1 & #$ (' V(')( ! ' m 1 &- - - - " ¤ O'65_7X! % a 1 F 7(* G N

' & ( & 7 ( m ( ' & ( m (* !

m !" ¢ ')( $ ! % $ 5n ! % F G a 1 0& 3 (' V( (*' ' m !' - - #$

1 & O'65_7X! % 1A- ¢ 3h

N ¤ W 7 ( m (*' & ( m ( ' & W &

! -Q1 m !" !" £) ¢ ')( $ ! % $ 5n'65_7X! % ( m ! " ( ! " 0& ' ] ( m ( " ¤ '! % m ( ! ' ! ! ' ( ! £ & N (' 7 ( 1 ( (* m ' '

m ' #7( (*! ' &

' $L ' ')( ( $ $ 5_5 7 7 0& £ & ¤ &-

TJ m (87! " 7 43 ¢ 3 )£ ¢ $9% ')( $ 5_ X7 ! % C7 ( m ( '! " & - & m ! ' 0& ¤ 3 395

20.

21.

22.

23.

24.

25.

) ' * '

' 5_7 7 6 ( $ ! % (*'! " &- & $ L J (* ! % 3 £ S ') 'g5 7! % 3 3 ¢ SEa ¢ 3h 1 ( ' ( 43 ¢ £ 3 ¢ J $&% 1 L "

&- 0& 43 £ £ & ')( $9$ % 5_7 F]3 O'65: G 0& & -!&

S

& SE- a 1 S 0&

=

'[email protected] B

¢ j + -21 lM1 -; . ¢ 3 V ¤ . . . E(:7 ¤ G j ++ -; & l + 1A-; & F

SEa & - 0& ST & & ¤ O'65n £ $ 5n L & - &- a 1 & ¢ X £ & 1A- & ¤ J O'65_7 ']" " (87 O '65_7 $ 5_7 L O]! " &- 0& ' '#" &-21 0& = '8>[email protected] J

5.8.5. Sums containing products of Un (z) 1.

2.

¢

X & 7 7(8 £ & 0 3 £ ¢ X £ 4 0& a ¢ & 5h' L " ¢ J

43 '8( $ ! % $ 5h' 5h ! % & ( ' L " 0& J ' ' % C7E(_ ! " ' 5h ! % ( 1 L' j 0& 3 J

5.8.6. Sums containing Un (ϕ(k, z)) 1.

2.

1 0 & ¤ a 1 0& j a 1 & 0 ¤

¢ ' & ¡ '[email protected] S $ L 4 3 = J S ¢ '$ L E & 43 £ & £ & a a 1 Q ¤ 4 3 S a S a J 3 3;

+

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

¢ ' 43 J $ L S ¢ ' $L S 4 3 J 1 ¢ ' 43 J $ L 1

; L1 :

¢ & 3 £ & S S l ¢ 3 ¢ S ¢ S 3 £ & S S F $ G l $ 5 3h& S S:F $ (8G 43 ¢ S ¢ i S l 3 ¢ & S

¢ ' S / $ L 4 3

J ¢ ' 43 J $L -2 1

1

43 ¢ S _ £ & 4 S S l

¢ S _ £ & 4 SEa

SEa 1 / 3 3 £ 3 ¢ S ? ¢ ' S S F $ G S l 3 £ & 4 S $ L 4 3

J 1 ¢ ' $L SEa 1 SEa F G £ ¢ S 4 3

1 $ l J 1

= '8> = '8>

= '8>

@CB

= '8>

@ B

= '*>

¢ ' $ L S S #F " ¢ $ G 3 ¢ S ? £ ¢ S 3 V& S 3

l J 1 = '*> ¢ ' $ ( & ¢ $ G £ 3 ¢ S ¢ S E $ 4 3 L S a F " n 5 $

1 l J 1 £ S E a S 3 1 = '8> ¢ ' 5 , 43 ¢ SEa 1 ? S l $ $ 4 3 L & S S )+* J 1 3 £ ¢ S = '8> ¢ ' SEa n 5 , $ $ 4 3 L & 1 SEa 1 ) * J 1 £ 43 ¢ S ¢ / S 3 £ 43 ¢ S ¢ 4 SEa 1 = '*> l 3 3 :

@CB

= '8> 3 £ & SEa 1

@CB

= '8> 1 S l ¢ 4

@CB

@CB

@CB

@CB

@CB

@CB

@CB

14.

15.

16.

, '

$ ¢ ' 43 ¢ S ? £ S $ L 4 3 & S * $ 5 , ) J 1 3 $ ¢ ' & SEa 1 SEa 5 , $ $ L

4 3 2 1

1) * J 1 £ 3 ¢ S ¢ S 3 £ ¢ l " ( 7! O' ¢ f ¢ # 5 7 $ L $ f &- -21 ) J

¢ S l S 43& = '*>

@CB

X3& S = '8> @CB $ m 5_7 $ m 5_7X! 5 , ( ! ' O'65_7X!A ' m 5_7X! ¤

5.9. The Hermite Polynomials Hn (z) 5.9.1. Sums containing Hm 1.

2.

& -21

3.

4.

5.

6.

7.

"

nk (z)

2(:7X! ' ¢ ')( $ ! % $ ! % & O'! % 3 * 2(:7 , ¤ ) ')( $ ! ' '" £ ')( $ ! % $ ! % 0& & ')(* $ ! % £ & - ¤ ' " " ' % $J L m ! " & 43 ¢ m ! ' 1 - -; O ¤ ( ( #$ 1 (*' L " 1 ' L ; 1 ¤ ')J ( $ ! % $ ! % - & J ' % - N &

1 1 !&

m !" ')( $ ! % $ ! % 3P

- & (' #$ 1- - &- - ¤ m' ! % ' - N & & 7 ( m ( ' 1 ( m (*' 1 !

( ' ( ' ' ' -21 7 ' % ')( $ ! % $ ! % 0& O'! % N 1 V7 ¤ J$ L J $ L

! (' m ( m !' m !" ')( $ ! % $ ! % ! " 0& ' % ! ' N m ( (*'6! 5_7 1 ¤

3 3<

+

8.

9.

10.

11.

12.

13.

14.

15.

16.

¢ 43

(']")"'&

$ ! %

; P1

( ' 1 ( ' ]&B B B ! ' ¢ ¤ & 4 3 '! % ES a N S 7 & B B B #7 ( "

1 2- 1 !

" ')( $ ! % $ 5_7X! % a 1 0&

" '65_7! % 3 ¢ a 1 a 2(:7 ¤ ,

1 ) * '" & ')( $ ! '%& £ a ')( $ ! % $ 5_7X! % a 1 0& & 1 O')(* $ 5_7X! % £ & - ¤ m !" ¢ ')( $ ! % $ 5_7! % - a 1 & 4 3 (' £ m ' ! % ' 1A- N & & 7 (

&<- - ( ' m ( ' & ( m *

#$

- ¤ &

!

m !" ')( $ ! % $ 5_7X! % 3P

- a 1 & (' #$ <& - - - - ¤ £ m ' ! % ' 1A- N & & 7 ( m ( ' & ( m (*' & !

( ' V ( ' O ' 7 ' -21 ')( $ ! % $ 5_7X! % a 0& £ '% ! % N 7 ' ¤ $ L * $ L

1

& ! J J

" "" ' ' % ¢ ! ¤ ! J $L m " a 1 0& 43 m ' &4 - -; 0

( '<( 1 L)" J ')( $ ! % $ 5_7X! % - a 1 &

( 1;- ! ( £ J & ' L % ' 1A- N

& &

&

( ' ( m !' m !" m £ ')( $ ! % $ 5_7X! % ! " a 1 0& ' % ! ' N m ( (*'6! 5_7 3 3 L

&

#$

¤

¤

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

, '

¢ (#' ")"'& _ 5 7X! % a 1 0 & 43 $ (' ( 1 (*' ]&B B B ] ! ' & 43 ¢ O '6 5_7X! %]SEa 1 N S -21 7&B B B #7 ( " ! " " ' $ 5 ! % a ST & 43 ¢ SEa £ S SE-;&a -21

¤ $ L J ( (*' m M7( m ! ' ' m 5 !" ¢ S 5n ! 5n'! % N m (*' 1 ¤ J $ L $ % a S & 43

! m 5 !" ' $ 5n 5_7X! % a SEa 1 0& $ L J ( (*' m £ 3 ¢ S C7 ( 5nm '! '! % N

m ( ' &!

' " " ¢ SEa £ SEa 1 1 -;

¤ 5 ! SEa J $ L $ % a SEa 1 & 43 ¤ ¢ ' $ ! % ! ' ! &-21

) , 43 J $ L $ % & 1 L ' J

'%& ' % a ¤ ¢ ' $ ! % a 5_7X! % a 1 & 1 1 ) , 43 J $L $ & L ' J

" " (' 1 ( ' m ( 1 O]! ' $9% ') ( $ ! % ! " " ¤

' ! % 0 & N : ( 7 (

m m

&- & 1 ! " " $9% ')( $ 5_7! % ! " m

&- a 1 & ( ' ( 1 (*' m ( 1 ! ' & O '6 5_7X! % & N 1 m (: 7 ( "

! " 7 . ' 5_7 F 3 G &- 0& j + - lM1 $ $ L J ( ' 1 (*' 5n £ & N ( " & 1 ! 3 3 P

¤

¤

¤

¤

+

27.

28.

7 . ' " 5 7 F]3 3 G &- a 1 & j + - lM1 $ $ L KJ ( ' & ('*( 1 5h £ & a 1 N ¤ ( " & 1 ! " 7 ')( $ ! % SEa &- & ' % S & & b # 0 ¤ Q $ L J

5.9.2. Sums containing Hm 1.

2.

3.

4.

5.

6.

; P1 ,

nk (z)

and special functions

'65_7 $ 5_7 L #& - &- a 1 & J 7 ' '# " " (8 & " $L ]!

& J

'65_7 '#" '#" 0& ¤

'

]! '#"

&-21 &

= '8>[email protected]

7 7 $ ! % 'T( $ 5_7X! % 0 f63 0& O'65:O! % fg3 3 ¢ (' ( '65 1 V7#7 ! ' Y £ & a 3 "

a 0& O'! % m (*')(:7X! N m (*' ] ! ( 7 7 $ 5_7X! % ')( $ 5_7! % 0fg3 a 1 & O'65 ! % fg3 3 ¢ ( ' ( ' ( 1 V7#7 O]! '%& Y £ & a & 3 "

6 ' _ 5 7 ! * ( ) ' : ( X 7 ! a % 0 & N m

&

m ( ' ] ! ( C7 ( m ! " ¢ C7 ( m ! " ¢ a ! O ' ! $ % 0fg3 & 43 1 % ')( m 5_7X! £ & 43 ( ' ( 'g5 m (*')(87 ( " 1 ( ' Y fT3 3 ¢ fg3 3 ¢ N ! m

& 1 (' V('65 1 m (*')(:7 m (*')(:7 N

m (* ' m (*' ! ( " ¢ C7 ( m ! " $ 5_7X! % 0fg3 a 1 & 4 3 43 ¢ a 1 '65_C7X7 ! % ( ')m (! ' 5 7! £ & 4 a 1 m 395N

¤

¤

¤

, '

(' V(')( 1 m (*')(:7 m (*' ! ( " (' V(')( 1 m (*')(:7 m (*')(:7 N

¤ m (* ' m (*' ! ( " ( 7! ' ¢ ' ")" £ 5_7! " a / & & 5_7X! ' a / £ & ¤ 7. 43 J $L ' £ & A& 3 £ ¤ / G $ L F 8. - / & J " ( (:7 ' ( $ ! % $ ! % j &- Q & 9. W W # ( ' ( ' ( $ ! £ J 1 L ' 3 ¢ N ¤ W

' % ! -Q1 (1 *' 1

" ( (:7 ')( $ ! % $ 5_7X! % j &- Q a 1 & 10. W W # ( ' ( ' ( $ ! £ a 1 J ' 1 L ! ' 3 ¢ N ¤ W

% -Q1 (1 *' &

' 7 ( X 7 ! ')( $ ! % ( 5 7! c 3 ¢ ¤ 11. " &- 0& ( P5_7! ' - c -; 0& Y

0fg3 3 ¢ fg3 3 ¢ & N 1

5.9.3. Sums containing products of Hm

"

1.

2.

3.

4.

nk (z)

' J( 1L $ ! % & 0 3# -21 -; ¤ $ L J " ( 7X! ' '%& ' J( 1L

5 7! O'65 7 1 -;

¤ J $L $ % a 1 & ' $ 43 A J L &- A& -21 & 1 £ + &-21 . 43A W £ E3 W / -;&-21 ' ¤ J $ L Q &- & 43V

395)

& A& ¤

+

5.

6.

' J $ ' J $

7.

£

8.

£

9.

£

L & L &-

a Q

1 Q

; P1 5

¢ £ a ¤

a 1 0& 43 Q

a 1 & 43 ¢ £ a 1 1 O ¤ ' 1 (*' L " J$ ! % ')( $ ! % Q &- 0& ' %

) 1 (*' L" J 7X! % ')( $ ! % a Q $ 5_

1

&- & ' '& ' ( % 7XO! '6 5_7X!

a 1 a F

1 ( 1 (*' L " J $ 5_7X ! % ')( $ ! % a 1 Q &- a 1 & ' '%& ( 7X'6! 5_ 7X! %

a 1 a F

1 5

(

, ¤

5 G ¤

5 G ¤

5.9.4. Sums containing Hn (ϕ(k, z)) 1.

2.

3.

4.

¢ ' 43 J $ L ¢ ' 43 J $ L

= '[email protected]

ST & ¡ S /

" S 43 ¢ S £ i £ & 4 -; (*') ( ( $ ! % ! $ 5n '! % ¤ a

( 7! " ' / 4 3 ¢ SEa 1 £ SEa 1 F $ L E S a $

1 J 1 S ¢ £ ¢ i £ & a 1 -; 43 S a ¢ ' $ L SEa / 4 3

J S 43# SEa a 395,

G S

( ! " ( ' ( $ ! % $ 5n '65_7! % ¤

(]! " $ 5:'! %

a ¤ S - & -21 Q

5.

, '

( X7 ! " $ 5 3 ¢

' J $ L ES a a SEa £ SEa a

1 / S ! " a a 1 $ ( n ¤ 5 ' ! % S 1 Q a -

£ S ¢ ' S $ 4 3 L S F $ G 3 & 4

J 1 S -; ¢ "'' &('

£ i £ S $ 5n'! % (*O')(* $ ! % £ & -

& 4 - 3 £ SEa ¢ ' SEa $ L E S a 4 3

F 7. 1

1 $ G 3Q & 1 J 1 S -; ¢ "'' &('

£ ¢ i £ S a $ 5n'! % ( O')(* $ 5_7X! % £ & -

& 4 - 1 3 ¢ '$ L 8. 3 f O SEa SEa F $ m 5 G J " ( ' % 5n '! % ( L" S 1 m f S a '65 J $ ! % ( $ ! % F G S - F G 1 ( J L ¢ ' $L f O SEa a 1 SEa a F 4 3 9. $ m 5 G

1 J " ( ' % 5:O'65_7X! % f SEa 1

& ( J L ( ( S L " 1 Y a F G a '6J 5 $ ! % ( $ ! % F m G S

- 1 6.

5.9.5. Sums containing Hm 1.

2.

nk (ϕ(k,

$ m ! " $ m ( ! ']")"" F $ G ')( $ ! % $ ! %

3953

¤

¤

¤

z))

' ]' " ' ( X7 ! ' % O'! % ' m ( ! F " m " $ "'& $ m ( ! ']")"" ')( $ ! % $ 5_7X! % a F

1 $ G ' '#" " '%& 1 # ( 7! ' % O'6m 5_7X! % ' ( ! a OF " m

1

1

¤

m ¤ G m ¤ G

+

3.

4.

5.

6.

7.

8.

$ ']" O')(* $ ! % &- F $ G $ & % 1 $ '#")" $ 5_7! "" O')( $ ! % &- F $ & $ % 1 $ '#")"'& $ 5_7X! "" O')(* $ 5_7X! % &- a $&% 1 $ '#" O ) ' * ( _ 5 X 7 ! $ % &- a &$ % 1 $ m 5_7X! '#" ')( $ ! % $ ! % F $ m 5 $ m 5 7! ']" ')( $ ! % $ 5_7X! % a 1 F

; P1 :

]! '#" O O') (* ! %

O]! ' ( 7X! ' G 3 O'! % O'! % A& ¤ F1 $ G ( 7! ' ! ' & 3 O '6 5_7X! % 3 '65_7X! % a 1 A& ¤ ! ']" ' >[email protected] ')(:7X! % =8 1F $ G O]! ' 7 G '! % ' m 5 7! ¤ O]! '%& O'65 7! % ' m 5_7! ¤ $ m 5_7 G

5.9.6. Sums containing products of Hm 1.

2.

3.

= '8>[email protected]

¢ ' ¡ 43 J $ L& ST & ( 7! " $ 5_7X! '#" 5_7X! % O'T(* $ ! % &- F $ 5_7 $ '%& O'6 5n ! % a

( 7! " $ 5_7X! ' $ 5_7X! % O'T(* $ 5_7X! % &- a 1 F '%& '65 ! % D a 43 ¢

nk (ϕ(k,

z))

G

=

a F $ _ 5 7

1 a 3

1 $ 5_7 G a 1 F a

1

G a 1) $ 5_7 G

'[email protected]

(8 5 , ¤

a F 5 GIH ¤

5.10. The Laguerre Polynomials Lλn (z) λ nk 5.10.1. Sums containing Lm

1.

(z)

( ! ' ('! " '! " 5 ! 5n' ! % S ;- & $&% $ % S 0& 395 5

= > '[email protected]

# . '/

2.

¢ ' a 0& 43 ¢ $ L 4 3 Sc J $ ' 5 _ 5 7! " 3P& Sc a $ L J % 5 7 5_7!

3.

4.

5.

6.

7.

8.

Sc a -; 0&

= > '[email protected]

0&

a ¤ Sc a & - 0& ' 3 3 E

1 ' ! " m ¢ (E5 : 5 7! " Sc a & $ L 3 J ( m 58' 5:7 ¤ (E5n7! (% (E( 5n( 7Xm ! 5n7! ' N ( 5*' 5n 7 ( m 5n! 7 L '

J (]! " 5n'! % ' ¤ ( Q(*'65_7X! " Sc a & 43 ¢ % ( &! ' SE $ c L ; & a J 5n'! % ( ! " ' ¤ ( P5 5_7! " Sc a & % ( P5 5_7! ' SE $ L c 0 & a J ( 5 7! ( ( ( ' m 5n' ¤ ' m 5 5n'! " m ! " ( P5 5 7! " 43& Sc a & $ % L N

J 5_7 m ! L J ( g5 5h' 5 7X! " ' ( g5 7X! " ( 65 5 7! " 3 Sc a 0& $ L J ' % % ( g 5 5h7X! '! ( % g( g5 5 7X! 7X( ! ( c 3& c & ¤ ' & ' SEa

Sc - & ¢ ' 10. 43 J $L ¢ ' 11. 43 J $L ¢ ' 12. 43 J $L 9.

SEc a 1 0 & 3 Sc - &

¤ SEc -;a &1 -21 0&

Sc -; 0&

= > '[email protected]

' ( Q( ! " C7 (Q(*'! " Sc - 0 & ( 9! ' Sc a ;- 0 & = > '[email protected] ( 6( ! " C7 ( m ( '! " Sc - & ( 9! ' ( # 6( m 5_7 ¤ ( P5_7! ( % m ! % ( '6! 5_7 L m '

N J 5 7 # 6( m * 395;

.

13.

14.

15.

16.

' (' ( J $ L

!" m 5 m !"

58'! "

3.

4.

5.

6.

( ' ] ( ( m 58' ¤ ` m ! ( "

nk (z)

I5_7X! ' 4 ')( $ ! % (P5_7! " c 0& ( 5 7! ' c F I5_7 G ¤ ')( $ ! ' ' ¤ & -21 ¢ ')( $ ! % ( P5_7! " c 0& ' % (P5_7! ' 3 43 3 d)3 ')( $ ! % ( ' (' ¤ ¢ 5_7#! 7 L 43 = ')( $ ! % @ ( P5_7! " c & N J m !" ')( $ ! % ( P5_7X! " - c & ( ' _ 5 7 ¤ a ! ' m c - - 1 c - - ' % -; N ! & & 5_7V( m (*'65_7#6 ( m (*'65_7 ( m !' m !" (' m ¢ ) ' ( ! ! ( 5 7 ! ' % ! ' N m ( (*'65_! 7# 5_7 L ¤ $ % " 4 3 c 0& " J '( 6( '! " ')( $ ! % ( P5_7! " 3P& - c & #$ ( 7X! ' 1;- ! ( ( P5_ ' % -; N & a 5_7 # P5_7 ¤ c 1 c

395: "

2.

¢ ' ('6( ! " m ! " Sc - & $ 4 3 L J ( 6( m ( ¤ (P5 m 5 % ! ! ' ( ]! N ( 7 (%V( 6Q( ( ( #7 ( ( m ' m ' ! ( " ` & 1 ^ 5n'! % ' - Sc - 0& % 3P& -; SEc - a 0& ¤ $ L 3 2 d J " (]! ' ( 5 5_7X! " $ 5 7! Sc a 0& $ L J " 7X! % 'g 5 7 D d c -21 0& 3 % ( P5n5 '65_5_ 7X! ' SEc -2a 1 a 0 & H ¤ S 1

5.10.2. Sums containing Lλ m 1.

Sc - & ( (]! % N ( & 1 ^

-

; N1 ,

# . '/

(']")"'&

( ' ( ' (% &BB B ¤ (]! ' (P5_7X! " c & ' % ( P5_7X! ' ES a 1 N S 2- 1 7 &B B B V7 ( " 7. ` ^ ! ( 6( '! " ) ' ( $ ! % ( P5_7X! " 3P& - c & 8. #$ ( 7X! ' 1;- ! ( (P5_ ' % -; N & a 5_7 #P5_7 ¤ c 1 c

' $ ! % ' O ] ! % ¢ ' ( P5_7! " c 0& ( P5_7X! ' c a F ,G ¤ 9.

43 J $ L ¢ ' 3 ¢ c -; & ¤ $ L 10. 4 3 c a2S & SEa J ' $ 5 !% % ¢ ' ( P5 5_7! " c a2S 0& ( 5 5_7X! ' Sc a 0& ¤ 11. 43 J $L m 5 !" ¢ ' ( P5 5_7! " c a2S 0& $ L 12. 4 3 J 7X! ( ( (*' m ¤ C7 ( m ! ' 5n( P'5_ ! % N J 5 7 m (*! ' L S ( Q(*'! " 5 ! 13. a2S $ % &c - 0& ( & ' "( S ( P5n'65_7X! ( 43 ¢ C$&7 % ( $ !( $ ! % c ¤ 7 ( G $ F E S a " ( ! ')( $ ! % $ ! % 3Pd 3 Sc - & 14. #$ ( ' ( ( 6( ]'65 1 (]% ' ! % N " = > '[email protected]

( & 1 1 "

! ( ! ) ' ( ! % $ 5_7X! % 3 d 3 Sc - 0& $ 15. ( ' #$ ( ( 6( ]'65 & (]% ' ! % N " = > '[email protected]

& 1 & ! (

¢ ' - S -21 0& S- S -;&-21 0& = > '[email protected] 16. 43 J $ L S

395<

.

17.

; N1 3

" '#" J 1&$ L % " & 43 ¢ ' % F#" G a F" G ¤ & -

1

nk 5.10.3. Sums containing Lλ m pk (z)

1.

2.

3.

4.

5.

6.

7.

( 7X! ' ¢ m 5 $ ! ']" a ')( $ ! % m 5 $ ! " c & m n 5 ' 65n')(:7 c - -; a 1 0& ¤ 43 ( ! " ' % ')( $ ! % ( P5_7! " 65_7! " c a a 0& ( P5_7! ' 65_7X! ' c 0& & ¤ m !" ¢ ) ' ( ! ( 5 7! " c a 0& $ % 4 3 ( O' m (Q(*' ' % ( m 5 ! ' 7! N (' V( 76 ( ` ' & 1 ^ m (*' ! " ' J '65 1 L " ( 5 7 ! " 43V& c a 0& $ L J '65 ( O' ' J % ( P5_ 1 7XL ! ' £ & N ( O' ( V( ' ' ((' 6 " ` ' & 1 ^ ! m !" a 3 # & & c ) ' ( ! ( g ' 5 ( P _ 5 X 7 ! $ % " m 1L" J

( O' 1 ( m (*' V'( 6(* ' m !' 7 (* m (*O' ( " ' % 1 ( m L ' ( P5_7! ' 3#

& N 1 ! J

1 ( m L)" J ')( $ ! % C7( m (*'! " ( P5_7X! " 3#& c a & ( ' % J ! 1 ( Pm 5_L ' 7X! 3V

N ( ' m ( ( O' ' ('( 6" (* ' ' ` m ' m & 1 ^ ! m !" a 3 # & & c ) ' ( ! ( g ' 5 ( P _ 5 X 7 ! $ % " m 1L" J

( ' ] m '7 % N !

m (*'65 1 P5_7

395L

¤

¤

¤

¤

¤

8.

# . '/

5 '65 1 L " ' J n ( 5 5_ 7X! " #& c a2a S 0 & $ & $ % L J (P5_7% ! (

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

' a 0& $ L Sc J -

N J

(

(*O' 5nO'65_7 ¤ 5_7 #P 5_7 L !

= > '[email protected]

Sc a &

#$ ( '65 ( ]! " & ¤ (P5_7! ( a ')( $ ! % $ 5_7X! % Sc 0&

%' % N

& # 5 7 !

' 7 5n Sc a 0& $ $ L J '65_7! 7 '65nO! c - & ¢ c a 0& 3 c a &-21 0& = > '[email protected] SEa

SEa 1 SEa

" ( 7! " S S S a ( $ ! % SEc - a S 0Q3 ¤ 9 $ % a2S $ 5 ! % c & 3 & " ' a > ' > @CB ')( $ ! % Sc a & $ L 0 & = Sc ! J 1 ' a & 43 ¢ 3 43Pd 3 Sc & $ L 3 Sc -; J = > '[email protected] B ( ]! " ( 5 7! ( ' > '[email protected] ( 6( 'g5 7! " Sc a & ( 6( 'g5 7! ( Sc -; 0& $ L = J " '65 1 L ( ( ]! a ¢ S J ')( $ ! % $ ! % S -21 0& 43 ! % ' % ST0/ ¤ $ 5 ( 5 5_7X! " 3& Sc a 0& '%& 1 3 c a 1 0& 3 (P 5 ( ! 5_7! c a a 1 & = _'8>[email protected] B ' S -;&-21 S -21 ( '65 #$ ( ]! " ( P5_7X! ( & a > '[email protected] ')( $ ! % $ 5_7X! % Sc & ' % N P5_7 % = & !

395P

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

' J $ L

.

)" " ( Q(* ! " a (1 Q (* (*' c S - &

( L " ( V( 6( V( 6( ! % N " ( Q ( (* ' ( & 1 ^ ! " 7 ')( $ ! % c - 0& c -; F Q3 G ¤ (' V( m ! ' ( &! ' ' % -; N 7( m (*' 0fI 4 3& - c - 0& J

J

4 3Pd 3 S&- a ')( $

; N1 3

(*' `

= > '[email protected] B

m ( ' ! Q(*'65_7 L ¤

¢ 3P& - c - & 43Pd2 43& -; c -; a 1 & ¤ ( &! ' (' &B BB 1 3P& - c - 0& ' % -; S N S J Q(*'65_7 #7&B B ! B #7 L ¤ 7 ' ¤ ! % C7 ( Q(*'! " c - 0& ' % ( 9! '

( 1;- V( ( 7X! " ( ! ' ')( $ ! % (Q(*'65_7X! " c - 0& ' % ( &! ' & N ¤

" ' ¤ ('! " & -21 ')')( ( $ ! ! ( ' &! ' ; $ % % P 3 & (Q(*'65_7X! " 43& c - & ( m !' m !" (' m V( ( " ¤ ')( $ ! % ! " - c - & ' % ! ' & N 1 m ( (*'6! 5_7 ` ^ " ) " ')( $ ! % m ( $ ! c - 0& (' #7 ' % ( &(*! ' '! 3P& -; N * ( 6 ' 5_7 #Q! (*'65_7 L ¤ m m

J )" " ' % ')( $ ! % $ ( m ! & - & ( m ! ' & 3P& -; -; 0& ¤ m 5 1 L' '( 9! ' ( ! "' 1 ( m (*' L" J ' % ( J

m ')( $ ! % & (*' L " - c - & 1L' J

J

(' m (' m Y £ fg3 ¢ N ! ! 'g5 7 ¤ N # 6 * ( 6 ' _ 5 7 5 # 6 ( ( m m

1 1

3;9N

# . '/

( Q ( 1 L" J

31. c - & * ( ' ) ' ( ! $ % & " L J ( &! ' (]! "' £ ¢ ¤ a #( Q(*'65_7X! ( 1 L ' d c -;&-21 & c -; 1 & J

' $ L 0& SEc a 0& ¤ 32. c a2 S J ' - c a2- S & $ L I f 33. J ( ( ¤ ( m 5_5n7! '' ! % (]! N ( (*( ' m ( ( ' V((' 6 " ` m & 1 ^ ! ( P5_7X! ( ' ( 9! ' " ' ' (6(*'65_7X! ( SEc - a 0& ¤ $ L 3 2 d & 34. c a2 S J ' ( Q( ! " $ # & L 35. Sc - & ( * ( ' J 1 L " ( J

( ( (*' V'( 6( ( ! ( (*O' ( " % N ` = > '8>[email protected] & 1 ^ ! ' 5 & A ! ( 9 ! ( &! ( & $ " a ( 5 _ 5 X 7 ! ( P n 5 6 ' _ 5 7 ! ( P n 5 '65_7! ( Sc a &

36. " " Sc - 0& J $L = > '[email protected] B " ( 5 7! ( ( 5 5_7X! " Sc a 0& 37. % ( #$ ' ( #$ c 5_7'g5 c 5_7 ( ! ' & ( ! '%& ! ! Y N ( P5_7X! ' & N '65 5: '65 5n

c 5_7 3 c

= > '[email protected]B W " ( ' J ' L 5 5_7! Sc a & $ L 38. ( * ( " J 1 L" J

( P _ 5 7 ! ( V( (*' ( > '[email protected] ! % N

J ( (* ' # 5_7 L = " ( ' m ( J L ( 5 7! ' a ! 7 ¤ ! ' % N m (:7# 1 5_ 39. $9% m " &c -

& 3;

.

40.

41.

42.

43.

44.

45.

46.

47.

48.

; N1 3

(' c * ( ' #$

7 # 6! ( O' ¤ ( g ' 5 c

7 F]3 G c - & F G F G

c - R a 1 & ¤ ('9! " ( O'! " F]3 G c - & ' m ¤ m ' ! % ' '( ! 9! ' - N (# 6 ( O'6! 5_7 L m ' m

J ( 9! " 'g( $ ! % m ! " - c - & #$ (' a c - a c - a 1 ! ¤ ' ( % 9! ! ' 3P& -; N m ' & & c a 1 ( ' c ( ' 5n7 m 5 ( '

' ( &! " 'Q( $ ! % c - & ( 5 4 , 5 5 ¤ 3 c - , c - ) , ) '( 9! " )" " ( &! '%& £) ')( $ ! % ')(P5_7X! " c - & ' % 43& -; ¤ 3 2 d J m 5n')( 1 L)" ( &! " G c - & ')( $ ! % m ! " ] F 3 ' 7 % N ( O' O'65n( m " (:7V( ` ¤ m! & 1 ^ O')( $ 5 m !A '( 9! " ')( $ ! % C7 ( m ( '! " - c - 0& " ' (' c - (*' c - a 1 (*' #$ '( 9' ! % ' m !

( m (*O! '65_7 ¤ m ' & N & c a 1 (*' c (*'65_7# 6

C7 ( m (*'! " ( &! " £) % ( 5n'! " 43& - c - & 3 d2 m ( 9! ' (' m (:7 & m % m ! ' '( % & ! ' 3P& -; N J m # Q(*O' ! L ¤ m ( &! ' 3P& - 4 P 3 2 d ( Q( 1 L" J ') ')( $ ! % J1 ( m ( ' L" ') ( $ ! % C7 ( m

('9! ' c - 0& ' % ;- N

3; ,

49.

50.

51.

52.

53.

54.

55.

56.

# . '/

$ ( ( 9! " ')( $ ! % ')(P5_7X! " - c (6(* $ ! : ( 7 &! '%& ' (Q ( * ( O'! (:7 3P& -; %

& ( ' #7 N c a 1 ( ' c ! a & (*' ¤

1 ( m L " ('9! " J ')( $ ! % C7 ( m (*'! " F]3 G c - & ( ('9! J 1 ' m % L ' ! ' F]3 G N ((*O ' ' m Q(*(*' ! '6 5_7 L ¤ m '

J m ( &! " £ - c - & ) ' ( ! ! ( % ( 6 * ( ' $ % m " & m L" J

(' m 5 5:' (87 ( m (%6(*' C ( & ! ' ¤ 6( 'g5 7 ( 1 ' % m ! ' m 5 6( 1 L ' F G & N 1 ! J

C ( & ! " £ - c - & ) ' ( ! C 7 ( * ( ' ! ( P 5 $ % m m " 1 L " J

( & ! (' ] m ( P5n' #6( m ( '65_7 ¤ ' G Q (*'65_7 ( " F ` ' % m ! ' m (P5 1 L ' &N 1 ^ ! J

m ! " ! " ( &! " £! ')( $ ! % C7 ( m % ( 9! " C7 ( ( &! " ') ( P5_7X! " - c - & 3 2 d ! ' '( 9! ' & ( ' V7 ( m ( % ( ¤ ' % C7 ( m ! ' ( & ! C 7 ( ( & ! 7 ( ( ' V 7 ( (*! ' L P 3 & N ; ' ' m m

J W " ( ' J ' L 5 5_7! Sc a & $ L ( * ( " J 1 L" J

( 5 7 ! ( V( ( ' ( > '8>[email protected] ! %

N J ( (*O' # P5_7 L = ' 7 $ L F G 3 d 3 F]3 G c a2- S & J ( (]! % N ( ( O' 5 5n7 '6( 5_" 7'( 6( ` ¤ & 1 ^ ! 7 7 $ @& [email protected] ')( $ ! % (P5_7X! " c - 0& (P5_7! ' c &

$ L = = J

3;93

.

57.

C7 ( m (*'! " ')( $ ! % (P5_7! " - c - &

; N1 3

V7 ( m (* ' ¤ m ! ' O '( ! % ! ' ;- N ( Q' (* '65_7 ( L

Jm !

58.

59.

5n'65 1 L" ' JP ( 5 7! " £ c - 0& $ L J

P5 1 J 5_ 7XL ! ' ' C7 (* m (*'! " 1 L " ')( $ ! % C7 ( m (*'! J " 5_7X! "

60.

61.

62.

63.

64.

( ' ('6( O' V(6(* ' ¤ ] ! ' O'! % & N 1 (6( ' ( " ` ^ ! c & W #$ ( ' 1 ( m ( ' ! ( & ! ' ' m ' % m ! ' 43V& -; N & m 5 a ! 5_7 ¤ 1 c - 1 c -

Q( $ 1 L " ( c L " '65 1;- c L " ')( $ ! % J (*J ' ')J ( 5_ 7 F G c - & c L" 1 J L" J

' % ( £ 3 ¢ a 1 J c L '%& F G D + c -21 . - F ,G9H ¤ 1 J L' O' ' ( $
£ R&-21 £ - - &-21 F G 3 7 ' ' L ¤

J O'65_7 £) ¢ ')( $ L - - a - 0&

1 J £ R a 1 £ ¢ - - &!- &4 F G ¤

a 1 ( Q(* ! " ' a $J L (1 Q(* (*' L " - c S - 0& ( J

( V ( 6 ( V( 6( (*' ! % N " ( Q ( ` = > '[email protected] B (* ' ( & 1 ^ ! 5n')(* $ ! % ( $ ! % ')( $ ! % ( P5_7X! " SEc a a & &-

( P5_7X! ( & ' ( P5_7X! ( P5_7X! c 0& c & = > '[email protected] ( ' S 3;!5

# . '/

65.

m !" 2d 5 7 L " ( &$ % c ( m _ J

66.

c a J 5n L '6" 5_7X! (9! " "'& &c - & ( ' a ( #$ c 1 m! ( 5 ' % 7! ' N

c a 1 Q (* m 5_7

) ' * ( ! ( ' ( $ % 1 " L 43 ¢ £ 3 ¢ J%

- &-21 & & $ &-

£ a 1 F 7 G -;&-21 F G a 1 (' c & #$ ( &! '%& cL" & 0d ) $&% ( 5n ¤ ! a J & '65_7X! " & c & ' % N

c c & a 1 &!- &

C 7 ( * ( ' ! n 5 O ) ' ( " m m 1 " L ')( $ ! J % ( 5 7! " F G c - & & ( ' ( #7( m ! ' m 5:O')( 1 L ' m ( ' V'( 6(* ' #$ m 1 £ 'J ! % ( P5_7X! ' & & N

m (:7 ] (* m ( O' ( ! ! ' ( 5 " m m 1L" ')( J $ ! % ( P5_7 ! " F]3 G c - & & m ! ' ')( m 5 1 L ' ( &! ' J '! % F3 G ( ' m ( O' ( m (*' Y N ( ' V7 (* (*1 ' Q(*! '65_7 m

& & m ¤

¤

67.

68.

69.

¤

¤

nk 5.10.4. Sums containing Lλ m pk (z) and special functions

1.

2.

¢ ' ¢ 43 J $ L c 0& 3 ¢ ¢ c -; & & -21 ')( 7 $ c -; & ¤ C7 ( m ! ' C7 ( m ! " ( P5_7X! " fg3 c 0& ' % ( P5_7X! ' ')( m 5_7! 3P& (' m (*')(:7 V('6(*' Y 3hf- ¢ fg3 3 ¢ N ` m (*' ! ( " & 1 ^ (' m (*')(:7 m ( ')(:7 V('6(*' ¤ 3 N ^ ` m ( ' m (*' ! ( " 3; ;

.

; N1 5

( 7! " ')( $ 5_7 ! % (P5 7 ! " 0fg3 c & 3. '& ( 5 7 7 ! fg3 3 ¢ '6 5_7! % 3 ¢ c & '& a 1 ' (' ('6(*' #7V7 ( ! 1 ¤ ' % ( 5_7X! ' m (*')(:7X! N

m (*' )" " ('9! ' ) ' ( ! ' % 3P& -; fg3 $ % 0f63 c - & 4. ( &! '#" 7 (*' V7#7 3 3 ¢ ')(:7X! % m ( '! 1A-; & N & J m (*'65_7 #Q(*'6! 5n] L = >_'[email protected] ( &! ' )" " ')( $ ! % 0f63 £! c - 0& ' % 43& -; 5. 7( ' V7#7 '2 Y 0fg3 £ m (* '!A(Q(*'65_7! N ! 5n * ( 6 ' 5_7Q(*'6 & &

7 (*' V7#7 '2 * ( O 6 ' _ 5 X 7 A ! ( Q * ( 6 ' _ 5 X 7 ! m N& & a & (*' # 6(*'6! 5n ¤

" ' m ( O' ( ( 7! " ')( $ ! % 0fg3 1 & & - 3P& 3 ¢ m (*' 1 N 1 m ( '6! 5_7 L ¤ 6. J ( ! ' ¢ ( 7! " a ' % 3 d2 d 3 _ & ¤ & $ % d & P 3 & 7. c -; & C7% ( 9! ' ( 7! " ')( $ ! % d 3 1 & c - 3P& O' % 0d 3 _ & ¤ 8. " / £ c 0& 9. ')( $ ! % Q(*O'65 & L " c - a a 1

J

( 7X! ' £ ¤ ' % ( Q( 1 L ' c a a 1 / J

" " 5 8 ( 7 j Q c a 0& 9 $ % 10. & " ' (' ( ' '( 6(*' #$ ' ! '! % (87 N ¤ W % : ( 7 ( * ( ' , & 1 ) 1 ! - W -Q1

-Q1 3;9:

11.

12.

13.

14.

15.

16.

17.

# . '/

$ 5_7 7 ( $ 5n6 ' 5_7X! % j Q a 1 & ' % F G ¤ & ( ' 1 $ 5_7 7 a

5n'65_7X! % j Q 0 1 & ' % N 7#7 ( ¤ $ &! ' $ 5_7 ( 7X! $ 5n'65_7X! % a 1 Q a 1 0& '! % F 63 G ¤ & $ 5_7 ( 7X! ' a $ 5n'65_7X! % 43& Q 1 0& O'! % / ¤ & 5 ! ' a ¢ '( 6(*'! " 5 G ¤

c 1 $ ! % 4 3 Q F c & P 5

& 1L ' J

¢ '( 6(*'! " $ 5 7! % a 1 Q &c - & 43 ' & 5 ]! c a a 1 F 5 G ¤ 5 1

1 L '%& J

( P5_7! ' (' 5n O ¤ . ( ! " $ 5_7X! % j + - lM1 c a 0& ' % N

# 5_7 ! L J &

nk 5.10.5. Sums containing products of Lλ m pk (z)

1.

2.

( 7! " ( 7! ' O' @ @CB ')( $ ! % (P5_7! " c 0& ( 5_7X! ' ' L c -; 0& ==

J ( 7! ' " ' ¢ $&% " " O')( $ ! % &- -21 0& ' (*' -;&-21 #& ¤ 4 3 1 J L ¢ $&% ')( $ ! % - c - & - 0& 4 3 (' a (*' a (*'65_7 ' ( 9 ! ( ! ' ' c c 43 ¢ - ¤ ! ' % & N & Q(*'65_7 )(*'65_7 P5)(* '65_7 $&% ')( $ ! % - c - 0& c - 43& #$ ' ( 1;- ( ( 9 ! ' % - N ¤ !

& c - a 1 c - 5 7 # 6( 'g5 7

3.

4.

3; <

.

5.

('9! " !£ ')( $ &$ ! % % ') ( 5_7X! " 43 Q& d 3 3 ('9! " ! £ ')( $ ! % ') ( 5_7X! " - c d 3

; N1 ;

c - Q c - &

43Pd2 a 1 Q& -; c - & ¤

0&

6. '( 9! ' & 1 L ' 3 ' % J F G c - & ¤

' ( ( ' ! ( ] ! " . ( 5 7! " c Q & ( 5_7X! ' j + c lC- c - - &-21 F ¢ G ¤ 7. &( ' '( V( )(*' ¤ ( ! ' " " ' % &N 4 3 : 3 4 3 Q 0 & L 8. c - Q J &" (' ¤ 65_7! ' a ( P _ 5 X 7 ! ' P _ 5 7 ! 65_7 L % 9. " c Q &- 0& 1N J ")" ' ( $ ! % Sc - 43& - 0& 10. ( ()'P5*7% '! ( % ( ! ' 3P& -; N (T( 'P 58 P7 58Q7 ()! ( 'P 5*7 L ¤

J " ( Q ( ! ( ] ! ' a & - S -21 3P& ' % 9 $ % = > '[email protected] 11. Sc Sc -; 0& & ( 7X! ' ( ]! " a a ' % 43Pd 3 Sc -; & = > '[email protected] 9 $ % 12. & P 3 & Sc c -; & ( 7! " c - 0& SEa 0& 5:' L c & ¤ 9 $ % 3 d 3 13. S SEa J & ( ! ' a ( 7! " ' % Sc & ¤ 14. $9% 3 d 3 Sc - 0& &-;- &- c 0& ( ]! " a 0& a 0& 9 $ % 15. Sc &7 65n'65_7 ( ( P5_7! ( % ' 6% 5_7X! ' N P5 5_ 5 7 65_7 ! L = > '[email protected]B

J 5n' ( ]! " a 0& c a2SE a & 9 $ % = > '[email protected] 16. Sc J L SEc a & &3;9L

# . '/

17.

J

18.

19.

20.

21.

22.

23.

(]! ' a > '[email protected] ' % Sc & $ L SEc - a 0& -;&- c 4 3& = -; & ( ]! " a 5n' > '[email protected] $9% Sc 0& c -; a & J L SEc -;a 0& = & ( 6( ! ' 5 $ S > '[email protected] ' % Sc -; & = J $ L SEc - a 0& &- - - c -21 43& $9% (Q(* $ ! '( 9! " ')( $ ! % ') ( 5 7! " 43 Q& - c - Q c - & 3 43Pd2 a Q& -; c - & ¤ 1 ( & ! " £) d2 ( P5n'65_7X! " F]3 G &c a - Q - c - & ( 9!' ' % & F ¢ G ¤ m 5n'! " ( &! " £ ) 3 d2 ')( $ ! % $ $&! %% ! C7 ( P5n'! " C7 ( m ( Q(*'! " Y - c - 0& c - 43& (' '65 V( m c 1; - c 1 ( #$ ¤ O' ! m % ! ' ! ( &! 5 '%& 9! N '

m ' m m 5 1 V( ! "

$&% ( 9! " £ ) d2 ( P5n '65_7X! " F 3 G - c - 3 Q - c - Q c a & & ( ' c c a 1 ( 9!' ' % & N " ¤ & 1 !

5 $

nk 5.10.6. Sums containing Lλ m pk (ϕ(k, z))

1.

2.

3.

¢ ' 43 J $ L Sc ¢ ' 43 J $ L c & ¢ ' 43 J $ L Sc &

= '[email protected]

& ¡ ¤

S ( ! " ¤ ' % % (( P5 5_77! ! ( -; 5 ' L ( 5:g : ' 5 7 ! " ' a J $ 3;9P

.

¢ 4. 4 3 1 ¢ 43 S

' J $ L S Sc F ' % ( P5_7X! ( % ( P5_7X! ( "'

( ¢ E S a $ G 3 1 % S -; 5 ' S -; a J $ : S & a 7 G c - & ' m 5

; N1 :

L 3 3 d2 -

= > '[email protected]

(]! " ' ¢ a a Q ¤ n 5 ' ! $ $ % L 4 3 5. c SE Sc a J m $ m 5 ! ']")"" m ')( $ ! % F G c -; F3 m G ¤ 6. F $ "" a ( 5:O! ']" ')(:7X! % = '8>[email protected] B 7. $&% &c - & 1 ( &! ' ( ]! "' ¤ $ m 5_7X! '#")"" ¢ ')( $ ! % - c - f & ' % ' m 5_7X! 8. 7 ( ! ' 7 $ " $ 5n'! % &-

& 3 # ' % ! -;&-21 F G ¤ 9. 1 7 7 $ "'& (]! ' $ 5n'! % & D -;&- F G 3 -;&-!& F G9H = '8> [email protected] 10. & &-

&-21 1 "" $ $ 5n'! % &- & #')(:7X! % ¤ 11. 1 $ "" n 5 ' ! ) ' * ( O ! # ) ' (:7X! % ¤ $ % % 12. &-

& 3 1 $ "" $ 5n'! % &-

& ]')( ! % 3 ]')(* ! % #'T(:7! % ¤ 13. 1 " ( ' V7 ( m m" $ " $ 5n'! % &-

& #' % ! N 7( m !V7<5 m ` 3 ¢ ¤ 14. 5 $ m ^ 1 7 $ 5 7! "" a £! ¢ $ 5:'g5 7! % 1

& ' % ¤ 15. & " " 7 ¤ $ 5 7! a £! ¢ : 5 g ' 5 7 ! ' # ) ' : ( 7! % $ %

% &- 1

& 3 16. 3 : N

# . '/

17.

18.

$ 5 7! "" 5:'g5 7! % $ $ 5_7X! "'& $ 5 7! 5 m $

19.

20.

21.

22.

23.

1

1

a 1 £! &" 5n'65_7X! %

7 ¤ ¢

& ' 7 % 3 7') (: 7X! % ]')(* ! %

&- a 1 4 )£ ¢ 4

& ' C7,7 5 ! N ( ' V 7 ! ( a m % m &- &

$ 5 9! "" (9! " a £! ( 9 ! ' & $9% (P5n'65_7! " c d2 4

& ' % m ¤ & ( $ m ! " $ m 5 ! '#")"" ']" ')( $ ! % (P5_7X! " c F $ G 3 ' % ' m ( P5_7X! ' ' m 5 ! c F3 '$ ]" 5n' $&% 3 d 3 &c - F $ G 43 ¢ ')(:7X! % &-21 ¤ ' ( 7X! " ¢ ')( $ ! % ( P5_7X! " f &-21 c F $ m 5 7G ' % ' m 5_7X!A( P5_7! ' ¤

¢ ' $ L f O SEa c 3 SEa J d ¢ 43fI S

¤

G ¤

F $ m 5 G ¤ m S ( Q( ! " : 5 ' ! $ % G Sc F G a F 3

nk 5.10.7. Sums containing Lλ m pk (ϕ(k, z)) and special functions

1.

2.

$ 5_7! '#" ')(* $ ! % (P5_7X! " & '%& " '65_7X! % 1 L ' & J

] ' " $ 5 7! ')(* $ 5_7X! % (P5_7! " '& " '65 7! % L ' J

F $ ' 3

G 5 5_7 c F $ 7G & I " 5 ! %' & Q(:7X! ' & c -2a 1 F 5 G ¤

a F $ 5_7 G c F $ 5_7 G

& - 1 '%& 9 " 5 ]! ' & c -2a 1 F 5 G ¤ Q(:7! '%& 3

& 3 :1

.

3.

4.

5.

6.

$ 5_7! " " $ ! % F $ 5_7 G

; N1 L

c a 3 & & " $ ! % $ 5_7! c a ¢ 4& ¤ &-

$ 5_7! "" $ 5_7! % a 1 F $ 5 7 G

c a 4 3 4& & " $ 5n ! % c a ¢ 4& ¤ &-

7X! " " & -21 $ 5_ $ ! % 3 &- 4 3Pd 3 F $ 5_7 G c F ')( $ G & ( _ 5 X 7 ! ' 3 ¢ &-21 ¢ &-21 '! % F '65_7 G ('6(*'! " ¢ g ' 5 7 $ ! % $ 5_7X! &c - F , ) "" & -21 $ 5_7X! $ 5_7X! % 3 &- 3 d 3 a 1 F $ 5_7 G &c - F ')( 43 ¢ &-21 ¢ &-21 ( OP'Q5_5_77! '! % a F 'g5 7 G 1 ( Q(*'! " ¢ 6 ' _ 5 7 $ 5n ! % &c - F T , )

'65_7G ¤ $ G '65_7 G ¤

nk 5.10.8. Sums containing products of Lλ m pk (ϕ(k, z))

1.

2.

3.

¢ ' 43 J $ L

=

¡ Sc &

( ¢ ' / Sc 3 / % S l 43 J $L Sc ¢ ' S 4 3 J $L 1

Sc F $ G

'[email protected]

= '8>

(P5_7X! ( ¢ S S l Sc ]F 3 $ G 43 % ( ¢ S 3 3 % !

@CB

3 :,

= '8>

@CB

0 ' /

4.

5.

6.

¢ ¢ &- -21 3:3 43 Q - c - 4 Q F $ 5_7 G &'%& ( ' : ( 7 V ( Q : ( 7 V( (*')(:7 ( ! '6_ " " 5 7X! % (P5_7!A 65:'g5 7! D & N L J '( 6(*'! " ¢ &-21 65_7X! " c F $ 5 7 G F $ 5_7 G & ' '& " ' ( 7 ! '65 7! % ( P5n'65_7X! 65 7! ( 5:'g5 7!Ij + c lC- c - - &- . F ¢ ' a 1 $ 5 7! "" ( g5 7X! " 43& c F $ 5 7 G a 4 3 & &!" $ 5 7 ! % ( 6 5 7! " a &-

3 ¢H ¤

G ¤

¢ 4& ¤

"

7.

J L 5_7 c - ¢ Q a 4 3 4 & $ & $ ( 5_9! 7X" ! % F3 G a 4 ¢ 4& ¤ & -

5.11. The Gegenbauer Polynomials Cnλ (z) λ nk

5.11.1. Sums containing Cm 1.

2.

3.

(z)

('! " '! " S 5 ! $&% $ % S - - 0 & ')( S '65 43 ¢ S ( % ! ' ! ' % £ & S - N '6 5_7

1 ! m !" ¢ ' C7 ( 9! " S c - 0& $ L 4 3 J ( (QS S V7 (Q( m ( 5n' 7% ( 6( m ( ! ' 1 ! ( 9% ! (C7C(% 6( ! ' ( Q ( 5n' #7(Q( m (

7 &N

S "1; - ¤

!

" ¤

(9! " (&! ( (P5 ! " c S a 0 & 4 3 ¢ SEa 1 -

1 ( J L£ . a ¢ Y D j SE+ -!a &4 l c &-21 3

<3 j ES + -!a &4 l c -!&4 . ¢ 3 £

H ¤ 1 1 3 : 3

.

4.

5.

6.

7.

8.

9.

¢ ' ( 9! " a ! 43 J $ L m " c S & ( Q( 43 ¢ S (&! ( m % m !'

!'

;

&N

(

P5 # Q( m 5 6( m 5 (*'65_7 1

!

5_7 ¤

( &! " ¢ ' ( P5 5 7! " c SEa a 0& $ L 4 3

1 J # ( & ! ( & . 3 ¢ a 1 j S + c -;a &-21 l a 1 £ 3 ¢ = > '[email protected] B & ( J L ( 9! " ( P5 5_7! " c SEa a &

1 &! ( & P5 -21 D (P(5n '! ( & c SEa a & <3 6(87 c SE-2a 1 & H ¤ ¢ ' ( 9! " a ! 43 J $L m " c SEa 1 & ( # P5 5_7#6( m 5 5: ( Q( (:7! ' 3 ¢ S £ ( 9! ( & m ¤ 6( m 5 (*'65n & % m !' &N

! ( &! " ¢ ' c SEa a 0& $ 4 3 L P 5 5 J 1 L " 1 J

( ( P5_7! ( . £ 1A- S d2 J 1 L ' j S + c a &-21 lM1 -; £ 3 ¢ ¤ 5 & ( 1L( &' J L J

( 6( L " ' J1 ¢ 3 d2 3 ¢ - c - 0&

$ L

SEa 1 J ( & ( &! ( & 5_7X! % ('6( ! '

Y ; ¢ 3h O S 3 3 F 1 Y j SE+ - a c a - S -21Rl -; & 3 :95

7 3 d 3 G ¢ h 3 O a 1 . 5(877 5n'8> @CB , = )

10.

' J $ L

0 ' /

(9! " &! ( & ¢ c a &

3 E S a P5 5 1 " 5_7! % 5 1 L ( & '

1 L J J

¢ ¢ Y 0d 3h

3 4 3 1 5 '*> @CB Y j SE+ c a a -2 1 l -; a 1 . £ 3 ¢ = n &

λ 5.11.2. Sums containing Cm

1.

2.

3.

4.

5.

6.

9! ' c 0& ' % P 5 1 J

5 $ ¢ 43 ')( $ ! % 5 J

( ' (')( (*')(

(87 1 ¤ F G &N

7( ' (% V( ')(*! 1;- L' &! 1 (9! ' & " ¢ 3h J ( PL 5n c & ' 5 6 ' _ 5 7 ! % "

1L" 1L' J

= '*>[email protected]

( 9! ' & $ 5 9!&'! " ¢ ')( $ ! % (Q(*'65_7X! " (P5n'65_7X! " c 0& ' % ( &! ' & ¤ 4 3 $ 5 P5_7X! & L" ¢ J

43 ')( $ ! % 5 1 L " ( P5n'65nO! " c a 1 & J

£ d2 ( 5 7! ' & ¢ 3h ¤ ' % 5 1L' J

( ( 9 ! C 7 ( * ( 9 !" " ¢ ' £) £! ')( (P5_7X! " S c & $ L 4 3 3 d 3 -

J 3 £ 3 £ d2 43 3 d2 a ¢ 3h c a & = > O'[email protected]B S - 1 Q(*'! " C7 (* Q(*'! " 43 ¢ $ L0d 3 £) ')( $ ! % ( (* c 0& ') ( P5_7X! " &-

J 3 £ S 3 £ 0d2 4ST3Pd 3 4SEa ¢ 3h

S c a2S & = '[email protected] 1 &- S

λ 5.11.3. Sums containing Cm

1.

pk (z)

nk pk (z)

")" ' & -21 ')( $ ! ')( $ ! % C7 (& ! " 63 ¢ - c - 0& ( ('! " Q(*'! " (:7 ¤ ' J % C 1 7( L &' ! F 7(8 G F G ' (*'65 1 Q " L J

3 :;

.

; 3

7 $ ')( $ ! C7 (*&! " ')( $ ! % C 7(%9! " F 7(8 G c - & F 3 d G ( 7 ) : 1

2.

3.

4.

5.

6.

7.

8.

m !" C7 (& ! " $9% )" " C7 ( &! "

,

= '*>[email protected]

( ' m (

7! ' 1 ! £ - ¢ 3h& - c - & m 5_ ¤ ' % &N

m 5 7 # 7( 1 -

¢ 3h& - c - & 7! % . 5 2 CP7 5_(87 ! D ¢ 3 ('65_ &! '%& j + -a c -21Rl c -;&-!&4 F E(:7 GIH 1 (9! " S a2S $ 5 ! % c a 0& &"( % C7 (&! ( S 43 ¢ $ L c - S F $ G SEa J ' (9! " $J L P5 ! " £ 0 ¢ S c a - 0& 7 S 9! ( F 5_ a &-21 lC- c - S -; . F 7<5(8 G G j + c S 5 1 ( L J

('])" "'& C7 ( &! " c - 0&

" 1 (*' &BB B ! 3 ¢ ( &'! '! % £ & SEa N S ( ' 7 ( Q(*' #7 &B BB #7 1

¤

¤

¤

¤

&L ' J 1 L" C7 ( &! " c - & 3 C7J (

&! '%& c -;a &1 -21 & ¤

' & -21 ¢ ')( ')! ( C$7 ! (9 ! $ % 9. " c - 0& 3 ( &! ' 3 ¢ ' % 1L' J

" ;5 7! ' ')( $ ! % C7 ( &! " c - & C7 (&! ' 10.

3 : :

('! " J 1 (*' L" C7(%6 (*'! " -

c -;

) *

I5_7 , ¤

= '8>[email protected] B

0 ' /

11.

7 ¢ ' ' 2- 1 ')( $ ! % C 7 (&! " c - & $ $ L L 4 3 J J

12.

13.

14.

15.

16.

17.

18.

O' '% ! % N ( ' # 7 ( ' ¤ &

1

!

m !" ¢ '*( $ ! % ! " C7E( 9! " c - 0& 4 3

( ' m . ' % ( m ! ! ' N ¤ ! ' & m ( ( ' 5 7 1

1 & L' J L" ¢ C7( &! " 3h

- c - & 3 # ('J 9 ! ' & ¢ 3h

-; c -;a & ¤ 1

( L' 7 ¤ 1 ¢ J ')( $ ! % C7 ( &! " 0 3 - c - & ' % (:7 ,

1 L' ) J

* ( ' 1 L " ')( $ ! % C7 J ( &! " C7 ( 6( '! " - c - &

( ( #$ 1;- 1 ' J % 1 ( &L ! ' - N ¤ '

1 1 &

&

! 1L" ¢ ')( $ ! % J m ! " C7(&! " c - & 4 3

7 ! j + c a !- &4 lC- -; a 1 . £ 3 ¢ ¤ m ' " ( " '#" (:7 L " 1 L " ' J1 ¢ ¢ C7( & ! " ')( J $ 5_7X! 0 3 - c - & 43 '65_7 J $ L

&L ' . £ ¢ ¢ Y ]C7 J ( 9! ' & c -;a & -21 0& 43 j + c a -;&!- &4 lC-;&!- &4 3 ¤ 1 ¢ C7 ( m ( '! " ')( $ ! % C7(&! " - c - & 4 3

( ' 6( ' #$ !'

-Q1 a 1 ¤ ' % C7 m ( 9! ' - N & m m ( 1 1 !

3 :<

.

; 3

( P 5_7 $ 1 ( L " 1 L " J ( Q(:7X!A $ (*P5 ! ')( $ ! % ' (% 5 J C7 (9! " 19. $ & J (9! ' & L " # C 7 Y ¢ 3h

- c - & 3 ' % ')( Q(:7X!A ')( P5 !AC7 (&! ' ¢ 3h

-;

( ' Q(*' #7 7(8 ¤ Y N ! & c a 1 ( ' c a (*'

20.

21.

22.

23.

24.

25.

( 9! " ¢ ' a 0& $ 4 3 L c S ( J 1 )L " -

J

(9 ! ( j + a c -21 lC-;&-21 . £ 3 S 1L( J

( 9! " ' ¢ c a 0& $

L 0 3 ( J

S -

1 L" J

( 9! ( 3 0 3 ¢ j + c a -21 l -;&-21 . £

S 1 L ( 1 J

=

O ' ' ( $ J 1 7

¢ = > '[email protected]

3 ¢ > ' !

> @CB

1L " 3 ¢ S a - 1 & L $ 5 J ! % ( 1 L)" O' ' J%

7 . ' L D 5n'! % j S + 9lC-;&-21 £ 3 ¢ <3 % j S & H = >_'[email protected] J ( 9! " S a2S $ 5 ! % c a &

&-

"( S % C7 ¢ $ L ¢ 3 SEa c - S F G ¤ ( 9! ( 3

SEa 7( $ J S 5n')(* $ " ( &! " ¢ ( $ L $&% &! " 3h

SEc a a & J &-

5n9! ' ' > @CB &! ' S c 0& c & = * m !" ¢ ' C7 ( &! " c - a S & $ 4 3 L

J 3 ¢ S ( m 5_5n7X'! ' ! % ( 9! ( N ( (*( ' m ( ( ' # 5 ¤ m &

1

! 3 : L

26.

27.

28.

29.

30.

31.

32.

33.

34.

0 ' /

¢ ' J 5 1L " C 7(%9 ! " c - a S 0& $ 4 3 L

J 3 ¢ SEa ( 9! ( j + -;&-21 l c -21 . ¢ 3 £

¤ SEa 1L( J

¢ ' ( P5n 5n'! " C7(&! " c - a S & c SEa & ¤ $ L 4 3 J

m !" ' C7% ( 9! " 0 3 ¢ - c - a S & $ L

J ( # (*' m ( 1 (Q( $ C 7 ( 5 ! ( & ! ( ' m ¤ W 5n'! % 3 ¢ S & N

m ( (*' 1 ! W

-Q1 J & L " . C7( &! " c - a & £ d2 j + &4 l c -;&-!&4 ¢ 3 £

¤

1 ('])" "'& C7 ( &! " c - a &

1 ( ' ( 1 (*' 9BB B 9! ' & £ 43 ¢ ( '6 a 5_7! % & 1 SEa 1 N S '( 6(*' #7 &B BB #7 " ¤ ! " " ' & ;5_7X! c -;a ;5 7 , ¤ ')( $ ! % C7 ( &! " c - a 0& C7 ( &! '

1 ) *

1 I 1 ( L' 7 ¤ ¢ ')( $ ! % C7 ( &! " 0 3 - c - a 0& ' J% & L ' ) (87 ,

1 J

J&L" ¢ C7 ( &! " 3h

- c - a &

1 3 ¢ £ d2; ¢ 3h O -; j + c -;&-21 l &4 . £ 3 ¢ ¤ ')( $ ! ' &-21 ¢ ')( $ ! % C7 ( &! " c - a 0& 43

1 ( '! " ( 1 (*' L " #( &! '%& a 1 3 ¢ J (* '! " - = '*>[email protected] '6 ( ' % & L ' J

3 : P

.

35.

36.

37.

38.

; 3

7 ¢ ' ' 2- 1 ')( $ ! % C 7 (&! " c - a & $ $ L L 4 3 J J

1 £ d2 O' '% ! % N ( ' 7 ( ' P 5_7 ¤ &

& !

' &L " C7(%9! " J ' ( $ 5_7! 3 ¢ - c - a & 'g5 7 0 3 ¢ -;&-21 $ L J

1 . ' L ¢ £ ¢ Y '( J 9 ! ' & c -;a &-21 0& 3 43 j + c a -;&!- &4 lC-;&-21 3 ¤ & 1 m !" ¢ ')( $ ! % ! " C7 (9! " c - a 0& 4 3

1 # 5 7 £ d2 ' % ( m ! ! ' N ( ' ( m (* ' 5_7 ! & ¤ ' & m 6

&L " ¢ ')( $ ! % J m ! " C7(&! " c - a 0& 4 3

1 9! j + c a !- &4 lC- -; a &4 . £ 3 ¢ ¤ m '

'65 7! % $&% C7(&! " c - a & -21 D ¢ 3 C7 (&! '%& c -;a & -21 0& H ¤

1 ( 1 (*' L " J ')( $ ! % C7 ( &! " ( Q(*' ! " - c - a & 40.

1 ( #$ L' 1;- P5_7 1 (% ¤ £ d ' % ( J & 5_ 7X! ' 1A- N &

& &

! ( L" &L" ( P 5_7 $ 1 J ( Q(:7X!A $ (*P5 ! ')( $ ! % ' % 41. ( 5 &J L " ( &! "'& $ J 1 ( L '% & Y ¢ 3h - c - a & 3 ' % ')(* Q(:J 7 !A ')(* 5 !A '( 9! ' ¢ 3h -;

1 (' #7 # Q(*'65_7 7 (8 ¤ Y N ! & c a 1 (*' c a (*' 39.

3<9N

42.

43.

44.

45.

46.

47.

48.

49.

0 ' /

m !" ¢ ' C7 (&! " $ L 4 3 J £ 3 ¢ S (

c - a SEa &

1 m 5_7X! ' (9 ! ( &

(

(*' m ( P5 _ 5 7 ¤ ( ( * ' m & !

¢ ' (P5n 5n'65_7X! " c ¤ c - a E C7 (&! " $ L 4 3 & & E S a a S a

1

1 J ¢ ' J 5 & L)" C7% ( 9 ! " c - a SEa 0& $ L 4 3 J

1 £ 43 ¢ SEa ( 9! ( & j + 1 -;9l c -21 . ¢ 3 £

¤ SEa &L ( J

' n 5 ! $ % C7(%9! " ( P5n ( 'g5 7! " £ - 0 3 ¢ - c - a SEa & $ L J

1 ( Q( ! ' ¢ ( Q! % (*( 9'! ! ( ( C 7 '( 6 (* ! ' 3h

-; c S -; & ¤ 5: $ 5 7! % ' C 7 ( % 9 ! " ( P5n ( 'g5:O! " £ - 0 3 ¢ - c - a SEa & $ L J

1 5_7X! % ( 9! ( & ( 6( (Q(*'! ( ('6J (* 1 (:7 ! L ' ¢ 3h -; c -; 0& ¤ & '

SEa 1 m !" ' C7(%9! " 0 3 ¢ - c - a SEa & $ L J

1 ( ( Q( #$ (*' m ( 1 C7 ( m 5 5n! '' ! ( %9! ( & £ ;0 3 ¢ S N ¤ W &

m ( (*' & ! W

-Q1 ( &! " S a 5 ! a2S $ % c &- a 1 & "( S % C7% ¢ ¢ ¤ ( 9 ! ( 3 J $ L 3 SEa a 1 c SE- a S a 1 F 7( $ G ( 9! " ¢ ' a a 0& $ L 4 3 c S ( ( J 1L" - 1 J

£ ( 9! ( & . j S + a c -21 lM1 -; £ 3 ¢ = > '[email protected] & ( J L 5n'! %

3<

&N

.

50.

; 3

(&! " ' ¢ c a & $ L

3 ( ( 1L" J

S - a 1 J

£ (&! ( & 3 3 ¢ j + c a -21 l -; a 1 . £ 3 ¢ S & L ( 1 J

= > ' ! > @CB O' 1L " ¢ a 1 0& ' ( $ L $ 5 ! J% ( (

3 J

S - a 1 1L" 1 7 O' J ' L D ' 5n% '! % j + 9 lM1 -; . £ 3 ¢ <3 7 % j SEa & H = > '[email protected] S

1 J ')(&! " (& ! '%! & 0& ¤ 3 £) d2 m $9(% C97 ! ( " ((* ' ! c 0 & " m m ' &-

WX\ X\]W . (:7 5n ' & 5n')( $ G &c !- & 0& ] d j + l F G ¤ $9% F3 3 ¢ ')( $&% $ ! % d2 7(* c a 0& ,

)

&- (' V(')( 1 V('65 1 #$ &! ' F G N ¤ (

' 8 ( 7 # 5 1 (* "

& 1 J L'

! 43 ¢ ')( $&% $ ! % d2 7(8 c a & ,

)

&- a 1 " ' (' V(')( 1 V(')( 1 #$ 9! '%& a 1 N " ¤ & ( O')(:7# 5 1 ! (*

&L ' J

$ 5n Q(:7X!A m ! " c -Q1 " ( &! " J 5 L (*' ¢ 3h

c a 0& a ( : 5 g ' & $ %

&-

c & m L" 1L " 1 L " J J J

( ' ]'65 #$

c a 1 ( m Q(:O'7X!! % ' & N ¤ & c a 1 # 6(* m 5 1 ! 7(*

$ 5n 6(87!A m ! " c -Q1 L " ( 9! " ¢ 3h c a 0& J5 ( : 5 6 ' ( ( ' a 9 $ %

&- a 1

c & mL " 1L " 1 L " J J

J

( ' ]'65 P5_7 #$

c a 1 ( m ¤ Q (:'675_! 7'%! & % N & c a 1 Q(* m 5 1 ! 7(*

3< ,

51.

52.

53.

54.

55.

56.

57.

0 ' /

5 7 $ (:7X! c -Q1 L " (9! " a ¢ h & 3 ¢ J & c & 3

5 '65 1 L " 1 (*' L " 9$ % Pn

&J

J

(' 5 ' #$

c -Q1 # : 6(8 '7!! % '%& N W ¤ a & c -Q1 c 1 ! &- &

58.

59.

5 7 $ (:7X! c & 3 ¢ 5n'65 1 J $9% P L " J

J

-Q1 L " (9! " ¢ 3

& c &a ( 1 (*' L " h

(' 6(87! '%&

c '65_7! % & N

c -Q1

λ

5.11.4. Sums containing Cm 1.

2.

3.

4.

nk pk (z)

a 1 & #$ -Q1 # 5:'65_7 W ¤

c a 1 &- &

!

and special functions

']" " (:7! ')(* $ ! % C7 (&! " #& - &- c - 0&

]! " ' 3 ]C7 (% 9! '#" c - a 1 & = '*>[email protected]

& -21 " " " ' ]! ]! ')(* $ ! % '( 9! "'& &- c - a 0& 3 # '( 9! ' c - a 1 & ¤

1 m 5_7! ' m !" C7 ( &! " fI c - & ' %

7 Y D 0f$ ¢ <3 m H N &

' I 3 m 5_7X! N &

( ' m ! m 5_7 1 7( ' m 5 7 m 5_7# 5 7 m 5n m 5n & !

2 m 5_7! ' m !" C 7 (&! " If c - a 0& ' %

1 7 ( ' m P5_7 Y D 0f$ ¢ <3 m H N m 5_7 ! ` & ^ 7(*' m 5_7 m 5_7# 5n ',(P5_7! 3 m 5_7X! N & m 5n m 5n !

3<93

¤

¤

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; 5

)" " M (87! ")" a ')( $ 5_7 ! % C7 (%9! " 0 fg3 c - & F 7(* G 1 5. '%& ( L ' & ( 1L 1 ¢ Y 0fg3 3 '65_J 7 ! % C7(*&! '%& 3 CJ7 ( 9! '%& c a -;&-21 0& 1 ( ' ]6( ' V7#7 #$ (1 L ' ; 1 ! ' % C7 (*J 9 ! ' m ( ' (87! F 7(* G N & Q(*'65 1 m (*'

C7 ( m ! " ")" " C7 ( &! " M(87! 0fg3 c - & 6. ( C7 ( ! ' ' % CJ 7 1 ( &L ! ' (*')m (:7X! F (8 7 G ' m ( ' ]6( ' ( ' (87 #7 Y fg3 3 ¢ fg3 3 ¢ N Q(*'65 m (*' &

1 m ! 1- (' Q(*' m (*')(:7 m (*')(:7 N & Q(*'65 1 m (*' m (*' 1- !

('! " ¢ C7 ( &! " F 3 7.

3 G j &- Q c - & ( ' (' # £ J 1' L % ' 3 ¢ N W W " & 1 (*' 1 ! ( (:7X!

('! " ¢ C7 ( &! " F 3 8.

3 G j &- Q c - a 1 & ( ' (' # 5 7 £ a 1 J 1' L % ' d2; 3 ¢ N W W " * ( '

& 1 & ! ( (:7!

_ 5 X 7 ! $ & J L (* " ' 0 3 ¢ j Q a &4 & 9. : 5 g ' 5 7 ! $ %

&-

1 J L" ( ' ]'65 & 1 'g' 5 % 7 N !A C7 (* ! & 7 #7 ! C7 ( $ 5_7X! & L " ¢ j Q 0 a &4 0& J

3 10. $ 5n'65_7! % ( 1 (*' )L "

&- a 1 J

( ' ]'65 1 '6'5 % !# N !A C7 (8 ! & 7#7 ! C7 ( 3
¤

¤

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11.

12.

$ 5 7!A $ ! % ¢ 3h O j ' 5 & L " 1 (*' L "

) * 6 J J

Y a 1 & £ ¢

&-

$ 5 !A $ 5 7! % ¢ j a 1 h 3 O

* ( ' 6 ' 5 ) L " 1 L " J

J

O'Q5 ! Y a 0& O'! % % ¢ 3

&-

13.

0 ' /

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' 5 L " ( 1 (*' L " h 6 J J Y a a &

&- 1

7 (

N & 7

*

N &

j a 1 ) * ¢ X £ X £ Y

N &

7<5

,

#$ ( ' ] '65_7 1 & 1 1 V7 ! C 7( !AC7 (8 !

( 7<5 , j a 1 * , ) #$ (' 'g5: & 7 & & C7 ( !AC7 (8 !

! 7( 7,5 , j a 1 * , ) ¢ 3 #$ (' '65 & 7 & & C7 ( !AC7 (8 !

! 7( 7<5 , a 1 ) * , j a 1 ¢ 3 R ¢ 3h

, j ) *

¤

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¤

& J 7 ! L % " (*' ¢ 3h O a n 5 6 ' _ 5 $

1) * 1 J L " ( 7! ' ¤ Y a &4 0& 7 * (

&-

1 J L' & J 7 ! % L ( " (*' ¢ 3h

15. n 5 6 ' _ 5 $ 1 )L " 7( J

7<5 a O'65 ! ' 7% ( Y a a

4 & 0 & , 1 ) * , &- a 1

1) * #$ ( ' ]'65 Y N ¤

1 & ! C7( !AC7 (8 !

( g ' 5 1 L)" ')J( $ ! % C7 ( &! " F E(:7G &- Q 16. ( ' ]Q(*' W W #$ ( 1 1 L ' L ' - ¤ ! Y c - 0& £ &A J ' % C7 J (* &! ' 63 ¢ -; N 6(*'65 1 1

14.

3< ;

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17.

18.

19.

20.

21.

22.

23.

; 5

( ' ( 1L" J ')( $ ! % C7 (&! " F E(:7 G &- a 1 Q c - & (' Q(*' W W (% L ' & 1 ' L - £ &A a 1 J ' % M7J (* 9! ! Q3 ¢ -; N

Q(*'65 1 &

'

* ( ' 1 L)" ')( J $ ! % C7 ( &! " F G &- Q c - & ' ( 1;- 1;- &<- " " ' ! % N 1 7(% ! ( 1 (*' L)" J ')( $ ! % C7 ( &! " F G &- a 1 Q c - & '& ( V( 1 a 1;- 1- " " '! % N 1 7 (

! "" 7 (8 ) O')(* $ ! % C7 ( &! " &- Q c - &

C7 (8 " ! ( ' (*' (1 L ' 1 J

¢ !

6( 'g5 1 1 ' % 1 L ' 3h

-; N

""J 7(* ) O')(* $ 5_7X! % C7 ( &! " &- a 1 Q c - 0&

C7 (8 " ! (' (*' (1 L ' 1 £ J

¢

6(*'6! 5 1 & ' % 1 L ' 3h -; N

"" J 7 (8 ) O')(* $ ! % C7 ( &! " &- Q c - a 0&

1 C7 (8 " ! (' ( ')( (1 L ' 1 ¢ £ d J

! a

5 1 1 Q(*'6 ' % & L ' 1 3h

-; N

J " "

7(* ) O')(* $ 5_7X! % C7 ( &! " &- a 1 Q c - a 0&

1 C7 (8 " ! ( ' ( ' ( (1 L ' 1 d J

¢ ! a

5 1 & Q(*'6 ' % & L ' 1 3h

-; N

J

3<9:

#$

¤

¤

¤

#$

#$

¤

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#$

¤

¤

24.

25.

26.

27.

28.

29.

30.

0 ' /

( Q(*'! " £ C 7( ! " 3 & - c Q - & & ' ( ;1 - V( c a 1- c - " " C( ' % ! N 1 7 ( ! ( Q(*'! " £ ¢ C7( ! " - 3h& - c Q - 0& & ( ' ] )(*' #$ ( P5_7! ' 1 ( ' L - ' % C7 (*J ! ' F 7 (8G N 5_7 ( ! 'g5

1

( O! )" " C7(! " F (:7 G c a Q - & &( ' J % C 1 7 (* L' ! F (87G N ( ')(*( '6Q5 (*' )(*' ' & 1 1 ,-Q1

! ( Q(*'! " ¢ C7( ! " 3h

- &c - Q - &

( ' (*' ( 5_7X! ' 1 ( ' C7 (8 " ! #$ L 1 ¢ ! 3 -; N

P 5_7 )(*'65 1 ' % J 1 L ' J

( Q * ( ' ! " C7( ! " ¢ 3h

- &c - Q - a &

1 (' V(')( ( P5_7! ' 1 ( C7(* " ! #$ L ' 1 £ ¢ ! a 1 0 3 -; N

5_7 ( 'g5 1 ' % J& L ' J

( P5_7X! 1 ( $ a L"J1L" J

- c 1 Q c - & G

F 8 ( 7 ) ' ( 5 C 7 ( & ! "

&& L" J

( L ' & J C 1 7% ( 9! ' - c F 7(8 G ( L" &L" a $ ( P5_7X! J 1 J

c 1 G

F Q c 8 ( 7 a ) ' ( 5 C 7 ( & ! "

1 & && L" J

9 1 ( L'& J ( &! ' - c -21 F 7(8 G 3< <

¤

¤

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31.

32.

33.

34.

35.

; ;

7 C7(! " F 7(* G c a Q - &

&(1 L ' J '! % , N 7 8 ( & 1 ) 7 C7(! " F 7(* G c a Q - a 0&

1 & 1 ( L ' J '! % £ & a 1 F 7(* G N & 1 43 Q - c - Q a 0 & &-

2.

3.

( ' #$ 1 * ¤ \] W -Q1

(' V (')( (*'65 1

!

(' V(')( V(')( 1 #$ ¤ \]W )(*'65 1 -Q1

!

( #$ 1;- V ( ¤ ' ! % ' £ & N & 1 7 ( )(*' ! ( 1 W W ( ! ! ' " 1- ! (P5_7X! " c Q a 0& ' % £ & N ¤ _ 5 7 # 7 ( (

'

&-

7X! ' £) 3 3 d2 R43 3 d2 c - a ¢ c 0& 3 ( P5_ ¤ ' % & &-

λ 5.11.5. Sums containing products of Cm

1.

( P5 $ !A( 5n'! " ¢ ')( $ ! % (Q(*'65_7X! " P5n'65_7! " 4 3 9! ' & ' % ( &! '

nk pk (z)

c &

1 ' J 5 L c 0& = = @ [email protected]

1 J L'

$ 5 9 ! $&% 5n')( 1 L)" ¢ ( ' L J " 9! " P 5n'65_7X! " c 0&

43 ')( $ ! % & * J £ d, ¢ 3 £ (9! ' P 5_7X! ' c -21 0& 5 1 Q(:7X! ' L' J

$ 5n Q(:7X!A Q(:7X! " (:7 a , S c - 0& $&% 5 1 L " 5: $ ! ( )" " P5n $ ! '#)" " ) J

7 5:' 9Q! (: > ' ( &(' J ' L SEc a & = 3<9L

= = @ [email protected]

!

c&-a 0 & = @ [email protected]

0 ' /

! " ¢ C7 ( &! " h c - Q a a & 3

&- 1 ( ' ( ' ( 1 # #$ ! ' & a 1 O '6 5_7X! % & N g 5 1 1 C7 (8 " !

! ! " ¢ C7 ( &! " 3h

c - Q a & 5.

&-

#$ (' 1 ( ' O'! ! %' N " & 65 1 1 ! C7(* !

! " ¢ a C7 ( &! " 3h c - a Q 6.

1

&- & (' 1 ( ' P5_7 #$ £ d Q O'! ! %' N " & g5 1 & ! C7(8 !

! " ¢ a C7 ( &! " 3h

c - a Q 7.

1

&- a 1 & ( ' ( ' ( #P5_7 #$ ! ' & £ d Q a 1 '6 1 C7 (8 " ! 5_7X! % & N g5 1 &

! $&% ( 5 $ ! 9! " P5n'65_7X! " c a Q c &

8. & & !' '%% & N ( ' P 5 ! (

1

$ 5 &! $9% P5 1 L " ¢ c a a 1 Q c 0&

J

9. 3 9! " P5n'65_7X! " 1 ( ' L "

&-

J

#$ (' # P5n'65 1 5_7! ' 2 P O'! % & N 5 C7 ( !AC7 (8 ! 1

! $ 5 9! $&% 5 1 L " ¢ c a a 1 Q c &

J

3 10. 9 ! P n 5 6 ' _ 5 X 7 ! ( * ( ' " " 1

&- a 1 L " J #$ (' # P5n'65 & !' & & '6 5_7X! % & N 5 C7 ( !AC7 (8 ! 1

! 4.

3<9P

¤

¤

¤

¤

¤

¤

¤

.

11.

12.

13.

14.

15.

; ;

& (* L " £ 7( J * - ) 7(* , &--

c 1 ( 'L" J

(' ')(*P5 C 7 * ( 9 A ! ( 9 ! ' & Y c - 0& '! % N& 7( #7(*

!

( 9! " $ ( P5_7X! $9% C7 * 5 O! " C 7 (&! " ')(* :

( 9! " & (* L " 7 ( $ ( P5_7X! $9% C7 * £ J

C 7 (&! " ')(* 5:O ! " ( 1 (*' L " - ) 7 (8 , &J ( ' ]')(* 5 ( 9! '%& Y c - &

]O'Q5_7! % & N 7( #7( ( P5n $ !A $ ! % J P5 (P5n'65_7X! " 1 ( ' L J

a Y c a 1

&-

"

1L "

9! " ¢ 3

O

a &4 Q 1 ( #$

WW ¤ 1;1;-

c a &4

a 1 Q # 1 (W $ ¤

W ! 1;1-

7(8 7,5 Q c ) * , c ) * , ( ' c c a 1 #P5n'65 1 ]&O! '' ! & % N !AC7 (8 & # 5 1 1 ! C7 (

(P5n $ !A $ ! % P5 1 J (*' L "9! " ¢ 3 O ( P n 5 6 ' _ 5 X 7 ! ( " 1 J L " 7(8 7,5 Y c a a a 1 Q c c , ) * ,

) *

&- 1 ( ' c c a 1 # P5n'65 & !'& # & '6 5_7! % N & # 5 C7 ( !AC7 (8 1 1

! (P5n $ 5_7!A $ 5_7! % J 5 & L " ¢

3 (P5n'65n ! " 1 ( ' L " 5 7! " J

7(8 7,5 a Y c a &4 Q c a c a , !1 ) * ,

!1 ) *

&-

( ' c a 1 c 5_7 #P5n'65 & (P5_7X !A'!% P5 ! ' ¢ 3h N

7 & C7 ( !AC7(*

& 5 1 P5_

! 3 L N

#$

! ¤

#$

! ¤

#$

! ¤

16.

0 ' /

( P5n $ 5_7X!A $ 5_7! % J 5 & L " ¢

3

5 '65n ! " ( 1 (*' L " P5_7! " (Pn J

7(8 Y c a a a &4 Q c a

1 ) * , c a 1 ) *

&- 1 (' c a 1 c P'65n5_O7! ! %'%& ¢ 3h N

7 &

& P5 1 # 5_

λk+µ 5.11.6. Sums containing Cmk+n (ϕ(k, z))

1.

2.

3.

4.

5.

6.

7.

8.

!

7,5 , 5_7P5n'65 #$ ¤ C7 ( !AC7 (8 !

¢ ' ¡ = '[email protected] 4 3 J $ L S c & ¢ ' S c ¢ & $ 4 3 L J ')( ! " P5 5n'! " ' S -; ¤ % &! ( & ' , G ,G F 3 F 3 a (*'! % P5 1 L ' $ 5n'! % 5n'65 1 L " J J

' ¢ 3 J $L SEc a & S '! " £ a a 0d2 3 £ & ( $ 5n n 5 '! % & S c Q ¤ a (&! ( ¢ ' S S c F ¢ $ G 3 ¢ S £ d2 4S S 3 % £ & S = '8> @CB $ 4 3 L l J 1 ¢ ' ¢ S £ d2 S S 3 9! % ( S S c F $$ 5n * ( G $ L 4 3 h 3 & 3 S l J 1 = '*> @CB (9! ( % ( &! ( ¢ ' c S / 43 ¢ S ! % £ & 4 S S l 3 43 ¢ S % $ L 4 3

J 1 = '*> @CB % ( 9! ( & ¢ ' c SEa / 3 ¢ S 5_7! % £ & SEa 1 S l $ L

4 3 2 1

1 J #( &! ( & '8> @ B 1 3 43 ¢ S = % ( 9! ( ¢ ' S '8> @CB c S F $ G d2 S S l 3 ! % £ & S $ L 4 3 =

J 1 3 L1

.

¢ ' ES a c SEa F $ G £ 0d2 SEa $ L 4 3

1

1 J 1 3 % ¢ ' c S / ¢ ! % 0d2 S $ L 10. 4 3

J 1 ( 7! " ' c SEa / ¢ 3 ¢ S $ L 11. , 7 5 $

1 J 1 9.

; :

1 S l (&! ( & 5_7X! % £ & SEa 1 = '8> @ B &! ( S 43V& S l 3 ! % = '*> @CB ( & % 5_7X! % 0d2 SEa 1 S S l 9! ( & 3 5 7! % = '8> [email protected] B

¢ ' $ L S c F G 3 (& ! ( ! % £ & S

S $ J S -; (P5 ! ( "' £ S O')(* ! " (* '! % ( P5_7! " £ & -

& - a $ 5:'! % ')(* = '*> @CB ¢ ' SEa c SEa F $ G $ 4 3 L

1

1 J ( &! ( & ' % ( &! ( & ( P5 5_7! ( "' £ S 1 (* '! % 3 5_7! % £ & SEa 1 & - a 1 S ')(* ! " Y -; a $ 5n'! % 'T(* ( &! " (* ')(* $ 5_7! £ & - = '*> @CB ( 7X! ( % £ ¢ ' S ¢ c S #F " $ G ! % d2 S S l

43 J $L ( 9! ( 1 3 ! % V& S = '8> @B ( ¢ ' $ & ¢ G 43 J $ L $ 5n ( c SEa 1 F " $ 1 ( & £ SEa S '8> @ B ( 7! 5_ 7X! % % £ d2 SEa S 3 ( &! 5_ 7! % 1 =

1 l $ ¢ ' 43 ¢ S % ! % £ d2 S S S $ L 4 3 & c S * $ 5 ,

l

) J ( 9! ( 1 % 3& S = '8> @CB 3

12. 43 ' % (&1 ! (

13.

14.

15.

16.

3 L,

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

0 ' /

$ ¢ ' & SEa 1 c 5 , $ $ L 4 3

2 1 E S a

1) * J 1 3 ¢ S 5 % 7! % £ d2 SEa S 3 #(&! ( % & 3P& S = '*> @CB

1 l ¢ ' d2 S S 3 &! ! ( % S S 5 $ $ 4 3 L & c S , l

) * J 1 = '*> @CB ¢ ' SEa 5 , $ $ L 4 3 & 1 c SEa

1 )+* J &! ( & 1 43 ¢ SEa 1 £ / 0d2 SEa 1 S l 43 ¢ S 5_7! % SEa 1 = '*> @B 7 $ 7 ¢ ' S S c * " $ , 43 ¢ S d2 SQ - S S $ L 4 3

l ) J (9! ( S 1 8 ' 3 % = > @CB &! '%& ¤ $ "" a ¢ £ ) ' : ( 7! % P5_7X! 9 $ % c 0d2 3 & &- & 3 1 m $ m 5 ! '#)" "" ')( $ ! % C7 ( &! " F OG c - ¢ & ' L ' 5 J !A C7 (&! c -; F ¢ 3 G ¤ ' m m " ) " ¢ ')( $ ! % C7 ( &! " f &- -21 c - f & ' 1 ( ' % ' 5 J 5_7X!ACL 7 ' (* &! ¤ ' m ! )" " ¢ ¢ O')( $ ! % C7 ( 9! " f &- -21 c - 4 f 4& " ' O'! % J ' 1 L ' 5_7!AC7 (%9! ¤ ' m (&! ']" $ ']" c a F $ G O')( O! % £ & &-

& $ % = '8>[email protected] 0 2 d

&-

1 (&! ' $ ']" a '8>[email protected] c a F $ G ')(:7X! % £ & &-21 9 $ % 2 d =

&- 1 1 3 L 3

.

27.

28.

29.

30.

7X! ¤ 1L " a ¢ 3 '6')5_ J

1 : ( 7X! % G

F

$ n 5 '! % 1 (*' L "

&-

1 J

" 1L " ¢ a J

1

F G , 7 5 $ a 5 n ' ! ( ' ( $ %

&- 1 1 L" 1 J

3 '6')5 (:7X!! % ¤ " 1L " a 1 F ¢ G

$ 5nJ ' ! % * ( '

&-

1 1 J L)" 7 O'65 ! 7! 3 '65_ D ')(:7! % ')(*O! % H ¤ " 1L " J a

$ 5n'! % ( ' ( , 7 5 $

&1 L" 1 J

3 '65 1 J ! % L " (*'

n 5 ' $ 1 1 J '65 L

6 ' _ 5 7 A ! 3 #')(

31.

" !AO'Q5 !%

33.

34.

1

a 1F !

" 1L " a J

1

, 7 5 $ n 5 ' ! ( ( ' $ % 1 L"

&- a 1 J 5 !AO'Q5 ! # '65 3 '65 !A #'6 ') ( !% 7 3 ( 1L " $ " J

(*' a 1 F ¢ 5 $ n 5 ' ! $ % m

&-

1 L" 1 "m J

"m 3 ' % ! 3 '65_7X! % ')(:7! N &

$ 1

¢ G

7 O'65 ! D ')(:7! % ')(*O! % H ¤

a 1 F ¢ G

&-

O! # '65_7X!AO'65 ! '65_7X! ¤ 7 ')( O! % 3 3 ')(87! %

32.

; :

F1 ¢ G !AO'65 O! '65 ! ¤ ')( O! % 3 ')(87! %

G

( ' : ( 7')( m ( m m !

" $ " 1 ¢ a " L 5 m $ 5n'! J % ( * (1 ' L " 7<5 $ &- 1 a 1 F

"J

" ( ' (:7'65 3 m ' % ! 3 '65_7X! m % '65_7! N m ( m m &

! 3 L95

V7 1 3 ¢ ¤ G V7 1 3 ¢ ¤

35.

36.

37.

38.

39.

40.

41.

42.

0 ' /

$ 5_7X! "" 1 L "'& J (*' a &4 ¢ 5 n 6 ' _ 5 X 7 ! $ %

&-

1 L" J

$ 5_7X! "" & L " " J

5 $ 5_7! $ 5n'65_7X! % ( 1 (*' L)" 7,? J

Y a &4 a

&- 1 $ 5_7X! "" 1 L "'& a ¢ J

&4

$ 5n'65_7X! % 1 (*' L "

&-

J

O'5 % £) ¢ # ' % ! ¤

¢ !£ ¢ O '6' 5 % ¤

£) ¢ #'6' 5_% ! 7 D ¢ £

H ¤

$ 5_7X! "" & L " " 5 '65_7X! % ( J 1 (*' L " 7,5? $ 5_7! $ n J 7 ]O'65 O! Y a &4 a ¢ £! ¢ £ D ' % #')(:7X! % H

&- 1 $ 5_7X! "" 1 L "'& a ¢ £) ¢ J

&4

$ 5n'65_7X! % 1 (*' L "

&-

O]'6' 5_% ! 7 J D ¢ 7 £ 4 ¢ X £ R £ # 4 H $ 5_7X! "" & L " " J

$ 5n'65_7X! % (')( 1 )L " 7,5? $ 5_7! J Y a &4 a ¢ £) ¢ 'Q5 O! O'65 O!A '65 ! £ &- 1 ' 7 % #7 O') (:7X! % ]')(* ! % & L " $ 5_7X! "'& $ 5n'65_J 7X ! % C7 (*'! " 5 7 ! 5 $ m #$ (' #7 '65 O 6 ' _ 5 7 m & ! Y a &4 ¢ £! ¢ ' % C7<5 ! N m &

&-

&- & a

" & L " $ 5_7X! "'& J

$ 5 7! 5 m $ 5n'65_7X! % ( 1 (*' )L " 7,5? $ 5_7! J

(' 'g5 V7 #$ 'g5 a ¢ ! £ ¢ Y &4 a ' % C7<5 m ! & N

&- 1 &- & a ! m

3 L;

¤

¤

¤

¤

¤

.

43.

44.

45.

46.

47.

48.

49.

; :

"" " $ 5 9 ! (9! " J P5 1 L " ' 5_7X! " ( 1 (*' L " 7,5? $ 5 9! &$ % ( 5n6 J " ' P5_7X! ' & ¤ Y c a a a 1 ¢ £) d2 ' % & L '

&- 1 J

"'& $ 1L " a ¢ J

* ( ' n 5 ' ! % 1 L)" &- 1 F G $ 1 J O'65_7X!V * ( ' ] 6 ' 5 ')(* ! % £ & N ` & ^ 7 (*' 'g5 & ` 3 ')(:7 & N ^ = '*> @CB 1L " $ "'& $ 5n'! J % ( (*' a 1 a F ¢ G , 7 5 $

&- 1 1 L" 1 J

* ( ' ] 6 ' 5 O 6 ' 5 V ! ')(* ! % £ # & N ` & ^ 7 (*' 'g5 ` 3 ')(:7 & N ^ = '*> @CB ( &! " 5 1 " L " £) a a ¢ £! d2 d2 -21 $9% ( P5n'6J 5_7X! " (*' c 1

1 L " &-

" ' J

P5_7X! ' ¤ ' % 1 L ' J

" # ' " ) " " ( ! ( ! $ $ m m ')( $ ! % C7 ( &! " c - F $ G

' ']" 1 ( ! ' m ¤ ( 7! ' % ' m ( !AC7 (9! ' c -; F " G

"$ '& ( m ! " $ m ( ! ']")"" c - a F $ G ')( $ ! % C7 ( &! " ' ']" 9 1 1 # ( 7 ! ( 7! ' m " ' & m ¤ ' % ' m ( !AC7 (9! ' c -;a FO" G

1 " $ 5 ! '#)" "" ( ! m m 'Q( $ ! % C7( 9! " _& c - ) $ 5* , *

' m ' 5 ( !Am C7]!( 9! c -; ¤ ' )+* m ( , m 3 L :

50.

51.

52.

53.

54.

55.

56.

0 ' /

( m ! " $ m 5 ! ]' ")"" ) ' ( $ ! % C7 (& ! " & a 1 c - a 5 , $

1 )+* ' & m " m ( ! ' & ( 7 ! m ¤ c -;a ' m 5 !AC7 (9! ' ( , m

1 ) * ( 7X! " (9! ' O]! ' ¤ ¢ ')( $ ! % C7 ( &! " f &-21 c - F $ m 5_7 G '! % ' m 5_7X! ( 7X! " ¢ ')( $ ! % C7 (&! " f &-21 c - a F $ m 5_7 G

1

9! ' & O]! '%& ¤ ( '6 5_7! % ' m 5_7X!

$ m 5:7! '])" "" 'g( $ ! % C7,( &! " 43& - ¢ f ¢ 4& c

)

7 75 $ m 5:7! , ( L' 1 O'J ! % ' 5n7X! F3 G ¤ m

$ m 5 7 ! ']")"" ')( $ ! % C7 (&! " 3& - ¢ f- 7 Y c - a

1 ) 7<5h $ m 5 $ 5 ¢ ' c F $ G d2 $ L

4 3 J 1 ¢ $ m 5_7X! '])" "" ')( $ ! % 9! " f ¢ 4 43

57.

¢ 43

58.

¢ 43

¢ & a 1

1 ( ' L '65_J 7X ! % ' 5_7X! ]F 3 G ¤ 7! , m ' ¤ ¢ 3 -; 3 ' %

5 7 $m _ $ m 5_7X! n 5 , ' 1 L' (]! ¤ O'! % J ' m 5_7! P5 1 L ' J

] ' " ' $ m 5_7! ( & ! ' ¤ ')( $ ! % C7 (9! " c - F $ m 5_7 G ' % ' m 5_7X!

1 L ' J

$ m 5 7! ']" ](9! ' & ' & ¤ ')( $ ! % C7 (9! " c - a F $ m 5_7 G ' % ' m 5_7!

1 & J L' 3 L<

# c )

.

; <

λk+µ 5.11.7. Sums containing Cmk+n (ϕ(k, z)) and special functions

1.

2.

3.

4.

5.

('<( 1 L)" ¢ J ')( $ ! % C7 (&! " &- a 1 F ')( $ 5_7 G 3 &- -21

1 '65_7X! '#" 7 Y c - F $ G 3 ' % 6 ' _ 5 7 , a

1 ) *

( 1 (*' L" 'g5 7 £ a G ')( J $ 5 7! % C7 (&! " c - F '65_7 Q 4 1 F

( '<( 1 L" £ a ¢ 1 3 &- -21 'g( J $ ! % C7, ( &! " &- a 1 F 'g( $ 5n7

1 I 7 Y c - a F $ G 3 ' % ¢ &-21 a ' n 5 7 ,

1 ) *

1 ( 1 (*' L " '65_ 7 ¢ £ a J ( &! " c - a F 'g5 7 G ')( $ 5_ 7X! % C7 / Q 4 1 F

1 1 (*' L " £ a ¢ J ( &! " &- F ')( $ 5_7 G 1 3 &- -21 ')( $ ! % C7

9 1 Y c - a F $ G 3 ' % ¢ &-21

F 'g5 7 G

1 1 ( 'L" 'g5 7 ¢ ( &! " c - a F '65_7 G ')( $ J 5 7! % C7 / O F

1 1 (*' L " ¢ ' ( J $ ! % C7 ( &! " &- F ')( $ 5_7 G 3 &- -21 1 '65_7X! '#" 7 Y c - F $ G 3 ' % g ' 5 7 ,

) *

1 (*' L " 'g5 7 £ J G ')( $ 5 7! % C7 ( &! " c - F '65_7 3 Q 4 F

1 ( ' L" ¢ J ( &! " &- F $ 5 7 G c - a F $ 5_7 4 3 V &-21 ')( $ ! % C7

1 ( ' (87 V( ' ( # #$ '%& '%& " '65_7X! % O'Q5_7! ¢ 3 N " 1 ( & 1 1

! 3 L L

G ¤ G

G ¤

G ¤

G ¤ G ¤

0 ' /

1 ( 'L" ¢ J 4 3 V &-21 ')( $ ! % C7 (&! " &- F $ 5_7 G c - F $ 5 7 G 6.

( ' (:7V(')( # #$ ' '%& " 1 ¢ ¤ '65_7X! % '65_7!A(Q(87! N " 3

( 1 ( & 1 !

( ( ' 1 L" ¢ ( &! " 7. 4 3 V &-21 ')J( $ ! % C7 Y a F $ 5_7 G c - F $ 5_7 G

&- 1 ( ' (87 V(')( # 6(87 #$ '%& '%& " & '65_7X! % '65 !A(6(87! N ¢ ¤ " 3

( 1 ( & 1 !

( < ' ( 1 L" ¢ J ) ' ( ! C 7 ( & ! G G

$ % 8. " &- a 1 F $ 5 7 43V

c - a 1 F $ 5_7 ( ' (87 V( ' ( # #$ '%& '%& " ¤ '65_7X! % O'Q5 ! ¢ 3 N " &

( & 1 1

! 1 ( L ' ¢ ¢ $ 5 7! " a J C7(! " F OG c 4 3 Q - & C7 ( ! ' 9. & )(*'! " a ¢ ¤ Y F3 G F]3 G Q c ) ' ( 5 7 ! * ( 6 ' 5 $ % 1L" J

( ) ' ( & ! " ¢ ¢ C7 (! " &c - F ' ( $ 5_7G 10. 43 3 &- -21 1 Y - F $ G 3 ¢ &-21 c F '65_7G

(') ( &! " '65_7 G C7( ! " ')( $ 5_7X! % - F '65_7 G ¤ 3 Q F

( 9! " ¢ a ¢ (')

11. 4 3 3 &- -21 C7( ! " c F ' ( $ 5_7 G 1 & 1 Y - a F $ G 3 £ ; ¢ &-21 c F '65_7 G

1 ' 5 7 ( 8 ' ( & ! " ¢ G C7E( ! " '*( $ 5 7X! % - a F ' 5 7 G ¤ / F

1 3 L P

.

12.

13.

14.

15.

16.

17.

$ 5_7X! "" a C 7(! " c 4 & Y - F $ 5_7 G

$ 5_7X! "" a C7 (! " c 4 &

¢ Q (')(:7 )(:7 ( 9! ' & ! " ]'65_7X! % )(:7X! N V( 1

! ¢ Q - a F $ 5_7 G

1 (')(:7 ( & ! '%& ! " ¢ '65_7! % 3 N 1

! " " $ 5_7X! C7(%9! " 43 Q c a 4 3 Q c - F $ 5_7 G

& Q $ ! ( % & $ ! " 5_7X! c a & $ 5_7X! "" C7 (9! " 3 Q c a 3 Q c - a F $ 5_7 G

1 & d2 Q

( P$ 5_5n7X ! ! " % c a & $ 5_7X! "" a ¢ 4 Q c - F $ 5_7 G C7(%9! "

& (')(:7 # Q(:7 ! '& V( 1 ! ]'65_7X! % ( Q(:7! N

$ 5_7X! "" a ¢ C7 (9! " Q c - a F $ 5_7 G c 4

1 &7 ( & ! ( 7X! ' % ' & D '65_7X! % '65nO! % a /

λk+µ 5.11.8. Sums containing products of Cmk+n (ϕ(k, z))

1.

2.

; L

3 ¢ ¤

¤

¢ Q ¤

¢ Q ¤

3 ¢ ¤

:& H ¤

¢ ' ¡ = '[email protected] 43 J $ L S c & ¢ 9! ' (9! ( ¢ ' S ¢ 9G H 43 ' % S l 3 % ! #& S 43 J $L D S c F" $ 1 = I' @ B 3 P N

3.

4.

5.

6.

&' 3 465 78

( 9! " ¢ £ -21 C7( ! " 3 Q (:7! ' ! " Q ' 5_7X! % )(:7! 6

c&-a 4 ¢ ¢ - F $ 5_7 G ( ' 8 ( 7 P5n')(:7 (87 ¢ 3 N W ( 1 V( 1 ( 6 &

!

(9! " ¢ ¢ £ ¢ -21 C7( ! " 3 Q c a 4 - a F $ 5_7 G

1 & ( ' : ( 7 P 5n')(:7 (:7X! ' '6Q5_ W 3 ¢ 7X! % Q& -21 & N

6( 1 1 (

!

(&! " ¢ &-21 C7 (! " c a F $ 5_7 G - a F $ 5_7 G

1

&-

(')(:7V(')( #$ 1W ( O&'6! 5:' & O! % £ Q a

-21 ¢ 3 N

( 6 ( O ' &

1 W

! (&! " ¢ C7 (! " c a a F $ 5_7 G - a F $ 5_7 G

&- 1

1 (')(:7 V( ' ( #$ ( & ! & W '6 5 ' & ! % £ Q a & -21 ¢ 3 N

( 6 ( O ) ' : ( 7 &

1 ! W

¤

¤

¤

¤

5.12. The Jacobi Polynomials Pn(ρ, σ) (z) (ρ 5.12.1. Sums containing Pm

1.

2.

3.

pk, σ qk)

(z)

. . . j S + a l 0 & 5_7 D j SE+ a a 9lV-21 & <3 j SE+ a -21RlV-21 0& H ¤ 1 1 7<5 . ¢ ' . a G j S + -;a 9l a & j $ L + 4 3 ] F 3 = > '[email protected] & l S J 5 65 5_7X! " (:7 a . '

5 5 7! " $ 5_7! F G j S + l & J $ L '65_7X!A 5 g 5 !AC7(*]! 5n'65_7X! . Y D - j S + -21Rl & <3 % 5 5_7% ! ' j SE+ a -21Ra lV-;&-21 . 0& H ¤ 1 3 P1

.

4.

5.

6.

7.

8.

9.

¢ ' 5 g5 5 7! " a . 5 _ 5 7! " j S + l 0& 43 J $ L ( !' 5 5_77! ! ( ( n 5 ' '65 7! ( j ¢ ' $ 5:' (87! % . 5 ! j 43 J $ L $ % S + l V- & 7 ')(:7X! % 65:5 ' ! % 5_7X! ' F (8 G j S + 9l -; ¢ ' 43 J $ L ¢ ' 43 J $ L ¢ ' 43 J $L

( ( ! " . C7 ( (*'! " j S + a l V- & 5_7 5 g5 5_7 ! ' . 2! ' F]3 G j S + ;- a 9l a &

0& = > '[email protected] B

= > '[email protected] B

' ( ( ! " 7 (8 a . (*6 ' 5_7! " F 7<5 G j S + l V- & J $ L '! % % 5n S + a -;9l-; . & ¤ ( ! ' ]F 3 7,5n G j E

(ρ pk, σ qk) (z) nk

2.

.

. ¤ S + a 9lV- &

. . j S + - l a 0 & j S + l a 0& = > '[email protected] ;- ( T( ! " 5 Q5 5:7! ' . . 5 5:7! " j S + a l - & 5 : 5 7! ' j S + a 9l & ¤

5.12.2. Sums containing Pm 1.

; !, ,

5 65_7! " . ¢ !£ ¢ ) ' ( ! 5 _7 ! " 5 65n'65nO! " j + l & $ % 4 3 - 5 g5 7! ' & 7 (8 ¤ '

% 5_7X! ' F G 5 g5_7! " . £) ¢ - $&% 5_7X! " j + l & 5 g5 7! ' & 5 g5_7X!A

5 g5:O! . ' % 'g 5 5_7!&7X!

+ a l 0& 5

g 5 7 I ! j # _ 5 7 ! % ' & 1 ( ' 5 g5n 5 g5n'65

5 g5 ! Y D ' % ' H ¢ 3h& N 5 g5 1;- ¤ &

5:

!

3 P,

3.

4.

&' 3 465 78

')(* $ ! % ( ')( 1 L " ( (*'! " . J $&%

j + lC- & &-

£ F G j + lC- -; . £ Q 3 ')( $ ! % . 5_7X! '#" " j + lM1A - 0& &-

' % . £ £ 5_ 7 ! j +

Q3 M l 1 ; '

43 ¢ £ 3 ¢ ')(* $ 5_7X! ' $ L ) ' ( $ 5_7!A J

¢ ¤

¢ ¤

. '65_7 $ 5 7! % ' ( $ L 5_7X! "'& j + a lC- -21 & ) J

1 ' O'65 7 ! % . £ 5_ ¢ ¤ X7 ! ' & j + alC- 1 - &-!&4 Q3 . S ( (*'! " $ 5 ! % j + l a & 6. a2S &( &(' " ( S ¤ ( 7 ! % 5n'65_7! ( 3 ¢ $ L 3 ¢ SEa j + lV- S . F $$ ((* : 7 G SEa J . . (:7 G j + - l & F E(:7 G j + -;9l a 1 0& ¤ 7. F (' m

. m 5_7X! ' ¢ m !" 5 g5_7 ¤ + l g5_7X! " j ' % & N m 5_7 g5_7 1 a & 8. 43 !

7 . 7 . ')( $ ! % m ! " j + - l 0& m ! ' j + a -21Rl - -; a 1 0& ¤ 9. 7 5 g5_7! ' 5_7 . ')( $ ! % 65_7! " j + - l 0& ' % g5 7! ' F G ¤ 10. 7 ( ! ' 5_7 . ')( $ ! % 65_7! " F 7 (8G j + - l & ' % 6 5_7X! ' F (:7G ¤ 11. . ( 7X! " $9% g5_7X! " j + - l & 12. '65_7! % .

5 ¢ & -21 D ¢ 43 ¢ ! '%& j + a -;&-21Rl -21 0& H ¤ 1 5.

3 P 3

.

13.

14.

15.

16.

; !, ,

m !" . ')( $ ! % ! " 65_7! " j + - l & 5_7 1 a ¤ ' % ( m ! ! ' N ( ' ( m (*5 '6 g 5_7 6! 5_ 7 ' & m 7 . ' O' -21 ) ' ( ! g5_7X! " j + - l & $ L * $ L $ % J J ( ' 5 g5_7 O' '% ! % N ( ' g ¤ _ 5 7 V 7 a

& ! 1

( * ( ' ! " . ')( $ ! % 65_7! " ( ( )(*'! " F3 7,5 G j + - l & ( 5 g5_7 #$ !' 1;- V(

' % g5 5 g75_ 7! ' F 7,5 G N & a 5_7 g5_7 a . W ¤ 1 ! +1

( ! . £) " 3 I 65_7! " F 7 (8G j + - l & ( ! '%& . 6 5_7! ' F 7 (8 G j + - &-21Rl & ¤

. 7(8G j + - l 0& (' V(')( m 5 ( '65_7 G & N 7( ' ( m (* ' 1;- ¤ !

(')(87! " ( ! " . £) ')( 5nO! " ( ( 2! " F 7 (8G j + - l & 18. 3 I ( ! ( ( ( ! 2' &! F 7(* a 1 D ¢ 3 ¢ ¢ ( ! ' ' & & j + a - &- l . & H ¤ G ' & 1 ( ! " . ')( $ ! % m ! " ( ( 2! " F E(:7G j + - l 0& 19. ' % ! ( ( ! ' ( ! F 7(8 G ' m ' (' a a - a 1

5 ( 'g5 7 #$ ¤ Y N a 1 (* ' (*'65_ 7 m 5 (*' 1- & !

17.

)£ C7 ( 3 I ' % C7 ( m ! ' ( ( m

m ! " ( ! " m ( ! " ( ( 2 ! " F ! '%& ! ' ( ( 2! ' F (:7

3 P95

20.

21.

22.

23.

24.

25.

&' 3 465 78

( ! " . ( ( ! " F 7 (8 G j + - l & ( ' (*' 5 (*'65_7 #$ ' % ( ( !( ' 2! F (: ¤ 7 G &N

' ( '65_7 (*O' 1;- * !

5

g n 5 6 ' _ 5 X 7 ! " . S $ 5 ! % j + a l a 0& a2S &"( S % ( ( )(*'! ( 3 ¢ $ L j + - S l- S . £) & ¤ SEa J 5 g5 7! " £ - ¢ & - j + a lV- . & 4 3 ) ' ( ! C 7 ( ( * ' ! 5 5 g 5 $ % m m " &L " C ( ! J ' 5 g5 7! '

£ ¢ ' % m ! ' m 5 5 g5 & L ' -; & -; J (' m 5 5 g5:'65_7 V( m ( ( (*' Y N ¤ (*'65_7V( ( )(*' 1 a &

!

5

g _ 5 X 7 ! " £ - ¢ & - j + a lV- . & 4 3 ) ' ( ! ! 5

( * ( 6 ' 5 $ % m " m L" J ( 2! '

7! ' 5 g 5 £ ¢ ' % m ! ' m ( ( ( & L ' -; & -; J ( ' ] m ( ( g5n')(*

5 (* m (*'65 ¤ Y N ( g ' 5 7 V ( ( * ( ' 1a &

!

5 65n'65 1 L " . J

')( $ ! % 5_7X ! " £ 0Q3 ¢ j + a l - & '! % . ' % O5_ + l a 0& ¤ X 7 ! j '

J m 5n')( 1 L " 5 g5_7X! " £ ¢ j + a l - . & ')( $ ! % m ! " 5_7X! " 6 3 5 65_7 ¤ ' 7 % N ( ' ] 'g 5: 5_m 7 (87

m ! 1;- &

3 P;

.

; !, ,

m ! " 5 g5_7X! " £ Q3 ¢ j + a l- . 0& ! 5 7 ! ) ' ( ! * ( 6 ' 5 $ % m " 1 "

( ' 1 ( m (*' V( (* ' #$ 43 ¢ m ! ' 5 g5_7X! ' Q3 ¢ N ¤ (1 m L ' 5_7! ' & 7 (* m (* ' ( ( (* ' !

J

1;- C 7 ( * ( ' ! " . m ( ( 2! " F 7 (8G j + - lV- & 27. (' m (:7 5 (*'65_7 ' % m (! ' ( ( 2! ' ! F 7(* ¤ G &N

' m (*'65_7 ! 1;-

" ' . . ')( $ ! % ( ( ! " j + - l V- & ( ( ! ' j + -;9l -; F Q3 G ¤ 28. 7 . ')( $ ! % ( ( ! " ' ( $ 5 m ! F (:7 G j + - l - 0& 29. ( ' V7

5 (*'65_7 ¤ ' % ( ( ! ( ' 2! F (: 7 G N _ 5 7 ' m & m (*'65_7 ! 1;-

5

) * ( 6 ' _ 5 7 ! " . ')( $ ! % (*'65_7! " ( ( ! " j + - l - 0& 30. ( #$ 1;- V( ( ! . W ¤ (' % ( ( ! ! ' F 7(8 G N & ( a + 1;- ' 1;- - ( (

( * ( ' 1 m . L" J : ( 7 G j + - l - 0&

31. F ) ' ( ! * ( ' ( ( 2 ! $ % " & J L" m 5 1 L ' ( ! ' J 7 8 ( G

F m ' % ( 1 L ' ( ( 2! ' ( ' m 5 J (*'65_7 (' m

5 (*'65_7 ¤ Y £ fT3 ¢ N & m ( 1 (*'65_7 ! 1;- & N m 5 1 ( 'g5 7 ! 1-

( ! " . 'g( $
3 P :

&' 3 465 78

('#)" "'& ( ( ! " F 7 (8 G 33. ' % ( ( ( ! ' ! F (: 7 G ' 34.

35.

36.

37.

38.

39.

40.

. j + - l V- & (' 5 ( 'g5 7 9BB B SEa N S = >[email protected] 1 (*'65_7#7 &B BB #7 ! 1;-

m !" . ')( $ ! % ( ( ! " F (:7 G j + - l - & ('

m ! ' ( ! ' 5 (* '65_7 5 g5 m (*'65_7 ¤ ' % ( ( ! ' F 7(* G & N

(*'65_7 #7 ( m (*' ! ,-Q1

( * ( ' ( ! " 1 . L" ')( $ ! J % C7 ( ( O'! " ( ( ! " F 7<5 G j + - l V- & 5 )(*O'65_7 1 a ¤ ' % ! ! ' ( ( 2( ! 2' ! F 7,5 G N ( O'

! '

' &

)(*O'65_7

1 ( (*' L " ( ! " . !$ J % C7 ( ( O'! " ( ( ! " F 7<5 G j + - lV- & ( O' 5 )(*O'65_7 ! ' ( 2! '

% ! ' ( ( 2! ' F 7,5 G & N

)(*O'65_7 ! .. m 5 $ 2! ']" 43 ¢ ')( $ ! % m 5 $ 2! " j + a ]l - + a 1 & ( 7X! ' 5n' 6 5n')(:7j + - -; a 1Rl a m m 5 $ ! ']" .. ')( $ ! % m 5 $ ! " F (:7 G j + - l - + a 1 & 5n' g7 5n')(:7 F (: 7 G j + -;9l a m $ ! % . ' 43 ¢ J $ L 5_7! " j + l - & 5 g5_7X! ' ¢ . ' %

3h& j + a 9lV-; F 5_7X! '

')( '

. . ' j + 2a - S l 0& j ES + a l -; & ¤ $ L J 3 P<

1a

¤

. -21 & ¤ . -21 & ¤ 7,5 ¤ G

.

; !, ,

m !" . ' g5 _ 5 7! " j + a2- S l 0& $ L J 43 ¢ S ( m 5_75:! ' '! g% 5 7! ( N ( (*( ' m (*( ' 6 5_5 7 g5 a 5_7 ¤ m &

! 1

_ 5 X 7 ! ' . "

5 g5: (*'65nO! " F 7 (8 G j + a2- S l & $ L 42. J ( !' ( ( ( (* j S + -;9 l . & ¤ : ( X 7 ! 7 * ( G F ' . . ' 5 7 G j + a2- S l V- & F 5_7 G j SE+ a l -; 0& ¤ $ L 43. F J ' ( ( ! " 7 (8 a . (*'65_7X! " F G j S + l 0& 44. J $ L ( 5 'g5 7! ( 7! ( j + -;9l . & = > '[email protected] B S ' m ( ( ( 5n')(:7! " 7 (8 a . m !" 45. F G j S + - l & J $L 5 g5 (*'65_7 m 5n' 5 7% ! ( N (

= > '[email protected] B &

5_7 m ! 1;-

a . ' . j + l 0& j S + a 9l V-; 0& 46. = > '[email protected] J $L S . ' j S + a l 0& $ L 47. J S F 7 (8 S a . 5 G 43 F (:7G j S + - - - -21Rl -; F (:7;G 1 = > ' ! > @CB 5_7 ¢ ' ¢ a a . 0& $ L 4 3 3 F G j S + l 48. J . 3 3 83 j S + -;a 9l & = > '[email protected] ' 5 7 a a . . G j S + l 0& j S + a 9l 0& $ L = > '[email protected] 49. F J 41.

3 P L

&' 3 465 78

' 5 g5 5_7X! " 5_7 a a . )(*'65_7! " F G j S + l 50. & J $ L ( ( ( ! ! ' j + a 9lV- . & = >_'[email protected] ' S 5 65_7! " 1 ( m L "

. ')( J $ ! % C7 ( m ( '! " 5_7X! " £ 63 ¢ j + a l V- & 51. (

5 g5_7X! ' ( O' m (*' V( (* ' 43 ¢ J 1 ' % m L'! 5_7X! ¢ Q3 & N m (*O' V( ( (* ' ¤ ' m ' ! 1;-

5 g5_7X! "

£ - ¢ & - j + a l V- . & 4 3 52. ) ' ( ! ! 5

( * ( ) ' ( $ % m " m L " J ( ! '

5 g _ 5

7X! ' £ ¢ ' % m ! ' m ( ( ( & L ' -; & -; J ( ' ] m ( ( g5n' 5 (* m ( 'g5 ¤ Y N * ( ) ' : ( 7 ( ( ( ' a &

! 1

C( ( ! "

. ' £ ¢ - j + a l V- & 53. S J $L 1 ( (*' L " C( ( ( ! " - $ ( V( (*' ( ( 5 g5J 5_7X! ( 5_7 S

G % F N

( (*O' V( ( (*! 1 a = > '*>[email protected] 5 g5 5_7X! " ' J 5n'65 1 L "

£ 63 ¢ j + a l - . 0& 5 5_7X! " $ L 54. a2S J 5 g5 5_7 ¤ 5_7X% ! ( N ( (*O' 5_5n7 O'65 5_7 7

&

! 1;-

C( ! " . a j + l - 0& 55. ')( $ ! % C7 ( m (*'! " m ( g5 1 L " 3# ( ' ] m ( g5:' ( m ( 'g5 7 5 g5_7X! ' J 5_7

( 2! '

¤ G ' % m ! ' m ( g5 1 L ' F & N (*'65_7( ( ( ' ! a J 1

( ( ! " m 1 L" . ')( $ ! % J C7 ( m ( '! " ( ( ! " F 5_7G j + - l - 0& 56. ( m L ' ( ! ' 1 43 ¢ ' J % ! ( ( ! F 5_7 G N ( ' (* Om ' ( ' ( O5 '6 5_(*7 'ga 5 7 ¤ ' m ' & m ! 1

3 P P

.

5 g5 5 7! " 5_ 7 ' a l a . & + G j $ L 57. F S ( ' g5 5 7! " J 1 ( * J g5_7! L " ( V( ( ' 5 g5 5 7 ( 43 ¢ S % N ( (*O' g5_7 a 1 = &

!

( 2! " a . +

l j 3 # & 58. ) ' ( ! ! ( ( * ( ' $ % m " & m )" L J

5 65n')(:7 (* m ( ! ' 5 g5_7X! ' F 7<5 G N ( ' ] (*m '6 5_7( ( ( ' ' %m !' m

5 ( 1L' &

! J

( (* ! " £ a a . ' $ L 59. - j + S - l & J 1 ( ( (*' L " J 5 7! ( V( ( (*' V( 5

g n 5 ( 7(8 S ! % F G & N ( ( (* ' ( ( (

60.

61.

62.

; !, ,

> '*>[email protected]

( (*' ¤ a1

( !

= 1;- > '[email protected] B

C7(* m (*'! " 1 L " J 7X! " ( ( 2! " F]3 7,5n G j + - l- . 0& ')( $ ! % C7 ( m (*'! " g5_

, 7 5 ! 5 g _ 5 7 ! ' ' m ' % ! g5_7X! F G ' m ' ( ' 1 ( m (*' ( a 1- - Y N & m 5 1 V( a a 1;- - K- ! . W

+1 a m ! " 1 L " . J +

l- 0& , 7 n 5 G j

] F 3 ')( $ ! % - a 1 L " g5_7X! " ( ( 2! "

J m ! ' ( 2! ' £ ¢ ' % 1;- - L ' ( ( 2! ' 43 -; & -; J

(' a - a 1 a K- 5_7 - Y N W & 7 ( a K- a 1 K- 5 7 ! + 1 a .

C7( m (*'! " . ')( $ ! % 65_7! " ( ( ! " F3 7,5 G j + - l - 0&

( O ' #7( m (*O'

5 )(*'65_7 ! ( ! O'm ! % ' ( ( ' ! F]3 7<5 G N ' m )(*'65_7 ! ( 1 a &

5N N

#$

#$

¤

¤

¤

63.

' J $ L J 7! ( 65_ !%

64.

43 ¢

65.

43 ¢

66.

&' 3 465 78

5 g5: 5_7! " 7<5

j + S a l a . 0& G F 5 65n (*'65 & L "

-

( 5

65n (*'6 5_7] 5n g5 5: N&

= > '[email protected] g5 7 5n 65 (* '65n 1 a !

')( $ ! % 7<5 . G j + l a & $&% 43 T3 F &-

(' V(')( V(')( #$ 7 ¤ 1 ¢ F (8

7 a

G & N ( ' (87 g5_ ! ,-Q1 ')( $ ! % 7<5 . G j + a l a 0& $&% - ¢ F &-

(' V(')(

1 5 g5n'65_7 ¤ 43 ¢ ¢ N (

) ' : ( 7

g5 7 ! 5_7 &

5 g5n '65_7 & - £ 3 ) ¢ 9$ % ( ( ( (* '(*! '" ! " F 3

G ( ( ( O')(:7X! '%& 7(* Y j + l . 0 & 3 ' % F G &!- & (' V( ( ' ( a a a 1 #$ ¤ Y N ( a a a V( a a & &

! & 1

- 5

6 n 5 6 ' _ 5 7 ! " 7 (8 & - ) J & )L $&" % 5n'65_7X! " & j + a l . & G F &-!& ( ' 5 g5n'65_7 #$ !' '%% & N ¤ & a &

1 ! &- &

C 7 ( ( ' ! : 5 O ) ' ( " m m 1L" . ')( $ ! % 65_J 7! " ( ( 2! " F 7,5 G j + !- & l - 0& & ! ' 5nO')( 1 L '

5 g5_7X! ' m J m '! % 6

5_7! ' ¢ & ( ' m ( #7 ( m ( ' V( (* ' #$ ¤ Y N (:7] (* 1 ( O' ( ( (* ' m & m ! a 1

67.

68.

5N1

.

5:O')(

69.

1L" . C 7 ( m ( '! " J m

) ' ( $ ! % 65_7! " ( ( ! " F 7,5 G j + -!& l- & & m !' m : 5 O')( 1 L' ( 2! ' J '! % ( ( ! ' F 7<5 G ( ' ( #7( ( ' 5 (*O'65 7 Y N (:7m ( 1 ( m ' (* '65_7 O 5n ¤

m & m !

5.12.3. Sums containing Pn

(ρ pk, σ qk) (z) mk

7 ) ' ( _ 5 7 ! $ % ( 1. ( ( 7 ! F 7 (8 '%& ( ! ' ' % ( ( ! ' m (

2.

3.

4.

; !, 3

( 2 ! " F E(:7 G a G 1 fg3 ')(:7! F 7(*

C 7( m ! " ( ( ! " F 7 (8 G ' % ( Y 0fg3 3 ¢ fg3

F

¢

3 3

0fg3 j ( ! ' C7( ( 2! ' m

and special functions

. 0f63 j + - lV- & ( ! ' & . 3 ¢ D '6 5_7X! % 3 j + a -;1 &-21Rl-;&-21 & H (' 5 (*'65_7#7 #7 G N ! 1;- ¤ ( 6 ' _ 5 7 ( ' ]

m &

+ - lm !' (*')(:7X! ('

.

0& F (87G 5 ( 'g5 7 m (*')(:7 3 ¢ N * ( &

'65_7 m (*' ! 1;- (' 5 (*'65_7 m (*')(:7 m (*')(:7 ¤ N & ( g ' 5 7 * ( ' ( ' m m ! 1;-

( '! " . G g5_7X! " j &- Q j + - l &

£ J 1' L % ' 3 ¢ N 65_7 (*' (' Va(' ( 5 a g5_7 (:7! " ¤ &

1 1 ! 1

. ( ! " " Ggj + - l V- 0& j F 7 8 ( C 7 8 ( ! & ' ( ! ' j + -;&-21 lV-;&-21 . & ¤ C7 (8 ! 5N,

5.

&' 3 465 78

7 ')(* $ ! % g5 7! " F (:7 G ( !' ' % 6 5_7! '

6.

7.

8.

Q j + - l . &

& -

. #$ ( ' (')( W 7<5 + ,a -Q1 ¤ ! F 7 (8G N

1 1 (*'65_7

7 . ')(* $ 5_7X! % g5_7X! " F (:7 G &- a 1 Q j + - l 0& . #$ (' V(')( W ] ( ! < 7 5 ' + ,a -Q1 ¤ ' % 6 5_7! F 7 (8G N ! ' 1

& (*'65_7

( ! " a . 65_7! " c Q j + - l & & (' V( 6( ' ( (*' #$ 7 ' % 5 6 g5_5_77X! ! ' 43 Q F 5_ ¤ G &N 1 ( ( ( ' (

' ! a . + 1 ('6(*'! " . ( ( ! " F 7 (8G c Q j + - l - & & ('

( P _ 5 7 ! ( 5 (*'65_7 ! - ¤ ' % ( ' ( ! ! ' F 7(* G N 5_7 (*'65_7 '

"

. J L ( 1; - ( ! " c a Q j + - lV- 0& 9. & ' % ( ( ( ! ' 2! F (:7G N (' V( 6(*( '6' 5_ 7 5 (*'65_7 ¤ ' & 1 1 ! ,-Q

5 g5_7! " . £) ¢ 5 g5:'g5:O! " a a a 1 Q j + l 0& 10. 5_7! "

& 5 65 5_77! ! ' & F ( G ¤ ' ( Q * ( ' ! " . g5 7! " F E(:7G c Q j + - l 0& 11. & ( ' ( ' ( + ,-Q1 . 5_7! ' 5_7 ( ' % ! ' g( P a ¤ 5_7X! ' F E(:7G N

5_7 (*! '65_ 7 1 5_7 (:7 . (:7G - V-;&-21 Q j + - l & F (:7G -; F 5_7G ¤ F 12. &-

5N 3

.

13.

14.

7 g5_7X! " F E: ( 7 G c a Q j + &( !' ' % g 5_7X! ' F

; !, 3

. - l 0&

( ' (')( V (')( 7<5 7 (8G & N 1 ( 'g5 7 1 - ! a . +1 5 7 (:7 . (:7G -;& - Q j + - lV- & F (87G -; F 5_7G ¤ F & -

#$

¤

7 ( . £ (9! " ( ( ! " F 7 (8 G c a Q j + - l- &

15. & (' (6( ' 5 (*'65_7 #$ ' ( 9! ' ( ! ' (:7 1 ' % ( ( ! F 7(* ¤ G N '

7 * ( 6 ( O ' * ( ' 5_7 1- &

6 ! 1;- 9! " . ¢ g( 5_ 7X! " c a Q j + - l 0& 16.

3

&-

#$ 5 g5_7 (' 1 ( '

9'! ! %' N " !AM 5_7X! ¤

P 5 g _ 5 7 C 7 (

& 1 ! 1

9! " . ¢ g( 5_ 7X! " c a a Q j + - l 0&

3 17.

&- 1 #$ ( ' ( ' ( 1 5 g5_7 &! ' & a 1 O '6 5_7X! % & N 5 65_7 C7 ( " !AM 5_7X! ¤ 1 ! 1

&! " . c a 0& j + - l V- 0& 18. P 5 P n 5 ' ! " &1L" J

9! ' j + c a -21 l c a -21 . & ¤ 5 1 L ' J

( &! " . ( ( 2! " F (:7 G c a Q j + - l - & 19. & . W #$ ( a a a a ( & ! ( ! W ' ' 19- -P 1 - ! + 19- ¤ ' % ( ( 2! F (*7G N

' & - a 1 - a 7( ( '

5N95

&' 3 465 78

5.12.4. Sums containing products of Pm

(ρ pk, σ qk) (z) nk

1.

2.

3.

4.

5.

6.

. . 5 g5n'65_7X! " 7 (8 65_7X! " G j + l - Q j + a l a & F & ( ' 65 5_7 5 g5:'g5 7 _ 5 X 7 ! ' ' % N& 65_7 5 7 + 1;- . + 1;- . ¤ ! 5 g5n'65_7X! " 7 (8 . a a .

( ( ! " F 7( G j + - l - Q j + l 0& & 65n'65_7 5 7 ! ' ' % N ( ( ' ( ( 5

¤ _ 5 7

& ! 1- 1; ( ' ( 2 ! 7 8 ( " . . 5_7X! " F G j + - l Q j + - l & & ( ' ( ' ( 65 #$ 7! ' 7,5 5_7 5_ ' % F G & N 5_7 5 7 + a 1 . + ,-Q1 . ¤ . !

X+ a 1 5 5 & L " ¢ ' a l a 1 . & j + a &- l &- -21 . 0& J + j

$ 4 3 L S S -; a J 1 ( ( ( ' L " J ( ! ' 5 1 L ' 5_7X! ( 5n 5nO! ' ( ( ! ( " ' 3 ¢ S J % ! 5 5 1 L ' 5_7X! ' ')(* J 5n'65 5n 5:'g5: #$ 1 Y N = > '[email protected] B

&

5:'g5 7 5n'65_7 ! 1;-

5 g5 5_7! " (87 a a . a .

+ l & j + -;9l a -; & j 9 $ % S , ) & 5:' L j + -;9l V-; . 0& ¤ SEa J 5 g5 5_7! " (87 . .

+ a l a & j + a a2S l a -; & j 9 $ % S , ) & 5n' L j + l -; . 0& ¤ SEa J

5.12.5. Sums containing Pm

(ρ pk, σ qk) (ϕ(k, nk

1.

¢ ' . j S + l & ¡ $ 4 3 L J

z))

= '[email protected] 5N;

.

2.

3.

4.

5.

6.

7.

8.

9.

; !, ;

¢ ' $ L j + l . & ¢ F3 G 3 ES a J S 5 g5 5n 'g5 7! " a a a a . ¤ Y a $ 5n'! % F ,G j S + - 9l Q ¢ ' S j + l . F $$ 5n (* G 3 ¢ S ¢ S S l $ L 4 3 h 3 & S J 65_7! ( S '8> @CB 1 3 = % . $m 5 ¢ '$ L 4 3 f & SEa j SE+ a l F $ m 5 G J S ( !" . - ¢ 43fI S F3 mG ( $ 5n + l a a F G ¤ ' ! j % a S "$ " ¢ F3 G j + a l a . ¢ & & $ % &5 g5n'65 7! ¤ 1 3 5nO! ']" # ')

(87! % m . $ m 5 ! '])" "" ')( $ ! % ( ( 2! " F 8G j + - l - ¢ & '

' 5 J !A ( L ( ! j + -;9lV-; . F ¢ 3 G ¤ ' m m ] ' " ) " " . $ m 5 7! ')( $ ! % ( ( 2! " F G j + - l- ¢ f ¢ 4& ' ( ! ' ( L ' % ' 5_7X!AJ ( ( 2! ¤ ' m " ( L . ¢ $ " ¢ J $ 5n '! % j + l

& 5 $ m & 1 "m " (')(:7 g5n' #7 3 ]' % ! 3 '65_7X! m % g5:'! N W 3 ¢ ¤ V ( m m &

!

. ¢ 65n'65_7! " $ 5n'! % F3 ,G j + l O&

& 1 ]7' % ! N (' g5n'65_7 #7 3 ¢ ¤ &

5N :

&' 3 465 78

10.

11.

65n'65_7X! " $ 5:'! % F 3 G

1 5 ' 5n7X! " 8 $ 5*'! % F]3 , G

1

g 5n'65_7X! . j + l ¢

& 3 ')(87! % ¤ &-

. j + l ¢

& & ' 5n7X! 3 Q58' 5:]7!A'g ( 58 ! ' % 58 ! 3 Q 5* 'g(*7X! % ¤

. 65n'65_7X! " $ 5:'! % F3 G j + l ¢

&

&1 65n'65_7!A 65n'65n !A g5n'65 ! ] g5:'g5 7 !A 65n'65nO! 3 #')( !% 7X ')(* ! % 3 g 5n'65_7X! ¤ ')(87! % 3

12.

13.

14.

15.

16.

17.

)£ ! " 65 m 5n'65_7X! " a . ¢ )£ I -21 9$ % m 5n'65_7! " F]3 G j & + - l I & 5 7X! ' ¤ '_ % a . ¢ £! ¢ £) ¢ 65n'65nO! " n 5 6 ' 5 7 ! $ %

F3 G j &+ - 1Rl & !- & '65nO! ' 7 % # 6V5n ')(:7X! % ¤ a . ¢ £! ¢ )£ ¢ 65n'65nO! " 5n'65 7! % F 3 ,G j + 1Rl 4

& 4 - $ & 7 # g5n'65n ! !A g5n'65 ' % 7 ')(:7X! % 65n'6O5n #')( O! % $ 5_7X! "'& g5n'65n ! " a . ¢ £! ¢ 5n'65_7! % F3 G j + 1Rl & $ 5 7! 5 m $ & " ( ' (:7#7 g5:'g5 #'65_7X! g5:'g5 7! ¢ 3 N W % m & 1- 1 a !

. $ ']" a F¢ $ G & $ % 3 g3 j + l &1 5 65n'65_7! ']" 5n'! 3

')(:7! % F G 5N<

! ¤

7 ¤

&-21 ¤

.

18.

19.

20.

. ( $ m ! " $ m 5 ! #' ")"" ')( $ ! % 5_X7 ! " j + lV- F ¢ $ G '#" ' 1 3 ' % ' 5 !A 5_7! j + l-; . F ¢ 3 m ' m . ( $ m " $ m 5 ! '#)" "" ' ( $ ! % g5 7! " 0 j + - l F 5 $ G ' 5 ' 5 m !A( g ! 5 7! j + -;9l . F m ' m m ( ( 7X! " . $ m 5 ¢ + l V ')( $ ! % 5_7! " j f &-21 F $ m 5_7G 7 (8 5 g5_7X! ' ' % '

m 5_7X!A 5_7X! ' F G

5.12.6. Sums containing Pn functions

(ρ pk, σ qk) (ϕ(k, mk

4 3 V

3.

4 3 V

G ¤

G ¤

¤

z)) and special

1 ( 'L" ' ( J $ ! % 5_7! " &- F ')( $ 5_7 G '65_7! '#" ' % $ G

F '65_7 G 1 (*' L " . J ')( $ 5 7! % 5_7! " j + lV- F ¢ '65_7 G = '*>[email protected] (*' ¢ &-21 ')( J 1 $ ! % L5_" 7X! " Y F $ 5_7 G j + lV- . F ¢ $ 5_7 G

&-

(')(:7V(')( 5 #$ ! '%& " 1 ¢ ¤ '65_7! % O'65 7!A

5 2! & N 1 3

! W

( 1 (*' L " ¢ &-21 ')J ( $ ! % 5_7! " Y a F $ 5_7 GTj + lV- . F ¢ $ 5_7G

&- 1 (')(:7V(')( 5 #$ '%& " & ! ¢ ¤ '65_7! % O'65 !A

5 2! & N 1 3

W !

¢ 1. 3 &- -21 1 Y j + l V- . F ¢ '65 7 ¢ £ G 3 Q F 2.

; !, :

5N L

4.

$ 5_7! "" 5 7! "

&' 3 465 78

. a a ¢ Q j + l- F ¢ & - ¢ D ! 7 '& . ¢ a ¢ c 4 Q j + l V- F $ & ]! " '6 5_ 7! % 5 ! ¢ 3

$ 5_7 G

7 -2a 1 1 ]F 3 G 3 '65 7! % H ¤ $ 5_7! "" 5 7 ! _ 5 7G 5. " ( ' (87 5 ¤

N ! (

. ¢ $ 5_7! "" a 5 7! " 43 Q c 4 3 Q j + lV- F $ 5_7G 6. & 7X! " F 3 G $ 5_5 7X! g% 5_5_ 7! " c a ¢ Q ¤ & ('6(*'! " . ¢ ¢ 7. &-21 5_7! " c F $ 5 7 GTj + l V- F $ 5_7 G & % ' & " (')(:7V( 6( ' (87

5 43 ¢ '65 7! % ( P5n'65_ ¢ ¤ 7X!A 5 2! & N 1 3 !

. ¢ ¢ &-21 - -; &-21 F $ 5_7Ggj + l - F $ 5_7G 8. &' " '& a ] ( 7X5 ! 1 - a - -;&-21 F]3 !G ¤ 6 ' _ 5 7 ! , G % 3 F 1 ( (*'! " ¢ a . F ¢ $ 5_ 7G &-21 C7(%9! " c - F $ 5_7 GTj + l 9.

& "'#" 65_7! '%& " '%& '65_ 7X! % 5 5

g5_7X! ' 5n'65_7!A( Q(:7X! ( ' (:7# 6(:7V( ( ' (87 W ¤ Y ¢ 3 N ( ( ( ( O')(: 7 ( &

1 !

¢ 5 g5n'65_7! " -21 C7 ( 9! " 10. F3 G Y c - F $ 5_7 GTj + a l a . ¢ ¢

&- ! " ( ')(:7 # Q(:7 5 g5n' _ 5 X 7 ! ' '65_ 7X! % ( Q(:7!A

¢ W 3 &N

¤ 5 g5:'! V( 1 ! (

5N P

.

; !, <

( (*'! " ¢ &-21 C7 (9! " '%& 5 g5_7X! ' & " L '65_7X! % 5 g5_7X! J 5n ' '65_7!

. c - a 1 F $ _ 5 7 GTj &+ - l a F ¢ $ 5_7 G W ( ' 8 ( 7 # ( (*')(:7 ( #$ ¢ ¤ ! ( ( (* ')(:7 1 3 &N

5 g5n'65_7! " ¢ C7 (9! " -21

12. F3 G Y c - a F $ 5_7 GTj + a l a . ¢ ¢

1 &" - (')(:7 #

5 g5n' 7X! ' ! '6 5 75_ W ! % 5 g5n'! & N

3 ¢ ¤ 1 (

!

9! " & -21 ¢ ¢ -21 &- ( 5_ 4 3 3 13. 7X! " Y c a F ')( $ GTj + l V- . F ¢ $ 5_7G ']" ( &&- ! ' 6 ' 5 7 ! ( &! '

5 g5_7X! ' . ¢ 43 ¢ a 1 5_7! j 6 ' _ 5 7 5 7 ! G G + 3 F F l ; ' ' " ( (*'! " '65_7 a Y ')( $ 5_7! % C7 % ( 6(*'! " ( ( )(*'! " F G c &- F 'g5 7 G ¤

11.

14.

& -21 ¢ ¢ -21 &- a 1 ( 9! " 5_7X! " 43 3 Y c a a F ')( $ Ggj + lV- . F ¢ $ 5_7 G

&- 1 ']" £ 43 ¢ a 1 '65_ 7X! 5_7X! (&! '%& j + lV-; . F ¢ '65_ 7G ' 7! ( & !

5 6 _ 5 ' ' ¢ 5_7! ' / F]3 ,G " ( (*'! '65_7 Y ')( $ 5_7! % C7 (%6 (*'! " " ( ( )(*'! " F G c a a &- F 'g5 7 G ¤

1

1.

¢ ' . ¡ $ L 4 3 j S + l &

J

5)N

(ρ pk, σ qk) (ϕ(k, mk

5.12.7. Sums containing products of Pn

z))

=

'[email protected]

2.

6 !

65 5n'65_7! " ¢ 6_ 5 7! " -21 F G Y j + a l a . 4 ¢ ¢ j + - l . F $ 5_7 3 ¢ G & ( ' 8 ( 7 65 5:' 5 7! ' ! " '65_ 7X ! % 665_ 5 5:'!A 5 ! & N

( 3 ¢ ¤ !

(*'! " & -21 ¢ ( 6 _ 5 7X! " ¢ -21 3 & 3 3. Y j + l - . F ¢ $ 5 7 G j + l a . F ¢ ) ' ( $ G ']" 5_7! ' & 6 ' _ 5 X 7 ! . 3 g5 7! ' 5 7 G j + l -; F ¢ '6_ 6 5 5 7 ! ' ¢ 65_7! ' F G " ( ( '! '65_7

. Y ')( $ 5_7! % ( )( ( " '! " 5_7X! " F + l a & - F ¢ '65_7 G ¤ G j 5.13. The Legendre Function Pνµ (z) 5.13.1. Sums containing Pνµ 1.

2.

k k (z)

£ ' (65_7! " ¢ j $ L h 3 O

& 4 3 a ( * 6 ' 5 J &L " J £ -; ( )( ! ' ¢ 3h

-; j -; 0& = C7 ( ( 1L' -; J

£ ' ( )( ! " ¢ j $ L

4 3 h 3 & ( * ( ' J 1 " L J £ -; (65_7! ' ¢ 3h

-; j -; & = C7 5 1L ' a J

]!

@ B

]!

@ B

5.14. The Kummer Confluent Hypergeometric Function 1 F1 (a; b; z) 5.14.1. Sums containing 1 F1 (a; b; z) 1.

m !' m m 5:' ¢ ' ( m !" ! " 1 N 1 5 ! $ L ! ' 1 N 1 5n' ! L ¤ $ 4 3 L J J J 5)

.

2.

3.

4.

5.

6.

7.

( m !" ( m #7( m ' ! " C7( (*'! " 3P& 1 N 1 5 ! $ L N #7 ( (*!' $ L J J J m m (*' ' (]! " ! " 1 N 1 5 ! $L 1 N 1 L ¤ $ L J J J ! C7 ( ! " ( 5_7! ' m m ( ¢ ' C7 ( (*'! " 1 N 1 ( ! $ L !' $ 4 3 L N

J ( J J C7 ( ! " m ¢ ' ( (*'! " 1 N 1 ( ! $ L $ L 4 3 J J (: 7m ! ! ' ! 3& N m ' ' 1 1J C7( ! " m !' m m 5n' ¢ ' m ( 5 7! " 1 N 1 ( ! $L m ( 5 7! ' 1 N 1 L $ L 4 3 J J J ! m ¢ m ( ' ¤ ' ¢ $ L 3 O - 1 N 1 ( ! $ L 3 O -; 1 N 1 (*' ! L J J J

; 5) ,

¤ L

¤ L

!(*'

5n' ¤ 5n' ! L ¤

5.14.2. Sums containing 1 F1 (a; b; z) and special functions 1.

2.

3.

4.

5.

m! ( m !" a ! 5 $9% " 3& &- ;- &-21 43& 1 N 1 J $ L ( m !" ! 9$ % " ( 7! " ¢ $&% ( 7! " ¢ $&% ( 7! " ¢ $&%

3&

3O

3O

3O

43 ¢ C 7 ( ' % ! ' N m (* ! ' L ¤ 1 1J m ! m !' m 5:' ¤ a 5 ' $ L % -21 3P& N N 1 1J 1 1J ! L & ( m !' m m (*' ' % 1N 1 L¤ -&- & 1 N 1 J ( ! $ L J ! ( m !' m m! ¤ a ' L -&- -; 1 0& 1 N 1 J ( ! $ L ' % ! ' 3P& 1 N 1 J 5n m ( $ a -&- -; 1 43& 1 N 1 J ( $! L ' % m ! ' ! N m 5:5n' ' ! L ¤ ' 1 1J 5)!,

6.

C 7 ( ! " ')( $ ! % m ( 5 7! " 43& - c - 4 3& 1 ' C7 % ( ( ! ' 5 ( m

!

m N 1 J (! $ L &! ' m # 5_7 7! ' -; N (*' Q(*! '65_7 L ¤ J

5.14.3. Sums containing products of 1 F1 (a; b; z) 1.

m ( ¢ O ' ( m ! " C7 ( (* '! " ! " m ( (* '65_7X! " 1 N 1 5n! O')( $ L $ L 4 3 J J 5n' !' ( ! £ m ! m ( ' ! J 1 L ' N m ' m ' & 5:'

m N1 1 J 5 ! $ L W ( m 5n' '65 1 #$ a 5n' 5n ' ! ¤ 1 1

5.15. The Tricomi Confluent Hypergeometric Function Ψ(a; b; z) 5.15.1. Sums containing Ψ(a; b; z) 1.

2.

5 ' ¤ m ¢ m : ¢ ' 5 '! L $ 5 ! $ L 4 3 0 fI n 4 3 L J J J

' ( ! " ( ! J $ L m "

m ( ( 5! $ L J

7! ' m !'

m (*' ¤ J ! L

3.

5 $ ' m 5 $! L $ L 0 I f J J

4.

m m 5n' ¤ ¢ ' $ ( ! $ L 0 fI 4 3 L J J J ! L

5.

m ( ' ! ¤ ' m! ¢ ( ' L * $ ( $ L -; L 0 g f 3 J J J

6.

¢ ' m ( 5_7! " ( * ( '! " 43 J $L

7.

m !" ' m 5 $ ! m ( 5_7! ' ( ' ! " 3P& 5 C 7 ( ! ' $ L J J $ L

J

m! ¤ ' L 5n

m m !' (:X7 ! ' ( ! $ L J

5)3

m 5n' ! ¤ 5n' L J

J

m ¤ (! ' L

.

; !; ,

5.15.2. Sums containing Ψ(a; b; z) and special functions 1.

2.

3.

4.

(]! " &$ % (]! " $&% ( 7! " $&% ( 7! " $&%

m a 5! $ L -21 43& J &m a -&;- -21 43& J 5 ! $ L

0fg3 ¢ & -;- & - a 1 0fg3 ¢ & -- &

m !' 5 ' m n ' % fT3 ¢ L J ! 7X! ' m 43 ¢ m ( ' 5_ % (! J m ! ( ! ' m! ' % ' 5n 0& J ( $ L J m ! ( 7! ' m (*' ¤ ( ' % J ! L J $ L

¤

¤ ' L L¤

5.16. The Gauss Hypergeometric Function 2 F1 (a, b; c; z) 5.16.1. Sums containing 2 F1 (a, b; c; z) 1.

2.

3.

4.

5.

6.

7.

m ! (#! ' m (K5:' ¤ ¢ ' #! " ! 5 ! $ L ' & N J 5n' ( L 43 J $L " N 1 J ! !" m 5 $ ! m !' m 58' ¤ ¢ ' ( m ( ' 5n7X! " N 1 m ( !' N 1 $ L L 4 3 J J J ! L ( m !" ( !" m ¢ ' ! " C7 (*')( m ( 5 ! " N 1 5 ! $ L $ L 4 3 J J ! ! ' 5n' ¤ ' ' m ! 5 ( ! ¢ 3h& N m 5n5n ' L ' m '

1J ! ( ! ' ! : 5 ' ' m " m m m ¢ ! " N 1 5 ! $ L ! ' N 1 5n' L ¤ $ 4 3 L J J J ! m m (*' ¤ ' ( !" ! " F 7 (8 G N 1 5 ! $ L ¢ 3h& -; N 1 $ L J J J ! L m 5 $ 5 $! m 5:' ¤ ' !" !" N 1 5 $ $ L L N

1 J ! L J J m 5 $ 5 $ ! ( !' m 5n' ' !" ! " 63 ¢ N 1 5 $ ! ' N 1 5n' L ¤ $ L L J J J ! 5)5

8.

9.

10.

11.

m !" !" ' !" * ( '! " $ L J C7 ( ¢ ' ( 43 J $ L ' J $ L ' J $L

m 5 $ 5 $! 5 $ N 1J !" m (*'! " N 1 ( ! $ L J ! m ! ' '

! $

m L N 1 J (*' L ¤ !

!' m (:7X! ' 43& N 1 J C7 ( ! " 7(8 m m 5 ( (*'65_7X! " F G N 1 ( ! $ L J C 7 ( ! ' ( ( ! 43& -; N m ' m

1J C 7 ( ! " 7 (8 m ( 5_7! " F G N 1 ( ! $nL J C( 7( 5_! 7X' ! -; N '

5n' n 5 ' 5:' L ¤ !

(*' * ( ' (*' L ¤ !

( ' ¤ m * * ( 1J ' ! L

5.16.2. Sums containing 2 F1 (a, b; c; z) and special functions 1.

2.

3.

4.

. ( m !" ( !" a !" j + -; -21Rl -;&- - a ¢ 3 £ &

& $ % & C7 ( ' % ! ' 0 Q3

L¤

. ( m !" ( !" a !" j + -; -21Rl - ¢ 3 £ & N 1 & $ % & 43 ¢

L¤

. m !" !" ! " 0Q3h

j + -; a A-21Rl a - ¢ 3 £ & N & $ % & 43 ¢ m !" !" ! " 0Q3h

& $ % Y N

1J

. j + -; a A -21Rl ;- a a - ¢ 3 &C7 m 5 $ 5 $! 5 $ L 3 ¢ 5)!;

m! N 1J 5 $ L m ¢ N * (

1J ' ! m! 5 $ L J C7 ( ! ' m (*' ' % N 1 (*' J ! m 5 $ 5 $! 5 $ L 1J C7 ( ! ' m (*' ' % N 1 (*' J !

L¤

£ & ( !' m ( ' (*' ¤ ' % N 1 ( ' L * J !

.

5.

6.

7.

8.

9.

10.

11.

12.

m !" !" ! " 0Q3h

j + - a A-21Rl & $ % & 5 Y N m

1J C7 ( ! " ¢ j + -; & - a 1Rl a 9 $ % 0 Q 3 & C 7 ( ! " ¢ $ 9 % 0 Q 3 C 7 ( ! " ¢ $ 9 % 0 Q 3 C 7 ( ! " ¢ $ 9 % 0 Q 3 ¢ C 7 ( ! " 9$ % j 43 ¢ C 7 ( ! " 9$ % j 43 ¢ C 7 ( ! " 9$ % j 43

; :1 ,

. -; a a - ¢ 3 £ & $ 5 $ ( m !' m (*' ' % N 1 5 $ ! L J ! . ¢ £ m -21 3 & N 1 ( ! $nL J ! ( ! ' m ' m 5n' ' % ! ' 3P& N 1 5n' J ! . m j + -; &- a 1Rl -; a a - ¢ 3 £ & N 1 ( ! $ L J & ( ! ( ! m ' ' m ' % !' N 1 J 5n' ! . m j + -; &- a 1Rl a -21 ¢ 3 £ & N 1 ( ! $ L J & ! ( ! ' m ' m 5n' ' % ! ' 3P& N 1 5n' J ! . m j + - l -; a a - ¢ 3 £ & N 1 ( ! $nL J & ( m !' m (*' ' % N 1 J ! . m ( $ ( $! + a - Rl -;&- - a ¢ 3 £ & N 1 ( $ L J & m ' ! % ' ¢ 3h& N m 5n'

1 J ! . m ( $ ( $! + -; & - a 1Rl - - a -; ¢ 3 £ & N 1 ( $ L J & m' ! % ' ! ! ' 0Q3h

N m 5n5:' ' 5n ' '

1J ! . m ( $ ( $! + -; & - a 1Rl - a -21 ¢ 3 £ & N 1 ( $ L J &5n' m ! ' ' % ( ! ! ' 3P& N m 5n '

1J ' ! 5):

L¤

L¤

L¤

L¤

L¤

L¤

L¤

L¤

13.

C 7( ! " ¢ j + -; &- a 1Rl ;- a a - . ¢ 3 $ & % 0 Q 3 & ( m' ! % ' ! ( '

! $

£ & N

1J !' 43&

m! ( $ L m N 1 J 5n' L ¤ !

5.16.3. Sums containing products of 2 F1 (a, b; c; z) 1.

2.

3.

m !" !" ' $ L m 5 5 1 L)" m 5 ( 'g5 J J

5J $ 5 $ Y N m 5 5 $ 5 !

1 m 1

m !" !" ' $ L m 5 ( 1 L)" m 5 ( ' ( J J

5J $ 5 $ Y N m 5 5 $ ( !

1 m 1

¢ 43 J Y

4.

' J $L m Y

1 L"

m! N 1 m 5 ( 'g5 N & m

1 L"

$ 5 1 m ] m 5 5n m 5 * ( ! '65 1 ¤

m (:7 ! N 1 m 5 ( 'g5 $ ( 1 (:7 m 5 : ( 7 ¤ N m 5n (* 5 ( ' !( 1 m & m

m ! " ! " 1 ( m ( (*' L " ' J

$ L m 5 5 1 L " C7 ( m (*'! " C7 ( (*'! " J

m 5n')( $ 5n')( $ m 5 $ 5 $! ! m 5 5:' ( $ 5 1

N 1 m 5 5 $ 5 1 N 1

' 5 5n' m m ! ! ' ! ! ' m 5n5 !! ' N m 5:5n' ] 5: 5n ! ¤ ' 5 : 5 g ' 5 ' ' ' m m m m &

1

m !" 5 1 L " m ( 5 ! " J m 5 ( '! " F 7 (8G N 1 m 5 1 5 ! $ m ( m ] m 5 ( 1! 1 ¢

3h& 1 & N m 5n (:7 m 5 ! ( ' ¤

5 $

N 1 m 5 (*'6 5)!<

.

5.

6.

; !<

m !" !" 1 ( m ( ( ' L " ' ¢ J

$ L 0 Q 3 m 5 5 1 L " C7 ( m (*'! " C7 ( (*'! " J J

5 $ 5 $ m 5 1 5 1 ! #$ m Y N !

1 m 5 5 $ 5 1 N 1 m 5 5:' ( $ 5 1

' m 5 5:' m ! ! ' ! !' m 5n5 !! ' ¢ 3h& &-21 N m 5n5n' 5n 5: ' 5 !1 ¤ ' m ' ' m m ' m 5 5n6 &

( ! C 7 ( ( * ( ' ! " " m m ' 1 L" ¢ J ( (*

' C7 ( (*'! " m 5 ! " $ L 4 3 m J & L" J

m 5 m 5 1! 1 Y N

!( $

1 m 5 5 $ N 1 m 5 5n') (:7! ' ! ' m 5 ( 1 L ' m ¢ J m ( 1 L ' ! ' m 5n (: 7X! ' 3h& &-21 J

m 5n')(:7 5n' m 5 5n')( 1 ¤ Y N : 5 n 5 ) ' : ( 7 5 n 5 '

m m &

!

5.17. The Generalized Hypergeometric Function p Fq ((ap ); (bq ); z) 5.17.1. Sums containing p Fq ((ap ) mk; (bq ) nk; z) 1.

2.

3.

m ! ¢ ' m !" ! " N a 1 ! ! 5 $ ` $ 4 3 L ^ J (

m ! ( m 5n' m !' ! ' a 1 N a ! 5:' ( ! m ^ ( ( ' & m ! ¢ ' 5 5n'! " ! a N a !" $ L 4 3 1 1 ^ ! 5 $ ` J '! % ( m ! 4 3 ¢ SEa % 5n ! ! ' a 1 N a 1 ^ ! 5n' m ! ( ' ! " '! " ! N a ! 9 $ % 1 ^ ! $ 5_7 ` ' ! O '! % m ' N a !9m 5n!9' 5n' 'g! 5 7 !' 1^ 5)L

` ¤

` ¤

` ¤

4.

5.

6.

m ! ¢ ' C 7 ( ! " m !" N a 1 ! !( $ ` $ L 4 3 J ^ m 5 ! (:7X! ' a N a m ' 1 ^ m ! m 5 $ ¢ ' m !" ! " a 1 N ! ! ` $ 4 3 L ^ J ( ! m ! ' a N a '

1 m ! $ 5 m ¢ ' m !" ! ! 5 ! 43 J $ L " a 1 N a 1 ^ $ ` ( ! m ! ' '

' m !" 7. J $ L 43& ! " ' (]! " m ! " 5_7 ! " 8. J $ L $ (:7X! '65_7 7X! m (:7!

m !95 $ m a 1 N ^ !95 $ ` !

m ! m 5 n 5 ')(87 ! m 5 (:7 ! ` ¤ m ! m m ( 5 7 ¤ ! ( (*6 ' 5_! 7 ` ^ m

m ! m ¤ a 1 N a 1 ^ ! 5:! ' ` m ! m (*' ¤ ! ! ` a 1N ^

m !95 $ m a 1 N ^ !95 $ ` !

m m !(:7 a 1 N ^ !(:7 ` 3 !

a 1N ^

m (*')(:7 m !(:7 ¤ !(:7 ` !

' m !" 9. J $L 4 3& ! " 1 ('! " S 5 ! 10. a2S $ % 43 ¢ S + a a 1 .

( m !95 a 1 N ^ !&5 $ ! m !" 43 ! " a

m !" ( a N !" 1 ^ C 7 "( % C7<5n'! ( C7

5)P

$ `

1 N ^

(

'g5 $ & m !95 !&5 $ m ! ( !0! ( S ( m !0! ( Y a N 1 ^

(*'65 $ & m !95 $ ¤ !95 $ `

!

$ `

43 ¢ J $ L ( (*' & m ! ( !( $ 5 m ! ` ¤ !

.

11.

12.

; !<

m ! " ( 'g5 $ & m !&5n $ J( 1L" !95n $ `

% ) 3 , ! " a 1 N 9 $ ^ ! a N a ( ' ( ' ( 1 & m !

1 ( ')(:7 ! ! m !95 $ m 5n $ (:7 m ! " m ! " ! " ! " N a & !95 $ m 5: $ 5n! $ ( m 5_7 ` ^ 1 m #$ m !95_7 a 1 5_7X! a N a ! m 1 !95_7 m 5 7 5_7 a & ( m 5_7

m !' & ' & 3 m ! '%& ! ' & ! '%& #$ m !95n'65_7 a 1 5n' ! Y a N a !95:'g5 7 5n '65_7 5n '6 5_7 a 5n' ( 5_7 m m 1 &

. ' ( g ' : 5 ] ) ' * ( ! $ $ m + & -21 $L a N ! `

J ^ ! ' £ &-21 a N (' 1 & m ! 3 7,5? ( 7X! ' ' @ ` !

^ = ! ' $L ' ')( ( $$ 55 77! a N ( 'g5: $ ] ') (*! $ 5n m ! `

^ TJ ! £ a N ( ' & m ! ! O

! ! ' $
1^ ! m ! 7 ')( $ ! % $ 5n'65_7! % N a ! 1 ( $! & 5 $

' m ! '6 5_7X! % N a !

! '65 & 1

¤

13.

14.

15.

16.

5,9N

¤

¤

¤

¤

¤

17.

18.

19.

m !95 $ ( ! " m ! " $ ! % ! " N a 1 !95 $ $! 5n ` ^ ' & m ! '%& ! ¢ 3 O ('Q]5n O! % ! ' & a 1 N a

m !95 $ " m !" $ 5 7! % ! " N a 1 !95 $ ] $! 5 ^ m !' & ' %& 3 O'65 ! % ! ' &

^

m !95n'65_7'65_7 !95n'65_7'65n '6! 5

` ¤

m ! N a 1 ^ ! ! ] ` m !95n'65_7 N a 1 ^ !95n'65_7O'6! 5 ` ¤

`

7 O ' 7 m !" G G J $ L F F]3 43

! "

m ( O'65n $ $ ( m !95n $ 1 ¢ ¤ Y a N !95n $

!

20.

( $ & m ! ¢ ' m 5 $ ! ']" ( P5 $ ! " ! a N a m 5 $ ! " $ 4 3 L 1 1 ^ ! #P5 $ ` J ' m ! 5n ' m 6(%5n9!')' (:7 a N a ! (#Q ( m (*! 6 ' 5_7 ` ¤ m 1 1^

5.17.2. Sums containing p Fq ((ap ) mk; (bq ) nk; z) and special functions 1.

2.

( 7! " ' ¢ $ & % $ L I f J C7( ' % m ! '

( $ & m ! a 1 N ^ ! ` ! ¢ a N a ( ' &! m ! (*m ' ! `

1^ m ¢ & -21 $& %m ' (*( '! $ " ! a N a ( $ ! m ! ( 3

1^ m ('65 $ & m !95 $ ' ( ! " m ! " !95 $ & ! J $L 1 L" ! " a 1 N a 1 J

a N a (' m ! 1 1 5,

m! ¤ ' `

! ¤ ! 1

.

; !< 3

m !" ( 'g5 $ & m &! 5 $ # 7 ' !95 $ ] ! ` $ L 4 3 & Q ! " a N a 3. 1 1^ J N ( ' ! m ! ¤ ^ ! ` 5.17.3. Sums containing p Fq ((ap ) mk; (bq ) nk; ϕ(k, z)) ( & m ! ¡ ¢ ' '[email protected] a N ! 5 $ L $ 4 3 = 1. ` 1 ^ J ! 2.

m m !" ' ( g ' 5 9 ! 5 $ $ m 2- 1 ¤ 9 ! 5 $ $ $ L a N ` ! " 1 ^ J ! 1 ( '65 $ m !95 $ $ $ "" m !" !&5 $ $ 5_! 7 ` ')( $ ! % $ ! % ! " a 1 N a 1 ^ 1 m m 5_7X! m 3 ')(* ! % 5 7! ')(:7! %

3.

= '8> [email protected] B

(' 5 $ I m !5 $ $ 5 7X! $ 5 7! "" m ! " '*( $ ! % $ 5 7 ! % ! " a 1 N !25 $ # ! $ h 5 ` ^

4.

m ' % 3 #'*(?7! % ¤

7

5.

( 'g5 $ m 9! 5 $ ' $ m 5 !" m !" 5m _7 ! " a 1 N !&5 $ $ m _ 5 7X! ` $ $ L J ^ !

6.

( ¢ ' S a N $ L 4 3 1 ^ J 1

( ' & m ! 1 C7( ! #$ ¤ ! a N a 1 ! 1 5_7

m ! ¢ ? S l ! ! ` 3 S m !( 3 ! ( 3P& S

5, ,

=

'[email protected] B

7.

8.

9.

¢ ' S a N 1 4 3 J $ L 1 ¢ S S 3 -;

' ( $ V m ! ! f O &- -21 a 1 N J $L 43 fI ! 1 ' 3 &-21 ' 5 a N ( ' ! V ( m ! ¤ m 1 !

( $ m ! ' ¢ -21 &- -21 a 1 N ! J $L fg3 ! a 1 ' ( ' 5 7 ! m m '65_7!& (*'! a N ! m 1 ! a (:7! m 5_7! ' ¢ 3 a '65_7! m (:7X!

11.

12.

m !( m ! ¢ E S a S ! ` 3 1 ! ( ! ^ m ! ( "' S -; ¢ a a 1. ! ( " ' a J $ 5n' L43 + ' ( ( 5_7! Y ' ( ( 5 7! " = > '[email protected] B " m

(

10.

! 1

( ' m ! (:7 ¤ 1 N !(87 ! a 1

( $ & m ! ' m !' ¤ ¢ ' ¢ f &-21 a 1 N ! ' m 5_7 ! ' 43 J $L ! a 1 ( m ! #$ ¢ ' ; 1 ¢ f 4 &-21 a N ! W 43 J $*L ! a . + 1 ! ' J 1 L 5_' 7 m ' ¤ !' m ( m ! ; 1

a 1 ¢ O '65_7 ¢ f 4 a N ! W 3 J $ L ! a . + 1 5,93 #$

¡\¤

.

; !< 5

5.17.4. Sums containing p Fq ((ap ) mk; (bq ) nk; ϕ(k, z)) and special functions 1.

( $ & m ! J 1 (*' L " ¢

')( $ ! % 3# &-21 &- F $ 5_7 G a 1 N ! $&% ! a 1 (:7X! '%& ' & " (')(:7V(')( 1 m ! (:7 3 '65_7! % O'65 7! ¢ ¤ " a N 3 ! : ( 7 (

m (87!

!

2.

J ( ')( 1 L " ' ( $ ! % 4 3V ¢ &-21

& $ % ( $ ! Y a F $ 5_7 G a N ! m

&- 1 1 ! a 1 (:7X! '%& ' & " (')(:7 V(')( & m !(:7 3 '65_7! % O'65 ! ¢ ¤ " a 3 N ! : ( 7 (

m (87!

!

5 7! '#" $ _ &$ % 3 d 3 3. '& " ' 5: ( 7! % ! ( E5* ' 5n7! m

$ 5_7! ""

4.

5.

4 3 1

$&%

( $ & m ! c&- F $ 5_7G a 1 N ! ! a 1 ( 7! ('g(*7]( ( 'g(*7# m !I( 7 a N !I( 7 ( " ` 3 ¢ ¤

^ (*7X! !

( $ & m ! 43 Q 1 N ! ! a ! " m ! 1 " a ¢ ¤ $ 5_7! % ! " c 4 Q & ( $ V m ! ( ( '! " ¢ $&% &c - F 'g( $ 5n7 G a 1 N ! 3 &- -21 ! 3Q ¢ &-21 c F '65_7 G ( $ & m ! ( ! ' '65_7 '65_7 '65_7X! % G a 1 N ! ¤ J $ L43 3 d2 F ! a 1

c&a - 4 3 Q a

5,!5

5.17.5. Sums containing products of p Fq ((ap ) mk; (bq ) nk; ϕ(k, z)) 1.

2.

('65 $ m ( $ & m ! ' 5_7 ¢ 6 4 3 J $ L N 1 ! '% & a 1 a ' 1 N ! ! 1 5 7! m ! ' & ( 7X! ( '6 _ 5 7!A ! ' m ( 5_7 (')(:7 m ! #7( (*' Y m '65 m 3 a N a

1 ! V( m (*' ! a (:7X!A ! ' 1 ( ' (:7 m m ! ' & a 1 N ! ! a ( ! $ ' m & -21 ¢ -21 3 &- a N ! a $ L ! 1 1 EJ ( ' & m ! ('65 $ & ! ¢ Y a N ! & 2 1 3Q a 1 N ! 1 ! a ! ( ! ' Em - ! ' a 1 '65_7 ¢ a a .M '65_7 1 G '65_7 ! ' 1 F KJ $ L 43 A+ C7 ( (*'! " ( $ & Y C7 ( (*'! a N ! " 1 m !

! ¤ 1

5.17.6. Various sums containing p Fq ((ap )+mk; (bq )+nk; z) 1.

!" ¢ ' !" $ L 4 3 J

2.

Y

3.

!

a 1

¤

( !" m 5 $ (K5_7! " & N 5 $ 7 L J ! C7 (#! ' m (*' (K5_7X! ' & N P(*' 7 L = K5 ( m ( ! @ B J ! ' £ £) m ! " m 5n'! " J $L - m 5 1 L " m ! " 1 L " ( 'g5 $ $ J 5 7 $ 5 m $ 5nJ '6 5n m ' % j N & $ 5 & $ 5 m 5 1 $ 5n m 7 0& D m ! ' 0& H

!

= = @ C7 [email protected] ('65 $ $ 5_7 $ 5_7 $ 5:'65 m $9% ! m 5n'! " ')( $ ! % $ ! % 1 L " m 5 1 L " N & $ 5 1 m 5 $ 5 1 ] $ 5n ! 7

J J

'! Y j & 0 ' % m ! % ' 0& ¤

5, ;

.

; L1

$&% ! m ! " m 5n'! " 4. ')( $ ! % $ ! % 1 L " m 5 1 L " m ! " ('65 $ $ J 5_ 7 $ J 5_7 $ 5 m $ 5n'65n m ' %! & 0 ¤ Y N

j & 0 5 5 5 n 5 ] n 5 7 $ $ $ $ m m ' 1 m 1 !

( 'g5 $ $ 5 $ 5 $ 5n'65n #$ ' ! : 5 ' !" m m " & m m £ $ L 5.

N

$ 5: $ 5 m 5 1 $ 5n m 7 m 5 1 m ! " & J ! J L" ' %

Y 0& D m ! ' 0& H = = @C7 [email protected]

n 5 ' ! ( 'g5 $ $ 5 & $ 5n $ 5:'g5 m #$ " & L " m J

6. $ 5_7 m 5 $ 5 1 $ 5 7 ')( $ ! % $ 5_7X! % m 5 1 L " N & J

!'! % Y & 0 ' % m ! ' 0& ¤

('65 $ $ 5 $ 5n $ 5 $ 5:'g5: #$ " " m m ' m ! " m 5:'! " & $ L 7. N $ 5_7 m 5 $ 5 1 $ 5n m ] $ 5 7 m 5 1 m ! " & J ! J L"

' % Y & 0 D m ! ' 0& H ¤ 5.18. Multiple Sums 5.18.1. Sums containing Bessel functions

; $ # $ ? ¡\¢!!£n¤4¤¤

S 7 & 1. <1

U a a WC[ $ % ZZZ 1 ' '#" ( & 7! ( " S ¢ ' % ]! 1 E S a 7

1 0& 2. <1

U a a WC[ U $ % ZZZ 1 ' ' % 7! ! ( S ( 5_$ 7 J 1 " S J % $ 3. -21

U a a X[, ZZZ 1

J

( $ L a 1 &-21 !£ & = '*>[email protected]

' >[email protected] L a 1 <1 £) ¢ && = * 3 ( " 8 M 5 B B B;5 ( ! '%& L

' % M BB BA ( ! Y 1 ¤¤4¤ S ¤

& 2 1 5,9:

5.18.2. Sums containing orthogonal polynomials

; $ # $ ? ¡\¢!!£n¤4¤¤ S j 0& S 0& ¤ 1. Ua a [ ZZZ 1 S S & ¤ 0& 2. U a a X[, ZZZ 1 S 7 0 3# S -21 ¤4¤¤ ¤ $ % 3. S

1 U a a X[, ZZZ 1 S 7 4. U a a [ $ % a 1 ZZZ 1 3 ¢ £ a2S 0 ¤4¤¤ S 1 & S -21 0 ¤¤4¤ ¤ S 1 1 S U a a [ a2S 0 c 5. -21 0 1 ¤¤4¤ S ¤ c ZZZ c U a a X[, ZZZ 1 " S 6. U a a [ $ % - ZZZ 1 5hB B B5 ( ! % S " ' % $ % &- &- = #7 &B BB ' ! 1 5hB B B5 ( ! 5 B B B;5 ( @ B S U [ c 0& c a Z Z Z a c & ¤ 7. U a a X[, ZZZ 1 S ( ! " (* ! " c & $ 8. Ua a [ ZZZ 1 ') (*(' (_(_B B B BB B((* ( ! ( ' ! c U a Z Z Z a c [ & ¤ ' S . + - l - 0& j + -;9l -; . 0& j 9. Ua a [ ZZZ 1 = 5hB B B5 ( ! 5 B BB5 ( @CB

5, <

Chapter 6

Infinite Series 6.1. Elementary Functions 6.1.1. Series containing algebraic functions 1.

"'( & ( m !" 5 ! $ % ! " " ( C 7 ( !0! ( S m !( ¢ 3 ¢ S + a a 1 . % C7 ( m !0! ( 4 3 J $ L N ^ !(

!

$ =

`

@ B

6.1.2. Series containing the exponential function

¢ 3 =

7 7 $&$ K $ ! ' (:7 3 7 3

4-4# $!

' 1 ( ' ' 1 ' ' 1 O'65

1.

2.

3.

4.

5.

6.

K

= = @&7]B [email protected]

$ !

7 5 _7

= = @&7]B [email protected]

7

$

K

7 7 7! 3 ' & 8 5_7

( 7X! ' 7 ' 5 7 ' _ 1

= = @&7]B [email protected]

7 $ 3 7 3 7 7 $ (:7

7X! '

7 7 3 ' & 8 (:7 3 O 6 ' 5 7 !

K

7( $ $ 5,9P

K

$ !

$ K $ !

= = @&7]B [email protected]

= = @&7]B [email protected]

= = @&7]B [email protected]

:1 3

6.1.3. Series containing hyperbolic functions

4-4# $! 1.

3.

4.

5.

6.

7.

9.

10.

= = @&7]B [email protected]

1 ' !(:7 7 $ $ K $ ! ¢ ' 3 3 3 1 $ ' ! 7 ' 7 3 $ 1 ' ! ¢ 7X 7 $& $ ' 3 1 $ $ ! 7( ' ! ' 3

K 1 7( ' ! 7 ( $ ¢ ' $ 3 1 ( 7X! ' £ ¢ 3 7 O 6 ' _ 5 7

2.

8.

K ¢ 3 =

' !(:7 7 $$ K $ ! ' 3 3 K

( 7X! ' O'65_7

¢ 3 1

'65_7X! 7

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= = @&7]B [email protected]

= = @&7]B [email protected]

= = @&7]B [email protected]

= = @&7]B [email protected]

= = @ 7]B 7 [email protected]

( 7 $ :

= = @&7]B [email protected]

7 3 7X

K

( 7X! ' O'65 7! ( &

6 ' _ 5 7 ! ( & !& ( " & S ( 7X! 5 7! % * ( &! % ES a 1 _

= = @ 7]B 7 [email protected]

53 N

P5_7X!

= @ B

6.1.4. Series containing trigonometric functions

( " S - "'& ! ¢ S 3 ] (:7X! % $ 5_7X! % "" S - -21 ¢ ¢ $ ( ! % 4 3 43 ! % £ 3 £ 3 £! 3 £ = ( ! ( ! = @ @CB ( " S - ¢ "'& $ ! ( " $ ! ¢ ¢ S 43 # (:7! % 3 3 $ 5 7! % $ 43 1 S Y - -21 3 ¢ ! % ¢ 3 £ a - SEa & £ 3 £ 3 £) 3 £ = ( ! ( ( ! = @ @CB $ 5 " $ 5 5 5 m ( X 7 ! 1 L 1 m L J J

m 5 $ 5 1 L ( $ 5 1L 5 m J

J

= = @ C7B 7[email protected] ( 7! " ( $ $ 5 m F f G 7 m m f <3 m 0f 1 = ( @CB ( 7 ! " ( @CB F

f G m f <3 7 f = $ 1

$ ! $ 1. $ ( " 1 ( " Y 4 3 ¢ S -21 ] (

2.

3.

4.

5.

6.2. The Psi Function ψ(z) 6.2.1. Series containing ψ(ka + b) 1.

2.

3.

(87 C

1 " $

C

1 " 7 $ 1

¢ 3

¢ 3

7

¢ 3 < 3

¢ 3 C

531

=

[email protected]

=

[email protected]

¢ 3 ¢ 3 3 & ¢ 3 =

[email protected] B

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

7 $ 1 7 $ 5_7X! $ 1 7 $ 5 m !A $ 1

:1 ,

<3 C ¤

¢ 3 5 ! ]

" $ (:7X! $

7 3

= = @ [email protected]

C

7 ¢ ¢ ¢ m ( ! ' <3 f <3

3 C ¢ 3 3 ¢ 3 ¢ ¢ 3 3 ¢ 3 3 ¢ 3

"'& 7 $ 5_7X! 3 ¢ 3 ¢ 3 $ 1 ¢ ¢ ¢ 3 C 3 3 7 7 < 3

$ 1 C <3 3 7 ¤ 3 C $ 1 ( 7 ! " £ 7 £ 3 3 7 7X $ 1 7 £ 7 $ $ 5_7! ¢ 3 , C <3 ¢ ) 1 7 £ 3 $ $ 5_7X! ) 3 , C 3 1 7 C 3 £ ] ¤ $ 3 1 53,

C

0f ¢ ¤

=

=

[email protected] B [email protected]

S -21 7 ¤ $ 1

£ F 7 G ¤

= = @&C7 [email protected] = = @&C7 [email protected]

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

7 $ 1 7 '& $ 1

7

"

7 3 3

' 5n 6

7 '%& $ 5 m ! 1 ( 7X! '%& '65 7! % 1

$&% £

C

<3

] ¤

C

£ 7 3 ¢ 3 £ ¤

7 ' . . ¢ . ¢ ¤ + a f ¢ <3 $ L + & - + a 1 0f f J C

"

7 $9% ! ¢

¡ ¡ 3

£/

3

¤

£ ¤ /

" $ 5_7X! % ¢ 9 $ % 7 £ 3 £ / / £ / E3 £ / £ / ¤ 1 1 " 7 G C £ £ ¢ 3 7 7<5 [email protected] $ F = 7 ( 1 ( 7! " 7 $ F G 3 £ C £ £ ¤ 1 £ C 7 7 G <3

3 = = @&C7[email protected] $ F 1 £ C £ ( 7 ! " 7 @&C7 [email protected] 7X 3 G F G = = $ 1 7 7 £ £ ¢ 5_7X! F G 3 R C £ £ ¤ $ $ 1 7 7 £ £ 5_7X!A $ 5_7! F G 3 3 C £ 3 £ ¤ $ 53 3

27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

37.

38.

:1 ,

7 7 $ 5n !9 $ 5_7X! F G 7 7 3 £¡ £ 3 £ £ 3 7 7 £ £ 2G

3 C $ (87! F 1 7 7 7 F 2G 3 , C 0 ¤ 5 7 ! $ 7 7 $ 5h !; $ 5 7!; $ 5 ! F G 7 7 ¢ g3 £ £ £ 7 #7 ( " 7 £ £ £ G 3? C N & ]! $&% F

7 7 7 G C ,

<3 £ C £ £ 3

9 $ % $ F 1 J1L" 7 7 G , 3 £ C £ £ 0 ¤

$9% $ F 1 7 J1L" 7 m ! " $ F G C £ £ D FOfT3 G 3 fI H 1 7 1L" $ 5_7XJ ! % $ 5_7X! F G 9 £ 3 ¢ C & £ 3 7 1L" $ 5n J ! % $ 5n ! F G 3 ¢ £ £ C ¢ 7 7 1L" $ 5n J ! % $ 5 ! F G F :3 G C 3 7 1L" $ 5 J ! % $ 5 ! F G F 7 3 G C 5395

£

£ ¢

¤

C

£ ¤

¢ £ C £ 3 ¢ ¤

¤ £ 3 ¢)¤

7 F fg3 2 G = m h7 @CB

£ £ £ < 3 ¤

£ 3 4£ £ 3 ¢

¢ 3 ¢

£ ¤

£ 7 7 ¤ 7 3

¤

39.

7 1 J $ L 5_" 7X! F G 3 ¢L£ 9 $ %

40.

1L" 7 G ¢ 3 J

F 5 7X! $ 5_7X! % $ _

41.

7 1L" $ 5_7X! % J $ 5_7X!A $ 5 ! F G F £ 3 G

42.

43.

44.

45.

46.

47.

1L" $ 5n !A J $ 5_7X!A $ 5 ! 9 $ % 7

C

£ ¤

£ £

3 C 3 7X & ¢ 3 £ ¤

7 F G ¢

3 £ ¢

& £ 3 ¢ ¤

C

C

£ 3 ¢ ¡ £ ¤

7 1L" $ 5_7X! % $ 5 J !A $ 5 !A $ 5 ! F G O 3 ¢ 83 " 7 9$ % 1 L " F J 7 £ / 1L" J 5_7X! % F $9% $

"

1 J 1 L "

£¤

G

£

£ /

/ < 3 £

7

G £ 3

C3

£ /

/ ¤

£ ¤

7 7 $ 5 m !A $ 5 m 5 7 ! F G 1 5_7! 7 5_7 ! £ f2 £ m m m

C3

C

£ £ 0 5_7 fI ¤ m

7 F G

7

£ £ £ L / E3 / '

53;

¡!¡

¤

48.

" 5 7 ! % $ 7 D

49.

50.

51.

52.

53.

55.

56.

57.

7

7 F G

J( 1L" J1L"

5_7X ! % $&% $ 1L" J $ 5_7X! % F 9 $ % 1 J 5_ L 7" ! % @ F $ =

54.

:1 ,

/

3 £

/

/

3 7 F G

¢ 3

£ ¤

7

£ 3 G

C3

7

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1 " & L " J $ L 5_ J X7 ! % @ F = 1L" $ 5_7XJ ! % $ 5n ! %

C3

7

7 F G

83

7 1L" &L" $ J 5_ X7 ! % J $ 5n ! % F G P3 J 1 L " 7 7 G 7X 83

$9% ! F