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! i ) 8$ ! x y !$ !9# 7, ( ( , "! $!$ * "$ ! x + iy !"!9# 7 $ !$ x = Re(x + iy) ,
y = Im(x + iy) .
:;;<
( * $$, = "$ ! x1 + iy1 x2 + iy2 6 , x1 + iy1 = x2 + iy2 ,
:;><
( #$ (, $ x1 = x2 , y1 = y2 ?= x2 = x1, ! y2 = −y1, "$ x2 + iy2 !$ !6# 7 ( x1 + iy1 !"!6# 7 $ x1 + iy1 @!$ "$ x + iy = x − iy .
'$!"$ ! !( $$ "$ !$
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; ! z1 + z2 $B "$ z1 = x1 + iy1 z2 = x2 + iy2 !$ !6# 7 "$ z = z1 + z2 = (x1 + x2) + i(y1 + y2 ) .
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. !"7 % (# $ $ !9# ! ! $ ( (! !7 !< z1 + z2 = z2 + z1D %< z1 + (z2 + z3) = (z1 + z2) + z3 > ! (! !7 ( !6 % !9 ( 7 ( #$B ( B $B "$ z1 = x1 + iy1 z2 = x2 + iy2 ! !$ ! "$ z, = z2 +z = z1 5 "$ !$ !6# 7 "$ z1 z2 !"!6# 7 $ z1 − z2 E" $( , z = z1 − z2 = (x1 − x2 ) + i(y1 − y2) .
:;F<
A " #! !" z1z2 $B "$ z1 = x1 +iy1 z2 = x2 +iy2 !$ !6# 7 "$ z = z1 z2 = (x1x2 − y1 y2 ) + i(x1y2 + x2y1 ) .
:;G<
. !"7 $ $ !9# ! ! $ 7 !< z1z2 = z2z1D %< z1(z2z3) = (z1z2)z3D < " : ( ( (! !7< (z1 + z2 )z3 = z1z3 + z2z3 C ! 7 ! ( !6 % !9, 7= #$ (!) $ $ ( 96 B! z2 = 0, ( ! !$ ! "$ z , = z2 z = z1 D ( 7 # * , *( :;G<, %B( H7!$ $ 7# x2x − y2 y = x1, , y2 x + x2y = y1 ,
:;I<
7! $ z2 = 0 ! ($ !6 ( !"$ H7 , #$ $!"$ x22 + y22 > 0 5 "$ z !$ !6# 7 "$ z1 z2 !"!6# 7 $ z1 !% z1/z2 - H79"$ $ :;I<, $!6 z 2
z=
z1 x1 x2 + y1 y2 y1 x2 − x1 y2 = +i . 2 2 z2 x2 + y2 x22 + y22
:;J<
4$ $ " ! $ $B "$ ( " 9 6 B * $"! !7 E #$ "$ $!"!6# 7 7 !! $B "$ , $ ( 9 * $" 9 !69 6 %! 7 * "$ ! z = x+iy " 9 =$$ (! $$ ($!!$ x y 8$ z = 0 ! $# 7 ( ( # "! ($! 6 =$$ @! =$ $ (! #K %( !$ !$ # , # !% $ ) , ! # ($! ) 9 =$$ 4$ # ! 96# 7 !6! ( !"! ( ( # $ 9 B $B "$ z = x + iy $ 9 " =$$, ! ! $ 9 B $B "$ $ 9 #$B , 7$B x y ! !% $ ! ($! ( ( ( 99# x y 7 $!"7 7 "$ ! =$ ! $ !$ 7 7) $$ ($!!$ :ρ, ϕ<,( ρ ) ( !# "$ ( "! ($! : , |z| = x2 + y2 ≥ 0 <, ! ϕ ) , 7$ 96 !( ) (! "$ ( (!$ ! 7 !% $ : , ϕ = Arg z < (!$ ! 7 $ ! ϕ ! !6# 7 ! 7 $ * ($$ $ :−∞ < ϕ < ∞< ' "! , 7 ( # * "$ ! $!"!6# 7 ( !" , !* $!"$ $K ( 7 z = 0 $K " 9 ( %(#)7 * ( (!, = 6 !$ 2π ⎧ y ⎪ ⎨ arctg + 2kπ : . .L !(!$< , x ϕ = Arg z = k − ; :;M< y ⎪ ⎩ arctg + (2k + 1)π : .. ... !(!$< , x arctg !"!6 * !"7 Arctg, % !"7, = % #K, −π/2, $=6 π/2 (! #K " $ Arg, = !"!6 9 # !"# !*!, %( $ $ !$ # $ arg, = %( ) !"!$ ( $ $B !"# Arg $ 9"$ # H7 (! $B ! 7$B ($!, ! $!$ ! $ * "$ ! x = ρ cos ϕ , y = ρ sin ϕ , z = ρ(cos ϕ + i sin ϕ) = ρeiϕ .
:;;N<
4$ $ ! ! ( % ! ( 7 $B "$ " $ ) !$ 7 ! 9 $* $" 9 9 ( ! 7 $B "$
, ( 7 ( % !6 z = ρ(cos ϕ+i sin ϕ) = z1 · z2 = ρ1 (cos ϕ1+i sin ϕ1)ρ2(cos ϕ2+i sin ϕ2) = = ρ1 ρ2 [(cos ϕ1 cos ϕ2− sin ϕ1 sin ϕ2) + i(sin ϕ1 cos ϕ2+ cos ϕ1 sin ϕ2 )] = = ρ1 ρ2 [cos(ϕ1 + ϕ2 ) + i sin(ϕ1 + ϕ2)] = ρ1 ρ2 ei(ϕ1 +ϕ2 ) .
. !! *" ( 7 ( 7 z1 ρ1 = ei(ϕ1 −ϕ2 ) . z2 ρ2
@$* $"! ! ( ! 7 $B "$ ! 6 " 9 $ $ ! ! *%!"$B ! ( 7 ( ( (! ! ( % !7 7 * "$ ! @!, 7= z = z1n , ρ = ρn1 ϕ = nϕ1 √ "$ z1 = z !$ !6# 7 n √ * "$ ! z, 7= z = z1n . # * !"7 $ $ !6, = ρ1 = ρ ϕ1 = ϕn E #$ !* * "$ ! z $!"!6# 7 ( !" , !) ! ( 6 # !"# !*! "$ ! z1 n
n
ϕk =
ϕ 2πk + , n n
% 9# "$ !, 7 $ ( ( n) ( ) 99# ( √ "$ z ( $B $B "$ (! ( 99# ρ, ! !*$ (79# 7 ! "$ , = 6 !$ ( 2πn # # $B !"# 7 7 n * "$ ! z ( ) 96 n @ "$ ! =$, = ( (!9# $ !"7 7, !K ! K$!B ! $ # * * $!, = $ !$ ) √ !( ! ρ " z = 0 '( ( !"7 ϕk $9# 7 $ k, = $!9# !"7 k = 0, 1, . . . , n − 1 n
n
$ %& $ √i%
π i O! $K !$ "$ z = i = e 2 $!6 2πk π i +i 2 , k = 0, 1 :$ ( 7 !(!$B # * "$ ! $!$ zk = i = e 4
;; < O ( $
√ π i 2 π π (1 + i) , z0 = e 4 = cos + i sin = 4 4 2 √ 5π i 2 z1 = e 4 = − (1 + i) = −z0 . 2
6z 6
π
= i = ei 2
π
z0 = ei 4
-
z=0
5π
z1 = ei 4
-$ ;;
√
i
$ '& $ √1& p % p
3 $ 6 # ( ! 7 1 = ei0 @ (, 7 (# $ !(,
$!6
2π i k zk = i = e p , k = 0, 1, . . . , p − 1
O ( $
z0 = ei0 = 1, 2π i 2π 2π + i sin , z1 = e p = cos p p ... 2π 2π −i i (p − 1) 2π 2π = e p = cos − i sin . zp−1 = e p p p
@ % ( (!9# K$! ! $ # * p−$!, = $ !$ ($$" * !( ! " z = 0
$ (& $
E #$
√ 1 − i 3%
π −i z = 1 − i 3 = 2e 3 , √
( 7 !"# !(! * 7 $
$!6 $!
2πk π √ −i + i 2 , k = 0, 1 zk = 2e 6
O ( $
√ π √ √ −i 3−i π π z0 = 2e 6 = 2(cos − i sin ) = √ , 6 6 2 √ 5π √ i 3−i z1 = 2e 6 = − √ = −z0 . 2
@ "! z !$ !6# 7 ' E , 7= 6 ε) ) "$ z, "$ 7 * ! !# $ E ! $ !(, "! z $$ |z| ≤ 1 6 K# 9, 7= |z| < 1D "! z = 1 6 K# 9 " 9 (! $$
( E ) " & # * +, E ) ' )- . ,/ , E 0) & - E . ,%
! $ !(, $! " |z| < 1 96 % ! # $! " |z| ≤ 1 96 % ! #, #$ "$ 6 K$ @ "! z !$ !6# 7 ' " G , 7= 6 ε) "$ z , "$ 7 * ! !# $ E @ "! z !$ !6# 7 " G , 7= ( # ε) "$ z 7# 7 7 "$, = ! !# % ! G , ! "$, = ! !# % ! G 1 1 ) " G , 7 $"! %( !"!$ Γ ( & ) " 1 -- 1 & ) " G = G + −
?= % ! # G $# ($ (7 * ! % * !( !, ! !$ !6# 7 " ' $ $ !( & "
!(! %( * 7(!$ ! $$ E , $ $ 99# % ! # G , !% ! % ! # G =$
E!"7 ' % ! G =$$ !(!! 7 , 7= !(!$ ! , = ! $# ( ( # " % ! G (7 "$ $! G !$ !6# 7 - - 59 ( ( # %( ! $ !$ $* 7( w = f (z) . :;;;< $! $B "$ w, = ( (!9# z ∈ G , !$ !6# 7 f (z) ( !"!$ w(z) = w(x, y) = u(x, y) + iv(x, y) , :;;>< ( ( u(x, y) ! v(x, y) $!" % ! G ( $B $B x, y, = ( (!9# % ! G z O! (!7 w = f (z) ! 96 ( ( # "!$ % ! G =$$ z ! "!$ % ! G =$$ w P% ! !!$, = ! 96# 7 ( %! 7 % ! G ! % ! # G ! ( $ $ % ( 7 " w % ! G ! $# 7 ( ( # (! !% ( #! " z % ! G Q7, = ( 96 ! ( %! 7 % ! G ! % ! # G , !$ !6# 7 " ) f (z) ' $ %( * 7(!$ ! $ !($, $ %! 7 z = ϕ(w) :;;A< 6 ( !" 9 % ! G, % z = ϕ[f (z)] .
2 f (z) ) ) " G & # 1 1 z )- " )
O # * !"7 $ $ !6, = ( $ ! 7 ( 96 !6 () !" ( %! 7 !
"# $
B! 7 f (z) $!"! ! (7 $ E - * 7 ) ( " 6 $$ {zn}, = %*!9# 7 ( (7 "$ z0 = () 79# 7 ( "$ z0 :zn = z0<, ( ( ( $B ( !"#
{f (zn)} 3# "
) ) zlim f (zn) = w0& ) & →z " & - f (z) z0& # n
{zn }
0
lim f (zn ) = w0 .
zn →z0
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)* ! 2 f (z)& # E & ) z0 ∈ E & # )- - z0
)& ) " ) f (z0) - f (z) z0& " zlim f (zn) = f (z0)% →z E!"7 4 z0 ) E & # ) ε z0& ) '1 E % n
0
?= "! z0 6 # ! 9 " 9 $$ E , 7 f (z), ! !) "7, ! !6# 7 9 " z0 ! 9 !$ !$ "$ 2
f (z) z0 & # " ε > 0 δ > 0& # 1 z ∈ E & # |z − z0 | < δ & ) |f (z) − f (z0)| < ε% . f (z) = u(x, y) + iv(x, y) $) $ !6 # ( u(x, y) ! 7 v(x, y) "! $ $B x, y
$ +&
f (z) = w = az + b ,
a b 5 %
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( ! !$ a = 0 Q7 :;;F< $!"! ( 7 B !"7B ! ) z $ 9 !"# 6 ! ! =$! z !"9 z ( (!6 #$ ( !"7 w, % f (z) 6 ( !" 9 ) 69 z E" $( , = %! 7 ϕ(w) = z = a1 w − ab = a1w + b1 !6 ! ! $ , = f (z) @ % , 7 f (z) 6 ( $ 9 69 z
! =$ ! 96 !6 ( !" ( ( # =$!$ z ! w R % 7 $$ * $"$ ! ( ( * 7 ( ) 9 ζ = az !6 ζ = |a|·|z|·{cos(arg a+arg z) +i sin(arg a+arg z)} = |ζ|{cos arg ζ +i sin arg ζ} .
O ( $ $ $ !6 |ζ| = |a| · |z|, arg ζ = arg a + arg z @ % 7 ζ = az %(#)7 "$ z ! $# ( ( # "$ ζ , ( # 7 * |a| ! % #K ! ( # z, ! !* ! $!$ !*! z ( (! !7 ! * ( (!! & !*! * "$ a S $"$ # * 7 6 " $($ 7* =$$ z |a| ! 6 =$$ 7 * ! "$ z = 0 ! arg a 4 !9"$ # ( $ :;;F<, 7 !! ! ! $ !$ $* 7( w = ζ + b, ! %!"$$, = * $"$ !# * 7 7*!6 =$$ z ! b @!$ "$ , ! 7 96 =$ z ) =$ w K 7B 7* (% , !
$ ,&
w = f (z) =
1 . z
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57 7 ! $!"! ! =$, $" f (0) = ∞ ! f (∞) = 0 ? (# $ !( ! ! $$, = f (z) 6 ( !"! ( $ ! 7 z, = ( %! !6 =$ z ! =$ w ! ! $$, = 7 f (z) 6 9 ! ) =$, ! $7 "$ z = 0 7 * $" !
6 $ 6 # !$ 9 9 ( ! 7 $B "$ z = ρeiϕ, w = reiψ = ρ1 e−iϕ 5 !"!9#, = arg w = − arg z, 1 |z|
E$! ( K7 ( 79# * 7(!$ ( %! 7, = ( 96 (!! 7, 7 # ( B ( %! # ζ = ζ(z), ( |ζ| = |z|, 1 arg ζ = − arg z , ! w = w(ζ), ( |w| = , arg w = arg ζ 4K ( %! 7 |ζ| !6 * $"$ (! # * (%$7 ( ( , $ 7 ) "! z B ($# " z, ! (* & ($$" , = ($# " z " w = z1 |w| =
$ -&
w = f (z) = z 2 .
:;;I<
57 7 6 ( !" 9 69 z, = $!"! ! =$ @ " ( ! $$ "$ ! ) !$ z = ρeiϕ, w = reiψ = ρ2ei2ϕ O ! * ( ! 7 * %!"$$, = "$ =$$ z, = !# ! , 7$ !(!6 ϕ ( (!$ ! 7 ( , B (7# "$ =$$ w, = !# ! , 7$ !(!6 2ϕ ( (!$ ! 7 ( @ "! z −z, ( 7$B !(!9#, ! !*$ (79# 7 ! π, ( (!6 ( !"7 w @!$ "$ , %! 7 $7 76# 7 %!*! !" 9 - * 7 % #K (! # ( %! 7, = ( 96# 7 69 w = 2 z 'B7 ! =$! z ! ( 9 9 B ($# =$ w 7 " B# ! =$ z , ( 7 7$B 0 < ϕ < π , $ !"7 z ( (!9# !"7 w )* ! 6" - -& -
-& )
(B $ !(!B % ! 9 ( $ % ! 7 % ! # z O!! % ! #, 7 !(!! 7 w = z2 , 6 7 ! =$! z, ! % ! 9 ( $ *6 ! =$! '($ , = *!$ % !) ( $ : ϕ = 0 ϕ = π<, B (7# ( 7 ) ( (! "! $ ( =$$ w '($ ! , = 7 w = z2 96 ( %! 7 $ # !) =$$ z ! ( 9 9 ! =$ w ' #! 7 z=
√
:;;J<
w,
= $!"! ! =$ w, 6 ( !" 9 & ( " =$$ w ( (!9# ( "$ =$$ z - * 7 ! =$ w (7 ! $ C , = $!6# 7 !! % 9 O! 6 ! √ " w0, 7! !6 !* ψ0, !( !"7 √ ψ z0 (w0) = re z1 (w0) = re (ψ+2π) (6 ! 9 z0 (w) z1(w) $ "$ w $ C 4$ # $ ( ! $B $ !($ K $ ! C $# ($ " w = 0 @ ( 7 %B ( $ C !* "$ w0 # 7 ( "! * !"7 arg w0 = ψ0 ' #! !"7 z0 (w) z1 (w) " w = w0 %(# ( 9 !$ B "! $ !"7 @!$ "$ , ! $ C # $ !( $!" ( ( !" w i 2
i 2
! "
)* ! 3# " D # w ) & #
" w = 0& " D - z0 (w) z1 (w)% 7 √ z(w) = w
(* $ !( $ ! C $# ($ " w = 0 @ ( 7 %) B ( $ C ( (! ! 7 !"7 !* "$ w0 # 7 ( "! * !"7 ψ0, ! $# 7 ! 2π @ !") 7 z0(w) z1(w) " w0 #! B $ 7 %B ( $ c %(# ( 9 !$ B "! $ !"7 #K () ! # $!6 z0(w0) = z0(w0)eiπ , z1(w0) = z1(w0)eiπ @ % 7 z0(w) K ! 9 z1(w), ! !$ )* ! 3# z0 ε & #
"1 z0 ε " - - 1 '& z0 ) - " - -& "1 1 % ' * 7 $ !)
( " 9 *! 7 6 "! w = 0 %
& $ '#
$
# * 7 7 %( ! ! # $ !! * 69 ( P 77 ($ ! ) $ ($# ( 6 $B ( )* ! B! % ! T =$$ z !(!! 7 f (z)
3# z0 ∈ G ) ∆z → 0 . , ' f (z0 + ∆z) − f (z0) , ∆z
) - f (z) z z0 ) f (z0)& "
f (z0 + ∆z) − f (z0) . ∆z→0 ∆z $ !( 7 f (z) !$ !6# 7
:;;M< # " z0 4( $ , = 7 *!$7 :;;M< ! $# ( % 7 !7 ∆z ( 7
f (z0) = lim
%&-&%& 3# f (z) = u(x, y) + iv(x, y)
z0 = x0 + iy0& (x0, y0) 1 u(x, y) v(x, y) 1 x, y& ' . !"# , ∂u(x0, y0 ) ∂v(x0, y0 ) = , ∂x ∂y
∂u(x0, y0) ∂v(x0, y0) =− . ∂y ∂x
:;>N<
O! $ $ 6 *!$7 :;;M<, = ! $# ( * , 7 ∆z → 0 4 !( ∆z = ∆x * 7 $!
u(x0 + ∆x, y0) − u(x0, y0) v(x0 + ∆x, y0) − v(x0, y0) + i lim . ∆x→0 ∆x→0 ∆x ∆x
f (z0) = lim
O ! !7 *!$" * !"7 * "$ ! $ $ !6 !) 7 *!$# * ( ! 7 "! $, !6 !
f (z0) = ux(x0, y0) + ivx(x0, y0) .
?= ! $ ∆z = i∆y,
u(x0, y0 +∆y)−u(x0, y0 ) v(x0, y0+∆y)−v(x0, y0) + lim = ∆y→0 ∆y→0 ∆y ∆y = −iuy (x0, y0) + vy (x0, y0) .
f (z0) = −i lim
4 99"$ ( ! $, 6 7 ! ( $ ( ) K# :;>N<
%&-&'& 3# (x0, y0) - u(x, y) v(x, y)
& -- 1 0 ' .+%,& f (z) = u(x, y) + iv(x, y) ) ) - - z z0 = x0 + iy0% '( ( ( !"7 ($ , $ $ u(x,y) v(x,y) "$ (x0, y0) # %$ ! $ ! $* 7( u(x0 + ∆x, y0 + ∆y) − u(x0, y0) = ux(x0, y0)∆x + uy (x0, y0 )∆y + ξ(x, y) , v(x0 + ∆x, y0 + ∆y) − v(x0, y0) = vx(x0, y0)∆x + vy (x0, y0)∆y + η(x, y) ,
:;>;<
! "
( ξ(x, y) η(x, y) 79# ( 7 $ x → x0, y → y0 K $(K, !
δx ! ∆y @ ( ! ( * 9 :;>;< :;>N< ! $!$ f (z0 + ∆z) − f (z0) = ∆z ∆x + i∆y i∆x − ∆y ξ(x, y) + iη(x, y) = ux(x0, y0 ) + vx (x0, y0) + = ∆x + i∆y ∆x + i∆y ∆x + i∆y ζ(z) = ux(x0, y0 ) + ivx(x0, y0) + , ζ(z) = ξ(x, y) + iη(x, y) . ∆z
'($ , = $ ∆z → 0 ! ( (! 6 $ 76 ( 7, ! K ! $K!9# 7 $$ @ 6 *!$" !"7 f (z0 + ∆z) − f (z0) = f (z0) , ∆z→0 ∆z lim
= ( ($# ($ # f (z) " z0
)* !& 3# f (z) 1 1 - " G & -- 1 " & f (z) ) ) " .'& " ,
P! $" (*!9# % $ # 7 $ H7 "$ $B !!$"$B % , ! $B ! !7B ) - * 7 $ !($ (7$B !$B 5 6 $ ($ ( 7 =$ $"!$B ( 7 !! !$B P $ ! ( $ !% !9# $B ! $ , (7 7$B $ * 7
$ .& 8 w = f (z) = az + b%
w = (Re a + i Im a)(x + iy) + (Re b + Im b) = = (Re a · x − Im a · y + Re b) + i(Re a · y + Im a · x + Im b) = u(x, y) + iv(x, y) .
4 $ ( 7 6 $ !7 K)-!! :;>N< ∂v ∂u = Re a = , ∂x ∂y
∂u ∂v = − Im a = − , ∂y ∂x
% ! 7 6 !! $" 9
$ /& 2 w = f (z) = z2%
w = x2 − y 2 + 2ixy = u + iv .
4 $ !! $" # 6 ∂u ∂v = 2x = , ∂x ∂y
∂u ∂v = −2y = − . ∂y ∂x
$ 0& 2 w = f (z) = z1 % w=
1 y x − i = u + iv . = 2 x + iy x + y2 x2 + y 2
4 $ !! $" #
∂u −x2 + y 2 ∂v = 2 , = ∂x (x + y 2 )2 ∂y
∂u 2xy ∂v =− 2 , = − ∂y (x + y 2 )2 ∂x
% %! 7 6 !! $" 9 9($ ! $7 "$ z = 0
$ %1& 2 w = f (z) = ez % w = ex+iy = ex (cos y + i sin y) = ex cos y + iex sin y) = u + iv .
4 $ !! $" # ∂u ∂v = ex cos y = , ∂x ∂y
∂u ∂v = −ex sin y = − . ∂y ∂x
- * 7 ! z1 ! z2, ( 7 7$B ez = ez E" $( , = !9# |z1 | = |z2 |, Im z2 = Im z1 + 2πk , % z2 − z1 = 2πk · i, ( k & 5 !"!6, = 7 ez 6 ($" 9 1
2
$ %%& 2 w = f (z) = ln z% O! !"7 *!$"! 7 $!"!6# 7 7 7, = 6 % 9 ( !$ "$ w !$ !6# 7 *!$ "$ ! z, 7= ew = z O (# * $ !( $ $ !6, = ln z 6 " !" 9 69 ?= ew = z1 ! ew = z2, ew +w = z1z2, !% ln z1z2 = w1 +w2 = ln z1 +ln z2 O !, 7= z = |z| · ei arg z , 1
2
1
2
ln z = ln |z| + ln ei arg z = ln |z| + i arg z .
! "
R % 7 !!$ ! %!*! !" # *!$! * "$ !, %( $ $ !$ ! # !"7 Ln z, ! = Ln z = ln |z| + i Arg z = ln |z| + i (arg z + 2kπ) . :;>>< w = ex+iy = ex (cos y + i sin y) = ex cos y + iex sin y) = u + iv .
4 $ !! $" # ∂v ∂u = ex cos y = , ∂x ∂y
∂u ∂v = −ex sin y = − . ∂y ∂x
3 ( K7 K)-!! "! $ $ 6# 7 ( ( !) $B ! $ !! $"$B - * 7 ( K7 $ K$B ( ! 7B $B "$ P !, "! * 7 !*) $* 7( z = x + iy → ρeiϕ, % , f (z) = u(ρ, ϕ) + iv(ρ, ϕ) @ ( ∂u ∂u ∂ρ ∂u ∂ϕ ∂u ∂u sin ϕ = + = cos ϕ − . ∂x ∂ρ ∂x ∂ϕ ∂x ∂ρ ∂ϕ ρ
P! *" %"$ 99# 7 $!$ ∂u ∂u ∂u cos ϕ ∂v ∂v ∂v sin ϕ ∂v ∂v ∂v cos ϕ = sin ϕ + , = cos ϕ − , = sin ϕ + . ∂y ∂ρ ∂ϕ ρ ∂x ∂ρ ∂ϕ ρ ∂y ∂ρ ∂ϕ ρ
@ (, $ 9"$ # ( K7$ :;>N< ! 9 ! 9 ) sin ϕ ! cos ϕ, $!6 !! $" K $* 7( ∂u 1 ∂v = , ∂ρ ρ ∂ϕ
1 ∂u ∂v =− . ρ ∂ϕ ∂ρ
:;>A<
P! *"$ "$ * ! $$, = ( # !* !! $" f (z) = R(x, y)eiΦ(x,y) H7! ( K7$ ∂R ∂Φ =R , ∂x ∂y
∂R ∂Φ = −R . ∂y ∂x
:;>C<
(# $ $$ ( K7, 7 ! ( %! !9# ! $ !!) $"$B , ! !, 7= f (z) = f (ρeiϕ) = R(ρ, ϕ)eiΦ(ρ,ϕ), ∂R R ∂Φ = , ∂ρ ρ ∂ϕ
∂R ∂Φ = −ρR . ∂ϕ ∂ρ
:;>F<
(
) #
* $
?= 7 f (z) 6 !! $" 9 % ! G , ! ! % ! 2◦ ?= f1 (z) ! f2(z) 6 !! $"$$ 7$ % ! G , B ! ! ( % ! 6 !! $"$$ 7$ % ! , ! 7 ϕ(z) = ff1(z) 2 (z) 6 !! $" 9 9($, ( f2(z) = 0 3◦ ?= w = f (z) 6 !! $" 9 % ! G z , % ! !"# G ! =$ w $!"! !! $"! 7 ζ = ϕ(w), 7 F (z) = ϕ[f (z)] 6 !! $" 9 69 z % ! G 4◦ ?= w = f (z) 6 !! $" 9 % ! G z |f (z)| = 0 (7 "$ z0 ∈ G , "$ w0 = f (z0) % ! G !"# f (z) $!"! %! 7 z = ϕ(w), = 6 !! $" 9 69 w 4$ # !6 ( K7 1◦
f (z0) =
1 . ϕ (w0)
:;>G<
B! % ! G =$$ x, y !(!! 7 u(x, y), = 6 ( 9 "! ) $ 9 !! $" f (z) @ ( 7 ! "! $! $!"!6# 7 " 9 ( !($$ ! , ! ( K)-!! ! !(! 9 ) 69 u(x, y) ! ( !" $!"$$ $ ($! ( v(x, y) 5◦
dv = vxdx + vy dy = −uy dx + uxdy ,
($ $ $ !6 ( 7 6◦ B! 7 f (z) 6 !! $" 9 % ! G - * 7 ( ( % ! ! =$ x, y ($$ $ $B u(x, y) = C v(x, y) = C , = 6 ( ! 7 "! $ f (z) O! ( * 9 ( K# :;>N< ! !!$, = B "!B (! % ! 6 ! ( $ 9 !! # grad u · grad v = uxvx + uy vy = 0 ,
@ % , ($$ $ $B u(x, y) = C v(x, y) = C 6 !6 * ! #
!" #!$%& $' $( '(
' # ) *
# ' (
+ , -
B! 6 !(! 9 (7! 6 !! $ ! C ! !(! 9 f (z) )* ! .*! ( f (z) ! $ C !$ !6# 7 n−1
f (ζk+1)(zk+1 − zk ) =
k=0
f (z)dz ,
:>;<
C
( z0 = a, z1, . . . , zn = b & ( # " , = %$ !9# $ C ! n ( 7 , a ! b & C , ζk & ( #! "!, = $# ! ! [zk , zk+1] $ C , *!$" !"7 %# 7 ! $ =7, = max |zk+1 − zk | → 0 ?= C & * !(#! $ !, ! f (z) & ! % ! 7, *! :>;< ! ($ 6 4$!7 !7 ! * ) *! ! ! $ ( $!7 !7 (7$B $ $B *! ( ( u ! 7 v "! $ f (z) , *! :>;< ! ( ! $$ $* 7(
(udx − vdy) + i
f (z)dz = C
C
(udy + vdx) . C
:>><
# $
5 ( K7 ! * !$ $!"7 *! ( f (z) $ C O # * $ $ !9# (7 ! $ , = 6 " $($ !) ( ( ($B ! $ $ $B *!
f (ζ)dζ = −
1. AB
2.
BA
f (ζ)dζ + C1
3.
af (ζ)dζ = a
:>F<
f (ζ)dζ . C
{f1(ζ) + f2(ζ)} dζ =
4.
f1(ζ)dζ +
C C f (ζ)dζ ≤ |f (ζ)| ds , C
:>C<
f (ζ)dζ . C1 +C2
C
5.
f (ζ)dζ = C2
:>A<
f (ζ)dζ .
f2 (ζ)dζ .
:>G<
C
:>I<
C
( ds & ($! ( $$ (*$ C
$ %'& ' "
I= Cρ
dζ , ζ − z0
:>J<
Cρ ) " ρ z0& "1 "1 - % 3 $ 6 # !!$" 9 0 ≤ ϕ ≤ 2π , $!6 2π I= 0
iρeiϕ dϕ =i ρeiϕ
9 ! (!7 $ Cρ ζ
2π dϕ = 2πi .
= z0 + ρeiϕ
:>M<
0
O ( $ $ $ !6, = !"7 *! ! :>M< ! $# ( ρ ! z0 !(! ! %( ! $$ *! $ ( , = 6 !! $"$$ (7 % % ! , $" % #K $ !( *!$9 % ! %( *) !$ * !(#! !! $ !, = !6 ! $ @!! $ !
# $ %&
!$ !6# 7 , ! ( ($ *! &
'&%&(& 2 3 4 91 0 " G )
f (z)% 4 )- - Γ& # " G & ) %
O*( :>>< ! ! $ !$
(udx − vdy) + i
f (z)dz = Γ
Γ
(udy + vdx) . Γ
E #$ 7 f (z) 6 !! $" 9 9($ ($ ! Γ, u(x, y) ! v(x, y) % ! , = % ! $ , !9# "! ) $ B( K * 7( @ ( ( $B $ $B *! ! "! $ ! ! !$ ( $ !! ! #! P (x, y) Q(x, y) ! % ! G , = % ! * !(#$ ) C , ! !
(P dx + Qdy) = C
G
@
(udx − vdy) = Γ
!
G
(udy + vdx) = Γ
G
∂Q ∂P − dx dy
dxdy .
:>;N<
∂v ∂u − − dxdy = 0 dx dy
∂u ∂v − dx dy
dxdy = 0 ,
= ( ($# ( 7 $ 57 ! ( 76 ! $$ H7 !"7$ !! $" ) KB "!B % ! !! $" *!$"$$ !"7$ 6 B! 7 f (z) 6 !! $" 9 ( H7 % ! G , = %)
! C E% ( # K9 " z0 %(6 !$
# $
ΓC
'$ G
ρ
r z 0 γ &% ∗
-
G
-$ >; $ K Γ, = $# G $# ($ " z0 ' ($ # * ! %(6 = ($ !$ γ , = $# " z0 :$ >;< - * 7 ( 9 f (z) . z − z0 Q7 ϕ(z) 6 !! $" 9 9($ % ! G ! $7 "$ z0, !, !"$#, ! !! $"! 9($ % ! G ∗, = % ! $ $$ Γ γ @ ( *( $ K *! ( ϕ(z) $ Γ + γ ( 96 9 f (ζ) f (ζ) dζ + dζ = 0 . ζ − z0 ζ − z0 ϕ(z) =
Γ+
γ−
O99"$ ! 7 * !7 (* *! , 9 # ! ) $ !$ $* 7( f (ζ) f (ζ) dζ = dζ . :>;;< ζ −z ζ −z Γ+
0
γ+
0
E #$ *! , = # "! $, ! $# ( $% ! γ , ! ! $ # !6 *! , = # ! !, !(! " %!$ 7 * !7 γ (7 * !( ! ρ " z0 ?= ! $ ζ = z0 + ρeiϕ, ! $!$
Γ+
f (ζ) dζ = i ζ − z0
2π
f (ζ)dϕ . 0
# $ %&
E ! *! ! $$ !$ "$ 2π
f (ζ)dϕ =
0
=
2π 0 2π 0
[f (ζ) − f (z0)] dϕ +
2π 0
f (z0)dϕ =
[f (ζ) − f (z0)] dϕ + 2πf (z0) .
3 76 !! !( ! ρ ( 7 E #$ 7 f (z) 6 !! $") 9 , 7 ! ( , 9 % ! G , ! ($ ( 7 ( # * ε > 0 ! !$ ! !"7 ρ, = |f (ζ) − f (z0)| < ε $ |ζ − z0| < ρ O ( $ $ $ !6, = $ ρ → 0 6 *!$" !"7 2π [f (ζ) − f (z0)] dϕ = 0 .
lim
ρ→0
0
' #! !6 K!$ #! 1 f (z0) = 2πi
Γ
f (ζ) dζ . ζ − z0
:>;><
.*! ! "! $ :>;>< !$ !6# 7 :', ! !! ! :>;>< !$ !6# 7 :' '($ , = ! "! $ $ K B (7# $K !"7 f (z) ! *!$ Γ, % !"7 ($ % ! !! $" $!"!9# 7 !"7$ ! *!$ '($ ! , = *! ! "! $ :>;>< !6 ( 7 ( # * 7 "$ z0 $ , = ! $# ! Γ #K (! # ! ! $ !$
1 f (ζ) f (z0) , z0 & ($ Γ , dζ = :>;A< 0, z0 & Γ . 2πi ζ − z0 Γ
$ "!
z0 ∈ Γ,
*!
1 I(z0) = 2πi
Γ
f (ζ) dζ ζ − z0
$"!
6, ! $ ( (! $B !B = ( ($ f (z) ! Γ # *! ! !(!$ !"7 @!, ! $ !(, 7= 7 f (z) !( #76 S# #(! |f (z1) − f (z0)| ≤ K |z1 − z0 |ν , 0 < ν ≤ 1 , :>;C<
# $
6 *! ! I(z0) 1 P.v. I(z0) = lim ε→0 2πi
Γε
f (ζ) dζ , ζ − z0
:>;F<
( Γε 7 76 % 9 "! $ ! Γ, = $# ! |z − z0| < ε 4$ # 1 P.v. I(z0) = f (z0) . 2
4 # ( !! $" f (z), % ! # $!"7 7 %! ) ! ! $ >;
'&%&+& 2 5 6 4 91 f (z) )
" G " G % 4 '1 1 " G ) 1 - f (z)& - ) n! f (n) (z) = 2πi
Γ
f (ζ) dζ , (ζ − z)n+1
:>;G<
z 1 Γ%
O! !"7 B( ( 7 6 "$ !6
f (z + h) − f (z) = h→0
h 1 1 1 1 = lim − dζ = f (ζ) 2πi h→0 h ζ −z−h ζ −z Γ 1 1 f (ζ)dζ = lim . 2πi h→0 h (ζ − z − h)(ζ − z)
f (z) = lim
Γ
4$ h → 0 7 ζ − z1 − h ( 7 B !"$# 7 *!$7 6 1 f (z) = 2πi
Γ
f (ζ) dζ , (ζ − z)2
ζ ∈ Γ
76 (
1 , ζ −z
:>;I<
# $ %&
7 n = 1 ( ( O! ( * 9 (! !!$" (, ! !9"$ 9 ( 7 (7 * n, ! ( $ ! ( $ # ( 7 n + 1 @!$ "$ , 7= 7 f (z) 6 !! $" 9 69 % ! G , % ! 7 f (z) !6 B( B 7( 57 ! $ # !! $" 6 (76 ( ( , = !6 K B( (7 % ! ' !# $ !() !7 K B( , !*! ! "$, $ $ !6 !7 B($B $= * 7( - * 7 % ( (7 (7 ! $ ! ($ ! ! $ !! $"
'&%&,& 2" 4 91 f (z) ) 0
" G f (z) & # G & ) % 4 f (z) ) ) " G %
(7 7 ! 6 % 9 ( $ K U ! !*! #$$ ! %!*! H7 % !
'&%&-& 27
!4 91 # f (z)
) & -- " % 4 ) %
? ($ $ !( ! !7 6 $ * 7 $* $" 9 sin z !*!(!6 , = sin z = = = = =
1 iz 1 i(x+iy) −iz −i(x+iy) e −e e −e = = 2i 2i 1 −y+ix 1 −y e e (cos x + i sin x) − ey (cos x − i sin x) = − ey−ix = 2i 2i 1 −y e (sin x − i cos x) − ey (− sin x − i cos x) = 2 1 y 1 (e + e−y ) sin x + i (ey − e−y ) cos x = 2 2 u(x, y) + iv(x, y) .
# $
4 $ !! $" # 6 ∂u 1 y ∂v = (e + e−y ) cos x = , ∂x 2 ∂y ∂u 1 y ∂v = (e − e−y ) sin x = − . ∂y 2 ∂x
@ % 7 sin z 6 !! $" 9 ! =$ E #$ ! 6 ! 9, ! %$ % 9 O !, !(# 7 ! !"7 z, ( 7 7$B | sin z| > 1 569 ! $ 9 $* $" 6 (79# 7 ( ( ) ($B (
) * "" *+*, #!$%&
+ , # " - # . ' . + # # # + * # * ( " " # " "/ 0 "
. $
- * 7 "! 7($ $$ "$ !$, % $!$ $ ∞
:A;<
ak ,
k=1
( {ak } & !(!! "$ ! ( #
)* !&n
{Sn =
k=1
.;%+, ) " & # ") ak } 1 % %B( 9 9 % 7(!
:A;< 6 ! limn→∞ an = 0 ?= %*!6# 7 7( ∞
:A><
|ak | ,
k=1
( $$ ( (!$$ " !$, , " $( , = %*!6# 7 7( :A;<, 7$ # $ !( %( " " 8! ( 7 ( ( 7 %
' ( )
7(! $$ " !$ * 7(!9# 7($ ( $$ " !$, 7 6 ( 7$ " $B( * 7(! 7 7( ( $$ " !$ ( !$$ !$ % 6 !" :' O*( !$ ! !%! 7( :A>< %*!6# 7, 7= , "$!9"$ (7 * an+1 ! N , ( K7 a ≤ l < 1 ( 7 B n ≥ N O*( !$ K 7( n :A>< %*!6# 7, 7= , "$!9"$ (7 * ! N , |an| ≤ q < 1 ( 7 B n ≥ N )* !& ?= % ! G $!"! "! ( # ( !") $B {un(z)}, $!$ $ n
∞
:AA<
un (z)
n=1
!$ !9# 7 Q ! #$ 7( :AA< !$ !6# 7 " % ! G , 7= ( 7 ( # * z ∈ G ( ($ "$ $ 7( %*!) 6# 7 ?= 7( :AA< %*!6# 7 % ! G , % ! ! $!"$$ ( !" 9 f (z), !"7 7 " % ! G ( 96 ( ( * "$ * 7( )* !&?= ( 7 %(#)7 * ε > 0 ! !!$ !$ N (ε), = $ n ≥ N (ε) # n uk (z) < ε f (z) − k=1
$ 6# 7 (! ( 7 B " z % ! G , 7( :AA< " ) % ! ! ( ! $ ( ! ! % )* 3 & ?= 9($ % ! G " $ ! # * 7(! :AA< # %$ ! ! " !$ !% 9 % * "$ * 7(!, 7( :AA< ") % ! G O! 9 !6 ! ! |un (z)| ≤ |an | , z ∈ G . :AC< E #$ 7(
∞ n=1
|an |
%*!6# 7, ( 7 ( # * ε
> 0
! !$ !
' * N,
∞
= |ak | k=n+1 #
< ε
$ n
≥ N
@ ( ! ( :AC< % ! G !6
∞ ∞ ∞ uk (z) ≤ |uk (z)| ≤ |ak | ≤ ε k=n+1
k=n+1
k=n+1
$ n ≥ N + 1 '($ , = !! '6K! ! 6 $K ( ! 9 ! 9
/ #
# * $ * 7(! $ ! # 7($ :AA< !*! # * $ O!! $ * 7 ! $ $ $ !( $B 7( , ( 7 7$B un(z) = cn(z−z0)n, ( cn & (7 ∞ n"$ !, ! z0 & !! "! =$$ 8 $ 7( n=0 cn (z − z0) 6 !! $"$$ 7$ ! =$ '($ , = % ! # % * $* 7( * ∞ 7( $!"!6# 7 6 cn @ , ! $ !(, 7( n=0 n!(z − z0)n %*!6# 7 $K " z = z0 7 $!"7 % ! % * 7( $ $ 6# 7 !) ! !
(&'&.& 2 8 !4 3# ∞n=0 cn(z−z0)n "
) z1 = z0& " ") z& # ) |z − z0| < |z1 − z0 |/ |z − z0 | ≤ ρ < |z1 − z0 | ") %
O 6 $, ( (7 7 %( $ ($$, ! %$$ ( #! ! $ $B $
.&%& 3# ∞n=0 cn(z − z0)n ")
z1& ") 1 1 z& # |z − z0 | > |z1 − z0 |%
- * 7 " B9 R ( ! |z −z0 | ( "$ z0 ( "$ z, ∞ 7$B 7( n=0 cn(z − z0)n %*!6# 7 ?= R = ∞, B "!B |z − z0| > R 7( %*!6# 7 B! R * % #K 7, ( ($ ! |z − z0| < R
' ( )
7( %*!6# 7, "!B *!$ %*!$ 7
|z − z0 | = R
7 %*!$ 7, !
)* !& 6" |z − z0| < R .R > 0, ) "
& R 5 "
@!$ "$ , $ ! $ $
.&'& ! ) R& #
|z − z0| < R ") & ") %
.&(& " ") - -%
.&+& " 1 & " 1 ) " 1 %
.&,& : ) ∞n=0 cn(z − z0)n
cn =
f (z)
-- 11 "
1 (n) f (z0) . n!
:AF<
4 !( z = z0
$! ( 7 $ * 7( f (z) = ∞n=0 cn(z − z0)n $!6 !"7 f (z0) = c0 4$($96 7( " !B, !( ∞ n−1 z = z0 $! ( 7 B( f (z) = n=1 cn n(z − z0 ) $!6 f (z0) = c1 P! *" ! $$ (! $!6 :AF<
.&-& 29 3 8 4 ) 1 R= , l
"
l = lim n |cn | . n→∞
? $ !( * 7 $ 7( f (z) = ∞n=0(z−z0)n, 6$ 7 * ( 99# ($$ O! ! 9 ! !%! $!6 , = 7( %) *!6# 7 |z − z0| < 1 ( (7 !! $" 59 9 * !$ 1 − (z − z0 )n+1 1 = . f (z) = lim n→∞ 1 − (z − z0 ) 1 − (z − z0 )
:AG<
''
+
. 0
@!$ "$ , ( $ !( !! , = ($ *! % ) $ 7( $!"!6 (7 !! $" 9 ! ! $$ $!7 (7 $ , ! !, "$ ! !! $" !) $$ (7$ $ 7( '( (# (!6 ! ! !
(&(&/& 2 4 2 f (z)& # )
|z − z0| < R& " " ∞ f (z) = n=0 cn(z − z0)n& ) %
4 %(6 Cρ !( ! ρ < R " z0, 7 $# " E #$ ($ # * ! 7 f (z) 6 !! $" 9, *( ( $ K ! ! $ !$
z
1 f (z) = 2πi
Cρ
f (ζ) dζ . ζ −z
:AI<
4( *! ! %$$ " $( 7 ∞
1 1 1 1 (z − z0 )n = · = . :AJ< ζ−z ζ − z0 1 − z − z0 ζ − z0 n=0 (ζ − z0 )n ζ − z0 z − z0 @ $ $ $ ! $ " $( ζ − z < 1 $ ζ ∈ Cρ, 7( :AJ< 0 %*!6# 7 ζ , #$ ! 6# 7 % $ "$ $ 7( ∞ |z − z0 |n 4( ! $ :AJ< :AI<, *6 " !B $!6 n+1 ρ n=0 ∞ 1 f (ζ)dζ f (z) = (z − z0 )n . n+1 2πi (ζ − z0 ) n=0
:AM<
Cρ
4 !"$ 1 cn = 2πi
Cρ
f (ζ) f (n) (z0) dζ = (ζ − z0 )n+1 n!
:A;N<
' ( )
$K :AM< $* 7( % * %! " z * 7( f (z) =
∞
cn (z − z0 )n .
:A;;<
n=0
O*( $ K :A;N< Cρ ! !$$ ( #$ !$ C , = $# % ! |z − z0| < R $# " z0 ($ ($ # ( ($$ %( O% $ $ 7( !$ !6# 7 !% 4
$ %(& $ 4 f (z) =
1 1 + z2
57 7 6 !! $" 9 9($ =$ ! $7 " z1,2 = ±i @ %(#)7 ! =$, 7 $# " z1,2 = ±i, 7 7 ! ( $ AAJ %$ !(!
7( @ ! - * 7 "! * |z| < 1 ' ($ # * *! 7 1 +1 z2 * 7(!$ 7 7 ! " !( * $" * , % , ∞
1 = (−1)nz 2n , 1 + z2 n=0
:A;><
= 7 76 % 9 K!$ !( -!( % # * 7( ( 96 1 O!! √ * 7 !( 6 7( @ ! ($ ! |z − 1| < 2 (6 $ !7 $! :A;N< $ ($# ( * ($B %"$ # 6 cn , $ 6 # ! $ ( ! 7 ∞ 1 1 1 1 1 1 (z − 1)n . − = = (−1)n − 2 n+1 n+1 1+z 2i z − i z + i (1 − i) (1 + i) n=0 π √ −i 3 $ 6 # π ! 9 9 ! $ $B "$ 1 − i = 2e 4 √ i ! 1 + i = 2e 4 $!6 K!$ #! ∞ π 1 n sin(n + 1) 4 = (−1) (z − 1)n . :A;A< n+1 2 1+z 2 2 n=0
', - (
(
1 #
$
-!K % !! , = !! $"! 7 9 $!"!6# 7 $ !"7$ ! *!$ % ! !! $" O!! %( !! , = !! ) $"! 7 9 $!"!6# 7 $ !"7$ ! ( # ( ) " , = %*!6# 7 ( (7 K# "$ % ! !! $" )* !& 9 !! $" f (z) !$ !6# 7 ( #! "! z = a, 7 f (a) = 0 ?= 6 "! !! $"! 7 ( 96 9, ( ( 7( @ ! " a 6$ # %$ $$ 9 )* !& (K * ( * ( 7 6! # * ) !( !$ !6# 7 7 a @!$ "$ , 7 7(! n 7( @ ! ( 7 !6 $* 7( f (z) = cn (z − a)n + cn+1 (z − a)n+1 + . . . , :A;C< ( cn = 0 ! n ≥ 1 5 (!6 $ # ( ! $$ !! $" 9 7 7(! n $* 7( f (z) = (z − a)n ϕ(z) , :A;F< ( 7 ϕ(z) = cn + cn+1 (z − a)1 + . . . ; ϕ(a) = cn = 0 :A;G< 6 !! $" 9 "$ a '! ( 7 7 (! ( 7 ! (7 "$ a O ( $ $ $ !6 !
(&+&0& 91 f (z) a ) % 4 ) a& f (z) ) '1 & a%
. 6 $ $ $ !6 ! ! !
(&+&%1& 2 : 4 3# - f1(z) f2(z) " G -1 an & # ") '- a " G & G f1(z) = f2 (z) .
' ( )
- * 7 9 f (z) = f1 (z) − f2(z) .
' ! 6 !! $" 9 G !6 $ 7$ "$ an, ! $ " a, = limn→∞ f (an) = f (a) O ( $ $ $ !6, = f (z) ( 96 0 (7 a
%1&.& 3# G - f1(z) f2(z) L ∈ G & " %
%1&/& 3# - f1(z) f2(z)& # )
" 1 G1 G2& # " G & ) ) F (z) & #
F (z) ≡
f1(z) , f2(z) ,
z ∈ G1 , z ∈ G2 .
. $ 6($ # $ $ !6, = !! $"! (7 % ! G ! 9 7 f (z) ( 9 !$ 9 %(#)7 ( % ! G , ! ! ! %(#)7 (, = $# G , ! # ! ( " G , = %*!6# 7 ( K# "$ !
. 2
-7($ @ ! " $ $ !$ ( 7 , = 6 !! $"$$ () 7$B * $B % ! 7B ' % ! 7B K $, ! $ !(, $ 7 !! $"! (7 "$ a 9($, ! $7 ! "$ a, ( ) ($# 7 * 7(!$ # % ! $* 7( 0 < |z − a| < R 7 !$B ! %( !$ !($ ( (!$B ! (H6$B 7B (z − a) f (z) =
∞
:A;I<
cn (z − a)n ,
n=−∞
= 6 !*! #7 !( @ ! !$ !6# 7 8 - * 7 % ! # % # * 7( 7 # * $K * $) * 7( ∞ n=−∞
n
cn (z − a) =
∞ n=0
n
cn (z − a) +
∞ n=1
c−n . (z − a)n
:A;J<
'
.
E" $( , = % ! 9 % 7( :A;I< 6 #! "! $! % ! % ) * ( (! ! "! $$ :A;J< E% ! 9 % K $ 6 * " a (7 * !( ! R1 ' ($ # * *! 7( %*!6# 7 ( (7 !! $" f1(z) =
∞
cn (z − a)n ,
:A;M<
|z − a| < R1 .
n=0
7 $!"7 % ! % (* $ %$ ! ζ = ∞ 1 n @ ( 7 ! !%( $* 7( n=1 cn ζ @ % $"!$ $ z−a 7(, = %*!6# 7 ( (7 !! $" ϕ(ζ) ζ ($ *! % !( !, 7$ !"$ R1 , % ζ < R1 4 !) 2 2 9"$ # ( ! !"!9"$ ϕ(ζ(z)) = f2(z), $!6 f2(z) =
∞ n=1
c−n , (z − a)n
:A>N<
|z − a| > R2 .
@ % , % ! 9 % (* * 7( (H6$B 7B $ (z − a) 6 % ! #, K! ( ! |z − a| = R2 ?= R2 < R1, 6 #! % ! # % $B 7( V R2 < |z − a| < R1 , 7 7( :A;I< %*!6# 7 ( !! $" f (z) = f1 (z) + f2(z) =
∞ n=−∞
cn (z − a)n ,
R2 < |z − a| < R1 .
:A>;<
4$ # f1(z) !$ !6# 7 "! $ 9 !( !!, ! f2(z) & "! $ 9 # * !( ?= R2 > R1, 7($ :A;M< ! :A;M< !9# # % ! % 7( :A;I< %*!6# 7 ( 7 !! $"
(&,&%%& 2 f (z)& # ) R2 < |z − a| < R1 &
8 %
"
4 %(6 # R2 < |z − a| < R1 ( ! ! CR1 ! CR2 !$ " a B! $ 6# 7 ! R2 < R2 < R1 < R1 7 "$ z , = !( #76
' ( )
R2 < |z − a| < R1, *( K ! ! $ !$
1 f (z) = 2πi
f (ζ) 1 dζ + ζ −z 2πi
CR
1
C −
f (ζ) dζ . ζ −z
:A>><
R2
1
! CR ( ! 7
z − a $ 6# 7 # ζ − a < q < 1 %( "$ ! ∞
1 1 1 1 (z − a)n = · , = ζ−z ζ −a 1− z−a ζ − a n=0 (ζ − a)n ζ−a
= $ ($# ( * 7( ( 7 K * *! ! :A>>< 1 f1(z) = 2πi
∞
CR
f (ζ) dζ = cn (z − a)n , ζ −z n=0
:A>A<
1
( 1 cn = 2πi
f (ζ) dζ , (ζ − a)n+1
CR
n ≥ 0.
:A>C<
1
! CR ( ! 7 2
ζ − a $ 6# 7 # z − a < q < 1 %( "$ K ∞
1 1 (ζ − a)n =− , ζ−z z − a n=0 (z − a)n
= $ ($# ( * 7( (H6$$ 7$ ( 7 (* * *!) ! :A>>< 1 f2(z) = 2πi
∞
C −
c−n f (ζ) dζ = , n ζ −z (z − a) n=1
:A>F<
R2
( c−n
1 = 2πi
CR
2
f (ζ) dζ , (ζ − a)−n+1
n > 0.
:A>G<
' (
E #$ (*! # :A>C< ! :A>G< 6 !! $"$$ * ) #, ! ( $ K $ * !7 ! ( # ( !$ O !, B %H6(!$, = $ ($# ( #!! 1 cn = 2πi
C
f (ζ) dζ , (ζ − a)n+1
n = 0, ±1, ±2 . . . ,
:A>I<
( C & ( #$ !$ , = $# # R2 < |z − a| < R1 $# % " a %
#
O!! $ * 7 ( !! $"$B " ! ) K * $ , 7$B K6# 7 !! $" # $B )* !& @ "! a !$ !6# 7 " f (z), 7= 6 !$ 0 < |z − a| < R 6 "$ :! $ 9"7 "$ a<, 7 7 f (z) 6 !! $" 9 ' ! " a 7 f (z) %$ $!"! 4$ # 6 $ $$ $ $B $ !($, ! ( 7$B ( 9) 6# 7 ! $!7 # !$B % $ $B " 1◦ 4$ !% $ ( "$ a 6 % *!$" !"7 lim f (z) z→a @!! % $ ! "! !$ !6# 7 E$!$ 7( !! $# " (H6$$ 7$ (z − a) ' # $ !( *!$" !"7 f (z) $ z → a ( 96 c0 @ !"!6, = ! ! $ f (a) = c0 , $" |c0| < ∞ '($ , = ! % $ "$, $ 7 f (z) 6 %)
9, ! %$ ( ! ! $* 7( f (z) = (z − a)m ϕ(z) ,
:A>J<
lim f (z) = 0, () ( m ≥ 0 & "$ , ! ϕ(a) = 0 ?= $ # z→a ! :A>J< "$ m > 0 !$ !6# 7 f (z) " z = a
$ %+& 2 sinz z ) " %
' ( )
, $ %(#)7 z = 0 ! ! $ !$ sin z z2 z4 = 1− + − ... , z 3! 5!
% 7( !! $# #$ ! $ # "! $ 2◦ 4$ !% $ ( "$ a 7 f (z) " !6, % lim f (z) = ∞ ' # $ !( "! a !$ !6# 7 m z→a - f (z)
(&-&%'& ! & #" a " - f (z)& "1
& #" 8 ' " * ∞
c−m c−1 + f (z) = + ...+ ck (z − a)k . m (z − a) (z − a)
f (z)
a
:A>M<
k=0
4 ' 0) ) %
(7 $ ($ 1 ! $!"$$ 9 g(z), % ( f (z), % g(z) = f (z) ! " ( ! $$ 9 9 *( :A>J< $* 7( g(z) = (z − a)mϕ(z), ( ϕ(a) = 0 @ ( a 9 f (z) ! ! $ !$ f (z) = (zψ(z) , ( − a)m ψ(z) 6 !! $" 9 69 @ % , ! $ H7 7$ 9 !$ !! $"$B & "! a, = 6 7( m !! $" g(z), ( "! 6 9 * 7( f (z)
$ %,& 2
1 ) e + 1 z = ± π(2k + 1)i, k = 0, ±1, ±2, . . .% z2
, 7 g(z) =
1
1 2 = ez +1 !6 $B "!B K * 7(, f (z)
% B(! $B 2zez ( 96 9 P !"$# 9 $ & K * 7( !K ! ! ( B % $ !B ($!$B 3◦ 4$ !% $ ( "$ a 6 *!$" * !"7 lim f (z) ' z→a # $ !( "! a !$ !6# 7 ) " 2
' (
(&-&%(& 4 a ) ) "
- f (z)& 8 a %
(7 $ ($
$ %-& 2 e %
1 z
) ) "
, $ %(#)7 z = 0 ! ! $ !$ 1
ez = 1 +
1 1 1 1 1 + + + ... , z 2! z 2 3! z 3
% * ! "! $! 7( !! $# % # # " E( !" !! $" ! ( $$ ! ( ! ! K$B ! $ ! B!! B % $ $B " 1◦ )* !& Q7 f (z) !$ !6# 7 !% , 7= ! !*! !6 % $ $B " '( ( AAJ ( #! ! 7 %$ ( ! ! ) ∞ $ 7( n=0 cn zn , = %*!6# 7 =$ !) ( $ $ ( 7 7= 7 9($ %$ ( ! ! $* 7( % * * 7(, ! 6 9 69 4$ !(!$ $B # * !$ $, !$ ) , sin z, cos z ! K 3!, ( % $B ! %(# $$ 7$ 2◦ )* !& Q7 f (z) !$ !6# 7 " !% , 7= ! !6 K$B % $ , 9 4$ !(!$ $B # * !$ ! (% )! ! # ( $B
*.$ ! " ' % " " "$ 0 ' * 1 " $ ( $ # " " + $ % " "
2 - $ # # #
-!K % ! , = 6 % $ "$ z0 7 f (z) %$ 6($$ "$ ( ! ! $* 7( 7( !! ∞
(
:C;<
cn (z − z0 )n ,
n=−∞
1 cn = 2πi
! c−1
C
1 = 2πi
f (ζ) dζ , (ζ − z0 )n+1
:C><
f (ζ)dζ . C
)* !&
z0
8' - - f (z) " ) & # )
, .& ( ( 1 2πi
< " ) C & # " - f (z) ) " z0 - f (z)% . !") C
f (ζ)dζ %
7 $ $ !6, = ! :C>< *6 ( 7 %"$ 7 $K! f (z) # ! % $ " 1 res[f (z), z0] = 2πi
:CA<
f (ζ)dζ = c−1 . C
O !, ! % $ " $K ! ($ ( 96 9 - * 7 (7 $ !($ 1◦ B! "! z0 6 9 K * 7( f (z) @ ( 6 "$ !6 !( f (z) = c−1 (z − z0 )−1 + c0 + c1 (z − z0 ) + c2 (z − z0 )2 + . . .
:CC<
4 $ %$( "! $$ :CC< ! (z − z0) 76 z → z0 :CF<
c−1 = lim (z − z0 )f (z) . z→z0
' (! $ !( 7 f (z) "$ $* 7( ( K7 ( B !! $"$B f (z) =
z0
%$ ( ! !
ϕ(z) , ψ(z)
:CG<
$" ϕ(z0) = 0, ! "! z0 6 K * 7( ψ(z), % 1 ψ(z) = ψ (z0 )(z − z0 ) + ψ (z0)(z − z0 )2 + . . . , 2
ψ (z0 ) = 0 .
:CI<
' #! $! $ ( 7 %"$ 7 $K 9 K * ) 7( res[f (z), z0] =
ϕ(z0) , ψ (z0 )
(
f (z) =
ϕ(z) . ψ(z)
$ %.& $ ' - f (z) = zn z− 1 %
:CJ<
, + &
√ n
2πk 1 = exp i n
E % $ $$ "!$ 6 6 "$ zk = :k = 0, 1, . . . , n − 1<, $" $ 6 9 $ K * 7( , 9 z f (z) ! ! $ !$ 7 f (z) = (! 7 g(z) = z n − 1 zk g(z) %$ ( ! ! $* 7( 7( @ !
1 g(z) = nzkn−1 (z − zk ) + n(n − 1)zkn−2(z − zk )2 + . . . , 2
g (zk ) = nzkn−1
2πk(n − 1) = n exp i n
= 0 .
@ % , zk 6 K * 7( g(z) @ (, *( :CJ<, ! ! $ !$
4πk zk 1 zk2 1 2 1 . res[f (z), zk ] = n−1 = = zk = exp i n zkn n n n nzk
:CM<
2◦
B! "! z0 6 9 7( m f (z) @ ( 6 "$ !6 !( f (z) = c−m (z −z0)−m +. . .+c−1 (z −z0)−1 +c0 +c1 (z −z0 )+c2 (z −z0 )2 +. . . :C;N< 4 $ %$( "! $$ 6 ! (z − z0)m (z − z0 )mf (z) = c−m +. . .+c−m+1(z − z0) +. . .+c−1(z − z0)m−1 +c0(z − z0 )m +. . .
'# ( % B "! $ B( 7( (m − 1) ( ( *!$" * !"7 $ z → zk ' #! $!6 ( 7 %"$ 7 $K 9 7( m dm−1 1 lim m−1 [(z − z0 )mf (z)] . res[f (z), z0] = z→z (m − 1)! 0 dz
:C;;<
* %!"$$, = ! :CF< 6 "! $$ $ !( !# $
$ %/& $ ' - f (z) = (1 +1z2)n %
, &
E % $ $$ "!$ 6 6 "$ z1,2 = ±i, $" %$( 6 ) 1 9 !$ 7( n , % 9 g(z) = f (z) % $ ( * 9 :! $ !( z = i< ! ! $ 7( @ !
g(z) = (1 + z 2 )n =
1 (n) 1 g (z = i)(z − i)n + g (n+1) (z = i)(z − i)n+1 + . . . n! (n + 1)!
$" B( K * n 7( " z @ ( *( :C;;< ! ! $ !$ res[f (z), i] = = = =
= i
( 99# 9
dn−1 1 1 = lim n−1 (z − i)n (n − 1)! z→i dz (1 + z 2 )n 1 1 dn−1 = lim (n − 1)! z→i dz n−1 (z + i)n n(n + 1) · · · (2n − 2) 1 = (−1)n−1 · (n − 1)! (z + i)2n−1 z=i (2n − 2)! 1 (2n − 2)! (−1)n−1 · = −i . [(n − 1)!]2 (2i)2n−1 22n−1[(n − 1)!]2
:C;><
# - #
+&'&%+& 2 ) 3 4 91 =.>, )
" G & " - "1 zk .k = 1, . . . , N ,& # " G % 4 f (ζ)dζ = 2πi Γ+
N
res[f (z), zk ] ,
:C;A<
k=1
Γ+ ) " G & 1 %
!*!(!6 , = 7= 7 f (z) 6 !! $" 9 ! % ! G , "$ *!$ Γ 6 % ! 6 ! $ #$$ : % , % $ $$< 4 () ! % $ "$ zk f (z) !$ γk , = $# K$B % $ $B " zk ' $! !$ "$ !
, + &
%!*! H7 % ! , = % ! !$ Γ γk :($ $ < 7 f (z) %( 9($ !! $" 9 '! ( # * ! ! $ !$
f (ζ)dζ +
N k=1
Γ+
f (ζ)dζ = 0 .
:C;C<
γk−
'$ $ 9"$ !"7 :CA<, $!6 ( 7 $ ' $ !$" !"7 $ :C;A< 7*!6 , = %!*!# B $ !(!B $7 76# 7 !%!*! K %"$ $$ $K$ f (z) %) $ $B "!B, = !# ($ % ! * !7, ! % (# %"$ 9 !$ *! "! $$ :C;A< - * 7 = ( ! $ 77 B! "! z = ∞ 6 # ! 9 % $ 9 " 9 !! $" f (z)
)* !& 8' - - f (z) z0 = ∞ )
& # ) 1 2πi
1 f (ζ)dζ = − 2πi C−
f (ζ)dζ , C+
C 5 & f (z) ) ) "1 & z = ∞%
'! ( $!"7 6 7( !! :($ :C><< ! ! $ !$ 1 res[f (z), ∞] = − 2πi
f (ζ)dζ = −c−1 .
:C;F<
C+
O ( $, !, $ $ !6, = 7= "! z = ∞ 6 ! 9 % $ 9 " 9 f (z), res[f (z), ∞] %$ ($ ( 7, "! 7 $K " ! % $ " ! ($ ( 96 9 @!$ "$ , :C;A< :C;F< $ $ !6 ( 7 ! ) $
+&'&%,& 91 =.>, )
#& " - 1 "1 .k = 1, . . . , N ,& z = ∞% 4 N k=1
res[f (z), zk ] = 0 .
zk
:C;G<
,' / ( 0
3 #4
B! ( % ! G1 G2 !9# #$B " , ! !9# # ( 7 *!$ γ $B % ! 7B 6 !(!$$ ( ( !" !! $" f1(z) f2(z), ( ( )* !& Q7 f2(z) 6 "
- f1(z) % ! # G2, 7= 6 !! $"! % ! G1 + γ + G2 7 f (z) !!, =
f (z) =
f1(z) z ∈ G1 , f2(z) z ∈ G2 .
:C;I<
O*( $ AC;N !! $" ( 7 :7= $ < $!"!) 6# 7 ( !"
#
* , # ,' - #
B! %B( %"$ $$ *! ( ( f (x) ! (7 ) ! : % "$ % < (a, b) x @%! ( $$ (a, b) (7 9 $ 9 C , = ! (a, b) 96 *!$9 % ! G , !! $" ( ) 6 f (x) % ! # G %( ! * !$ "$ !! $" * ( 7 f (z) ! ! !$ $K$ b
f (z)dz = 2πi · R ,
f (x)dx + a
:C;J<
C
( R & ! $K f (z) % ! G ?= *! C (!6# 7 %"$ $$ !% $!$$ " K!$ *! ab, !(!" ! ! !$ H7! 9 (7$B $ !(!B ( 9 f (z) %$!9# !$ "$ , = % $B(! 7, = !(!! ! (a, b), % ! ( 9 "$ 7 9 "! $ 9 @ ( K!$ ! !B ($# 7 ! ( * 9 ( ( * ( 7 ( $B 7 $B "! $ $ :C;J< $ !( "$B ( (a, b) $"! * 7(!9# 7 ($$ ) , = " K$99# 7 !$ "$ , = % #! *!$" * B ( $!$ *! ! ! (a, b) @ *! "! $ !
, + &
! %"$ 9 !$, ! $K !$ * *!$" !"7, $" !"! K $7 76# 7 $ 9
C
+&+&%& ; ! - * 7 *!
2π R(cos θ, sin θ)dθ 0
2π I=
:C;M<
R(cos θ, sin θ)dθ , 0
( R & ! ! #! 7 B !* O %$ !
1 1 1 1 dz 1 1 , cos θ = (eiθ + e−iθ ) = z+ , sin θ = z− . z = eiθ , dθ = i z 2 2 z 2i z
4$ θ ( 0 ( 2π ! ! z %*!6 !$ & ) |z| = 1 ( (! ! 7 @!$ "$ , *! :C;M< B ($# *! ! (
1 I= i
1 dz 1 R(z + , z − ) . z z z
:C>N<
|z|=1
E #$ ! ($ ( 7 ! ! # ! ! $ !$ 1 a0 + a1 z + . . . + an z n 1 ˜ R(z + , z − ) ⇒ R(z) = , z z b0 + b1 z + . . . + bm z m
:C>;<
˜ 6 !! $"! 7 ($ ! |z| = 1 9($, ! $7 ) ( R(z) " N ≤ m # % $ $B " zk , = 6 7$ !$! :C>;< @ ( ! ( $ C>;C ! $!$ #! I = 2π
N
:C>><
˜ res[R(z), zk ] .
k=1
˜ ?= mk & 7( 9 ! zk R(z) : " $( , = Nk=1 mk ≤ m<, !9 ! ( * 9 :C;;< ! $ !$ $* 7( I = 2π
N k=1
1 dmk −1 mk ˜ lim (z − zk ) R(z) . (mk − 1)! z→zk dz mk −1
:C>A<
,, (
( ) $
$ %0& 6" 2π
I= 0
dθ , 1 + a cos θ
:C>C<
|a| < 1 .
O %$ ! z = eiθ $!6 1 I= i
|z|=1
1 dz 2 dz · = . 1 a z i az 2 + 2z + a z+ 1+ |z|=1 2 z
O!$ !6 ( ! K * 7( z1,2 √ −1 + 1 − a2 a
$# ($ *! :C>>< $!6
=
|z| = 1
−1 ±
:C>F< √ 1 − a2 a
@ #$ z1
@ ( ! ( * 9 $
1 2π 1 √ = 4π , z I = 4π res = . 1 az 2 + 2z + a a(z − z2 ) z=z1 1 − a2
=
:C>G<
$ '1& 6" π
I= −π
ln | sin θ| dθ , 1 + a cos θ
|a| < 1 .
:C>I<
@! 9 ! 9, 7 (# $ !(, $!6 1 I= i
|z|=1
1 2 dz 1 − z2 dz · = Re · 2 . ln 1 a z i 2 az + 2z + a z+ 1+ |z|=1 2 z
:C>J<
@ % $ $ ! ( K7
iθ 2 1 − e2iθ e − e−iθ = ln = Re ln 1 − z . ln | sin θ| = ln 2 2i 2
@ ( ! ( * 9 $ :C>>< $!6
1 1 − z2 · 2 , z1 = I = 4π res ln 2 az + 2z + a √ 1 − z2 1 − a2 − (1 − a2 ) 1 2π = 4π ln · =√ · ln . 2 a(z − z2 ) z=z1 a2 1 − a2
:C>M<
, + &
+&+&'& ; !
∞ f (x)dx −∞
O! 6 ( $K ( %"$ 7 ! $B *! $* 7( ∞ I=
:CAN<
f (x)dx −∞
$ %( * 7(!$ $ !( , $ 7 f (x) 6 !(! 9 ! ( %$ !! $" ( ! ! B9 ! =$ !$ "$ , = ( 7 %( !( #7$ (7$ ( (! $ ! !(! ! !( %$# 7 !
7 % % 91 f (z) ) 1 # Im z > 0
- 1 % 91 R0& M δ& # 1 1- #& # |z| > R0& ) |f (z)| <
4
M , |z|1+δ
:CA;<
|z| > R0 .
lim
R→∞ CR
:CA><
f (ζ)dζ = 0 ,
CR ) " |z| = R& Im z > 0 1 # z :$ C;<
y6
CR R
K -
z=0
-$ C;
x
, ! ( :>I< ! $ $ $ R > R0 6 ! ( $ $ ( K7 MπR Mπ f (ζ)dζ ≤ |f (ζ)| dζ < 1+δ = δ −→ 0 . R R C C R→∞ R
R
,, (
( ) $
< #! % % ?= $ $ $ 9# 7 (7
ϕ1 < arg z < ϕ2 =$$ z , ! :CA>< ( CR !, = $# #
!6 $ * !
< #! ' % $ $ $ 9# 7 $ !(, $
7 f (z) 6 !! $" 9 " ((! "$, 7 "! z = ∞ 7 76 % 9 # $ " (* * 7( f (z) , # $ !( !( f (z) 7( !! z = ∞ !6 $* 7( c2 c3 ψ + + . . . = , z2 z3 z2 |ψ(z)| < M , ($ $ $ !6
f (z) =
$" ! :CA;< $ δ = 1 ! ; !B ($# K$ ! !7 $ %"$ (7$B ! $B ∞ *! $
f (x)dx
−∞
+&+&%-& 91 f (x)& # −∞ < x < ∞&
" 1 # Im z ≥ 0& -- f (z) ) + )
"1 % 4
∞
f (x)dx
)
−∞
)
∞ f (x)dx = 2πi
N
res[f (z), zk ] ,
:CAA<
k=1
−∞
zk 5 " - f (z) 1 #%
O! $ $ 7 f (z) B ! =$ !6 " #) # % $ $B " zk , $" $ !( #79# |zk | < R0 - ) * 7 !$ , = !(!6# 7 (! ( −R ≤ x ≤ R :R > R0< ! ! CR, |z| = R, B ! =$ @ ( ! ( $ $K 6 ! ( $ 9 #
R f (x)dx + −R
f (z)dz = 2πi
CR
N k=1
res[f (z), zk ] .
:CAC<
, + &
E #$ 6 $ ! $ $ ;, *!$" !"7 (* * ( (!! "! $ :CAC< $ R → ∞ ( 96 9 4$ # ! ! "! $! # * $! ( R ! $# < #! % +&+&%-& P! *"! ! !6 $ !(, $ !! $" ( 7 f (z) $ 9 ! =$ !( #76 $ $, = 6 !! *" 9 $ ;
$ '%& ∞
I= −∞
dx . x2 + 1
:CAF<
P! $" ( 7 (*! # !( #76 $
π $ U % $ 9 " 9 B ! =$ 6 "! z0 = exp i 2 , $"
! 6 9 K * 7( @
1 1 , z0] = 2πi π = π . I = 2πi res[ 2 z +1 2z z=ei 2
:CAG<
E" $( , = #! * $!$ $"! !! , #$ ∞
I= −∞
dx = x2 + 1
+∞ π π arctg x =π = − − 2 2 −∞
$ ''& ∞
I= −∞
dx . +1
x4
:CAI<
P! $" ( 7 (*! # !( #76 $ $ CC;G U % $ $$ "!$ B ! =$ 6 "$ z0,1= exp i π+2πk , 4 :k = 0, 1<, $" %$( "$ 6 9 !$ K * 7( @
1 1 , z0 ] + res[ 4 , z1] = I = 2πi res[ 4 z +1 z +1 √ :CAJ<
π 2 1 1 = = 2πi . + 3 4z 3 z=ei π4 4z z=ei 3π4 2
E$!$ #! !! ( 6 $ ! # 9 !(!"9
,, (
( ) $
$ '(& ∞ I= 0
cos xdx . x2 + a2 y6 ia
−R
:CAM< 7 %"$ 7 # * *! ! $ 6 # ! 9 (*! # , % !! ) $" ( 7 $* 7(
CR
t K
-
-
z=0
eiz f (z) = 2 z + a2
-
R x
! % * !7, %! $ ! $ C>, ( 7 7 * R > a 1 E #$ ! CR 7 f (z) !( #76 |f (z)| ≤ R2 − , a2 ! !( #76 $ $ ; 4 $ 9 f (z) !B ($# 7 " z = ia ( $ R > a ! ! $ !$ -$ C>
R −R
eixdx + x2 + a2
eiz dz e−a . = 2πi z 2 + a2 2ai
CR
'( 99"$ ( "! $ $ $ 9"$ ! # , !( K!$ *! ∞ 0
π cos xdx = . x2 + a2 2aea
+&+&(& ; !
∞ eiaxf (x)dx
& 7 =
−∞
7 ' 2 = 4 91 f (z) ) 1
# Im z > 0& " - 1 " 1 & arg z ) |z| → ∞%
, + &
4 a > 0 )
lim
R→∞ CR
:CCN<
eiaζ f (ζ)dζ = 0 ,
CR 5 |z| = R 1 #%
! * 7 !7 f (z) ( 7 !"!6, = $ |z| = R !6 !
|f (z)| < µR ,
|z| = R ,
( µR → 0 $ R → ∞ ' *! :CCN< %$ ! ζ $ 6 # ( K7
= Reiϕ
$ 0 ≤ ϕ ≤ π2 . ' #! $!6 sin ϕ ≥
2 ϕ π
π π eiaζ f (ζ)dζ ≤ µR · R eiaζ dϕ = µR · R e−aR sin ϕ dϕ = C 0 0 R
π/2 π/2 2aR π = 2µR · R e−aR sin ϕ dϕ < 2µR · R e− π ϕ dϕ = µR (1 − e−aR ) −→ 0 . a 0
0
R→∞
< #! ( ' ?= a < 0, ! 9 !$ $ !
$ ( 7 (*$ $ ! =$ P! *" ( 7 !9# $ !(!B a = ±iα :α > 0<, $ * !7 ($# 7 ( ( ! :Re z > 0< !% :Re z < 0< ! =$ < #! + ' ! W (!! ! $K!6# 7 ! ( $ 9 $) !(, $ f (z) !( #76 $ $ ! =$ Im z ≤ y0 (7 ($# 7 !! *" (# $ ! ζ = Reiϕ + iy0 < #! , ' ! W (!! ! $K!6# 7 ! ( $ 9 (7) $B $ !(!B, $ f (z) !( #76 !% $ 3! $ $ $) ($$ %( , ! ($ #$, = H7! !7 ( (! *
,, (
( ) $
$! eiaζ , = $ a > 0 !% "6 K $( !(!7 B ! =$ ! W (!! !B ($# "$ ! !7 $ %"$ ! $B *!
+&+&%.& 91 f (x)& # &
" 1 # Im z > 0& -- ) "1 1 # ∞ iax ) ? % 4 −∞ e f (x)dx& .a > 0,& ) ) ∞
iax
e
f (x)dx = 2πi
n
res[eiaz f (z), zk ] ,
:CC;<
k=1
−∞
zk 5 " - f (z) 1 #%
O! 9 $ % $ "$ zk f (z) !( #79# |zk | < R0 - * 7 !$ , = !(!6# 7 (! ( −R ≤ x ≤ R, R > R0 (*$ CR ! ! |z| = R B ! =$
O*( $ $K C>;C ! ! $ !$ R
iax
e
f (x)dx +
iaζ
e
f (ζ)dζ = 2πi
res[eiaz f (z), zk ] .
k=1
−R
n
CR
O! 9 W (!! *!$" !"7 (* * *! ! "! $ $ R → ∞ ( 96 9 O ( $ $ $ !6 ( 7 $ 4$ ( ( CC;G ! CC;I $ ! 7, = 7 f (x) !6 % $ $B " ! ( '$7 76# 7 = (7 !" ( $7 ( 79# ! !$ $ ( %"$ 7 ! $B *! $ !(, $ 7 f (x) !6 ( #! % $ $B " ! ( 4 ! ! !( $ !(
$ '+& 6" ∞ I= 0
sin αx dx , x
α > 0.
:CC><
, + &
3 $ ! K$ # ! 9
*! $* 7(
∞
1 I = Im 2
−∞
(*! # , ! $ !$ )
eiαx 1 dx = Im I1 . x 2
:CCA<
'($ , = *! I1 %B( * 7(!$ 7 * !"7 ! * *! , ! ! ∞ I1 = V.p. −∞
⎧ −ρ ⎨ eiαx
eiαx dx = lim ⎩ x ρ→0
−R
R→∞
y6
CR
x
R dx + ρ
⎫ ⎬
eiαx dx . ⎭ x
:CCC<
- * 7 B ! =$ Im z ≥ 0 !) $ Γ, = !(!6# 7 ( () [−R, −ρ], [ρ.R] ( B ! Cρ, |z| = ρ, ! CR, |z| = R :$ CA< ' ($ # * ) eiαz ! !! $" ( 7 , ! ! z , !6 % $ $B " @ ! ( ! $ K ! ! $ !$
-
−R
K
−r 0
-
-$ CA
r −ρ
f (ζ)dζ = Γ
−R
-
R x
eiαx dx + x
R ρ
eiαx dx + x
Cρ
eiαζ dζ + ζ
eiαζ dζ = 0 . ζ
:CCF<
CR
E ! ( (! ! ( ! $ W (!! 76 ( 7 $ R → ∞ ' # ( (! ! Cρ ( !6# 7 (H6 ! 7 :! 9 * ($$!<, $ ! ζ = ρeiϕ $!6
I3 = Cρ
eiαζ dζ = i ζ
0
eiαρ(cos ϕ+i sin ϕ) dϕ −→ − iπ . ρ→0
π
:CCG<
- * 7(!9"$ !"7 :CCF< !B ρ → 0 R → ∞ *( :CCC< :CCG< $!6 ∞ V.p. −∞
eiαx dx = iπ , x
α > 0,
:CCI<
,, (
( ) $
($ ∞ 0
sin αx π dx = , x 2
:CCJ<
α > 0.
$ ',& 6" ∞ I=
sin ax dx . sh πx
e−πx
0
:CCM<
'# ( 9 f (z) =
cos az + i sin az eiaz −πz cos az + i sin az = = e , e2πz − 1 e2πz − 1 eπz − e−πz
% ( 6! 7 ! "! $! ! ( x ( 96 (*! # ) 4 *6 9 9 , = %! $ ! $ CC - * 7 !"7 6 y
ib I
bR
?
+i
6
R II 0b r
b
-
R
-
x
iax eia(x+i) −a e = e 2πx = e−a f (x) , f (x+i) = 2π(x+i) e −1 e −1
% , ! %H6(!$ *! $ B ! $ *!$7B ($, ! !
-$ CC
1 − e−a
- * 7 *! ! ( 7 R, R + i R+i R
eiaz dz = i e2πz − 1
1 0
R f (x)dx . (A) r
eia(R+iy) dy = ieiaR−2πR 2π(R+iy) e −1
7$ 76 ( 7, $ R → ∞
1 0
e−ay) dy , e2πiy − e−2πR
(B)
, + &
.*! ( 7 !, = $# ! 7 , %"$ 96# 7 !$ "$ ri
(1−r)i
f (z)dz = − (1−r)i
1 =− 2
1−r
1−r
f (z)dz = −i ri
−iπy
e r
e−ay 1 dy = − sin πy 2
1−r r
r
e−ay dy = −i e2πiy − 1
1−r e−iπy r
e−ay dy = eπiy − e−iπy
(cos πy − i sin πy)e−ay dy = sin πy
1−r 1−r i 1 ctg πye−ay dy + e−ay dy . =− 2 2 r
r
E #$ ! ! $# #$ 7 ! "! $! *! !, %"$ $ #$ 5 (!6 #! 1 2
1−r 1 −a(1−r) 1 −a −ay −ar e e − 1 . (C) e dy = − −e −→ − 2a 2a r→0 r
! ( 7 ! % $ "$ z = i, = !"! I , %$ ! z = i + reiϕ dz = ireiϕdϕ @ ( (*! #! 7 %$ ( ! ! $* 7( f (z) =
eia(i+re
iϕ
)
e2π(i+reiϕ ) − 1
−a
=e
eiare
iϕ
e2πreiϕ − 1
.
- !( "$ #$ !$ ( % ! "! $ !# * $! ( ( 7($ @ ! "$ #$ & iϕ
eiare = 1 + iareiϕ +
!$ & iϕ
e2πre = 1 + 2πreiϕ +
2 1 1 iareiϕ + . . . = 1 + ia(z − i) + (ia(z − i))2 + . . . ; 2 2
2 1 1 2πreiϕ + . . . − 1 = 2π(z − i) + (2π(z − i))2 + . . . . 2 2
' #! ! ( 7 (*! #! 7 !% !6 $* 7( f (z) =
1 e−a · + P (z − i) , 2π z − i
,, (
( ) $
( P (z − i) & ! $ #! " z = i 7, (!
f (z)dz =
I
e−a 2π
I
−a
e = 2π
1 z−i dz
+
π
− 2 0
P (z)dz =
I
ireiϕ dϕ e−a + O(r) . (D) + O(r) = −i reiϕ 4
P! *" % $ "$ z = 0 II
1 f (z)dz = 2π 1 = 2π
II 0
π 2
1 dz + z
P (z)dz = II
ireiϕ dϕ 1 + O(r) . (E) + O(r) = −i reiϕ 4
E #$ ($ ! (*! #! 7 !6 % $ , *( $ K *! ! ( 96 9, "$ ( 96 9 * 7 ! "! $! O ( $ $!6 ! "$ ) #! ∞
∞ sin ax 1 dx = 2 Im f (z)dz = I = e−πx 2 Im [−B − C − D − E] = sh πx 1 − e−a 0 0 e−a 1 1 −a 1 1 + e−a 1 2 e −1 + + = · − . = 1 − e−a 2a 4 4 2 1 − e−a a
/ """ # $
2 * 2 * $ # 0 2
" 0 ) % ) + 3 $ 0 * ( ,0 + 3 $ $ /
!
)
#
2
4 7 ! ! ! 6 (69 $ $B ! ! ! ! * $$ ( ( , $ ! ! $$ ( ( # !$ ( B ) $B ! %H6 ! $ !( 7$ ! !$, 7$ 7$ = ' 7 ! ! ! $# (7 *! # * 7, $ # , ! $ !(, H7 (7 $B( !(!" : $"!) * ($! # * 77<, = 7 76 % 9 9 ( f (t), ! $# 7 ( ( # 7 F (p), 7! !( #) 76 (7 ! *%!" 79 P% 79 "! $$B B($B ! $# 7 ( ( # $"! ($! # 77 )* ! @ 8 ! $# ( ( # f (t) ( t 9 F (p) p ! ( * 9 ) 7 ∞ F (p) =
:F;<
e−ptf (t)dt .
0
1 (
.
4$ ( , = *! 6 ( 7 ( # f (t) @ %B( !!$ ! , ( 7 7$B 7 6 ( * 7(!$ f (t), = 6 $!"$$ ( 7 B !"# ( −∞ < t < ∞ !( #79# ! $ 1◦ 4$ t < 0 7 f (t) ≡ 0 2◦ 4$ t ≥ 0 7 f (t) ! ( # % ( 7 t !6 % #K " # # " $ K * ( 3◦ 4$ t → ∞ 7 f (t) !6 % K $( # !7, % ( 7 # * ! 9# ! (H6 ! M ! a, = ( 7 B t > 0 $ 6# 7 ! |f (t)| ≤ Meat . :F>< @ "! $ 7 *!$7 !"# a !$ !6# 7 f (t) O !, ( 7 f (t) = tn ! "$ !7 ( 96 9 '($ , = *! :F;< 6 ! $ *! , = ! $# ( ) p 7 ( !!! O # * $ $ !6, = *! %*!6# 7 $ B !"7B !!! p ! $ !(, 7= 7 f (t) $ t → ∞ 76 ( * *!$" * !"7, = (76# 7 ( 7, ! Re p < 0, *! %*!6# 7
,&%&%& < .A%+, ") " Re p > a& a 5
- f (t)& x0 > a .A%+, Re p ≥ x0 > a ") % 7 ( # * p = x + iy $ x > a ! !!$ ! ε > 0, = x > a1 = a + ε, $" |f (t)| < Mea1 t @ ( ∞ ∞ M |F (p)| = e−pt f (t)dt ≤ M e−xt ea1 t dt = , x > a1 , :FA< x − a 1 0
0
% *! %*!6# 7 $ x > a ?= x ≥ x0 > a, !! *"! ! (!6 ∞ |F (p)| ≤ M 0
e−(x0 −a1 )t dt =
M , x0 − a1
= ( ($# % #
:FC<
2
.
Q7 F (p), = $!"! ! ( * 9 :F;<, !$ !6# 7 " 8 f (t) Q7 f (t) !$ !6# 7 F (p) O H7) f (t) ! F (p) %( $ " !"!$ !$ "$ f (t) F (p)
!%
:FF<
F (p) f (t) .
!*!(!6 , = ( !% #K ! $ $ !) 6 !! $" OH7 6 , "$ 6 !! $" 9 69 %! 7 ! ! ! F (p)
,&%&'& $" 8 F (p) - f (t) ) ) - - f (t)%
p
"
Re p > a&
a
5
(7 $ ($
,&%&%& < #! 7 6 O!( %! 7 (7$B !$B ( 1◦ 6 B B!
f (t) = σ0(t) =
0, 1,
:FG<
t < 0, t ≥ 0.
@ ( ∞ f (t) F (p) =
e−ptdt =
0
1 , p
! 7 F (p) $!"! % ! Re a > 0 @!$ "$
f (t) = σ0(t) =
2
◦
0, 1,
1 t<0 , t≥0 p
Re p > 0 .
:FI<
!(! 9($ %( $, = 7 f (t) !"!6 ( % f (t) · σ0(t)
@ f (t) = eαt .
:FJ<
1 (
.
E%"$ $ *! :F;< $!6 ∞ F (p) =
e−pteαt dt =
0
3◦
1 eαt , p−α
:FM<
Re p > α .
f (t) = tν ,
:F;N<
ν > −1 .
' # $ !( *! :F;< !6 $* 7( ∞ F (p) =
e−pttν dt ,
:F;;<
Re p > 0 .
0
4 " $ !(, $ ! p $!6 ( !"7 p = x > 0 @ ( ∞ F (p) ⇒ F (x) =
e−xt tν dt =
0
1 xν+1
∞
e−s sν ds =
0
Γ(ν + 1) , xν+1
:F;><
( Γ(ν + 1) & *!!)7 ! '! ( 6($ !! $" * ( )
7 7 F (p) % ! Re p > 0 !6 $* 7( ∞ F (p) =
e−pttν dt =
0
Γ(ν + 1) . pν+1
:F;A<
4$ # $ !( ( %$B !"# ν %B( %$!$ * %!*! ) 1 !" pν+1 , 7! 6 % ( !! $"$ ( 7 % ! # 1 ( xν+1 ( x > 0 @!$ "$ ,
Re p > 0
tν
Γ(ν + 1) , pν+1
ν > −1 ,
Re p > 0 .
:F;C<
O !, ( 7 $B ν = n ! ! $ !$ tn
Γ(n + 1) n! = , pn+1 pn+1
Re p > 0 .
:F;F<
2
.
!
) # 4
,&'&%& 2 * #!4
1 ) * # Fi(p) fi(t)& Re p > ai & (i = 1, . . . , n)& F (p) =
n
αi Fi(p)
i=1
n
αi fi (t) ,
:F;G<
Re p > max ai ,
i=1
αi ) & ai 5 - fi(t)%
$ '-& 2cos ωt4
? $ !( ! !7 6 ! $ $ :FM< $!6 1 1 iωt e + e−iωt cos ωt = 2 2
P! *" sin ωt
ω , p2 + ω 2
1 1 + p − iω p + iω
=
p , p2 + ω 2
Re p > | Im ω| .
Re p > | Im ω| .
:F;I< :F;J<
,&'&'& 91 F (p) f (t)& Re p > a&
1 p F f (αt) , α α
∞ 0
1 e−pt f (αt)dt = α
α > 0, ∞
− αp τ
e 0
Re p > a .
:F;M<
1 p . f (τ )dτ = F α α
,&'&(& 2 * *> !4 91 F (p) f (t)& Re p > a
)
fτ (t) =
0, t < τ , τ > 0, f (t − τ ) , t ≥ τ .
:F>N<
4 fτ (t) Fτ (p) = e−pτ F (p) ,
Re p > a .
:F>;<
1 0
∞ Fτ (p) =
−pt
e
∞ fτ (t)dt =
−pt
e
∞ f (t − τ )dt =
τ
0
−p(t +τ )
e
f (t )dt = e−pτ F (p) .
0
$ '.& 26 !4
- * 7 %! 7 B ($ :($ $ /XYZ>;M<
f (t) =
:F>><
0, t<τ, nf0 , nτ ≤ t < (n + 1)τ , n = 1, 2, . . .
7 $!"7 f (t) ! ! !$ 9 [ !(! σ0 f (t) = f0 [σ0(t − τ ) + σ0(t − 2τ ) + . . .] .
'$ $ !6 ! $ # , ! 9 !7 $!6 1 1 f0 e−pτ f (t) F (p) = f0 e−pτ + f0e−2pτ + . . . = . p p p 1 − e−pτ
f6
f6
2f0 f0 0
:F>A<
0
τ
2τ
3τ
4τ
-
t
-
τ
2τ 3τ 4τ
-$ F;
t
$ '/& 2 !4
-$ F>
P! *" ! $!$ %! 7 ($" :$ /XYZ>>N<
f0 , 2nτ ≤ t < (2n + 1)τ , n = 1, 2, . . . f (t) = :F>C< −f0 , (2n + 1)τ ≤ t < (2n + 2)τ ,
, "! ! $K f (t) = f0 [σ0(t) − 2σ0(t − τ ) + 2σ0(t − 2τ ) − 2σ0(t − 3τ ) + . . .] ,
2
.
!
1 −pτ 1 −2pτ 1 −3pτ 1 − 2e + 2e − 2e ... = f0 p p p p f0 2 − 2e−pτ + 2e−2pτ − 2e−3pτ . . . − 1 = p f0 2f0 1 − e−pτ + e−2pτ − e−3pτ . . . − = p p 2f0 1 f0 1 − e−pτ f0 f0 pτ = th . − = p 1 + e−pτ p p 1 + e−pτ p 2
f (t) F (p) = = = =
:F>F<
$ '0& 2 ! * !4
@ ! ! 9 !7 ( 76 $!$ !*! # ( 7 %! ) 7 ($" 4 (# * 7 $ !( , $ 7 ( f (t) %$ ( ! ! $* 7(
f (t) =
ϕ(t) , 0 ≤ t < τ , 0, τ ≤ t.
:F>G<
4 !"$ %! 7 ϕ(t) Φ(p) ! ϕ(t + τ ) Φτ (p) 4 $K :F>G< $* 7(
f (t) = ϕ(t) +
0, 0≤t<τ, −ϕ(t + τ − τ ) , t ≥ τ .
'$ $ !6 ! $ # , ! 9 !7 $!6 f (t) F (p) = Φ(p) − e−pτ Φτ (p) .
B! 7 ϕ(t) 6 ($" 9 ( τ , % Φτ (p) = Φ(p) @ ( $ :F>I< !6 Φ(p) =
ϕ(t + τ ) =
F (p) . 1 − e−pτ
:F>I< ϕ(t), ( :F>J<
,&'&+& 2* #! 6 4 3# f (t) )
" Re p > a f (t) F (p)& Re p > a&
f (t) pF (p) − f (0) ,
Re p > a .
:F>M<
1 0
5 6 (! $B ! $ %! 7, 7! ( 76 !$$ ($) 9 !7 $*$! ! ! 7 %! 7 ! ! .*6 "! $!B $!6
∞
f (t)
∞ ∞ e−pt f (t)dt = e−pt f (t) + p e−pt f (t)dt = pF (p) − f (0) .
0
0
0
,&'&,& 2* #! 6 56 ! 4 3# f (n)(t) ) " Re p > a f (t) F (p)& Re p > a& f (n) (t) pn
f (0) f (0) f (n − 1)(0) F (p) − − 2 − ...− p p pn
,
Re p > a .
:FAN<
(7 6 !! *"$ (# Q ! :FAN< % $ =6# 7 $ !(, $ f (0) = f (0) = . . . = f (n−1)(0) = 0 f (n) (t) pnF (p) . :FA;<
$ (1&
0 :' )*
:FA>< :FAA<
a0 y (n) (t) + a1 y (n−1) (t) + . . . + an y(t) = f (t) , y(0) = y (0) = . . . = y (n−1) (0) = 0 ,
f (t) ) ) t ≥ 0%
?= ! $, = f (t) = 0 $ t < 0, ! %( !$ %! 7 f (t) F (p) B! 7 y(t) B( !( #79# $ !7 %! 7 @ ( ( $ 77 :FA>< ! e−pt *6 t ( 0 ( ∞ ' #! $!6 ∞ n Y (p) a0 p + a1 pn−1 + . . . + an = F (p) , Y (p) = e−pt y(t)dt . 0
4 !"$ *$B ( !B 7 Pn (p) $!6 Y (p) =
F (p) . Pn (p)
:FAC<
2
.
@!$ "$ , 7= ! %( !$ ( $ $*! y(t) * ( ) %! 9 Y (p), $B(! !(!"! %( H7!!
,&'&-& 2* #! 4 91 f (t) F (p)& Re p > a%
4
t ϕ(t) = 0
1 f (τ )dτ F (p) , p
:FAF<
Re p > a .
t
∞ f (τ )dτ
0
0
=
1 p
e−pt dt
∞ 0
t
∞ f (τ )dτ =
0
∞ f (τ )dτ
e−pt dt =
τ
0
1 e−pτ f (τ )dτ = F (p) . p
,&'&.& 2* #! 4 91 f (t) F (p)& Re p > a% t ϕ(t) =
t1 dt1
0
0
4
tn−1 1 dt2 . . . dtn f (tn) n F (p) , p
Re p > a .
0
57 ! $ # ( ($# 7 !! *" ( )* ! O* 9 f1(t) f2(t) !$ !6# 7 7 $!"!6# 7 ( K7 t
t f1(τ )f2(t − τ )dτ =
ϕ(t) =
:FAG<
0
f1 (τ )f2(t − τ )dτ .
ϕ(t),
=
:FAI<
0
,&'&/& 2* #! * 4 3# f1(t) F1(p)& Re p > a1 f2(t) F2(p)& Re p > a2 &
t f1(τ )f2(t − τ )dτ F1(p)F2(p) ,
ϕ(t) = 0
Re p > max{a1 , a2} .
:FAJ<
1 0
∞ 0
e−ptdt
∞
=
t
∞ f1(τ )f2(t − τ )dτ =
0
e−pτ f1 (τ )dτ
0
∞
∞ f1(τ )dτ τ
0
e−ptf2 (t − τ )dt =
e−pt f2(t )dt = F1(p)F2(p) .
0
$ (%& $ - F (p) = (p2 +pωω2)2 %
O*( :F;I< :F;J< ! ! $ !$ p cos ωt , p2 + ω 2
ω sin ωt . p2 + ω 2
@ t F (p)
sin ωτ · cos ω(t − τ )dτ = 0
t sin ωt . 2
:FAM<
,&'&0& 2 > ! * #!4 91 F (p) f (t)& Re p > a%
4
F (p) −tf (t) ,
:FCN<
Re p > a .
F (p) =
d dp
∞
e−pt f (t)dt = −
0
∞
e−pt tf (t)dt −tf (t) .
0
59 ! $ # ! !*! #$$
,&'&%1& 2 > ! * #!4
91 F (p) f (t)& Re p > a% 4 F (n) (p) (−1)ntn f (t) ,
Re p > a .
:FC;<
2
.
,&'&%%& 2 ! * #!4 91 F (p) f (t)& Re p > a
f (t) t
∞ 0
f (t) t
) " % 4
f (t) dt = e−pt t
∞ F (q)dq ,
Re p > a .
:FC><
p
4 !"$ ∞ I(p) =
e−pt
0
f (t) dt , t
($96 $! : ! "! $ !!< d I (p) = dp
∞ 0
f (t) dt = − e−pt t
∞
e−pt f (t)dt = −F (p) .
0
'!B6 , = I(∞) = 0, $!6 ∞
∞ F (q)dq = −
p
I (q)dq = I(p) − I(∞) = I(p) . p
$ ('& $ " - 1t sin ωt%
E #$ sin ωt p2 +ω ω2 , 1 sin ωt t
∞ p
ω p π − arctg . dp = p2 + ω 2 2 ω
O !, $ ω = 1 π 1 sin t − arctg p . t 2
:FCA<
1 0
@ (, !B 9"$ ! $ # F>G, ! $!$ %! 7 *! # * ! t si t = 0
sin τ 1 π dτ − arctg p . τ p 2
:FCC<
,&'&%'& 2 * 5!4 91 f (t) F (p)& Re p > a% 4 λ ) '* F (p + λ) e−λt f (t) ,
Re p > a − Re λ .
:FCF<
∞ F (p + λ) =
e−(p+λ)t f (t)dt =
0
∞
e−pt e−λt f (t)dt e−λt f (t) .
0
,&'&%& ! * #
! ( $= ! $ ( *!9# !$ %! 7 %!*!# B 7 7$B $ %! $ $ " :t > 0< ;< 1 1p , Re p > 0D + 1) , ν > −1, >< tν Γ(νpν+1
A< tn pn! , n & !! #, n+1
Re p > 0D Re p > 0D
Re p > Re αD C< eαt p −1 α , F< sin ωt p2 +ω ω2 , Re p > | Im ω|D G< cos ωt p2 +p ω2 , Re p > | Im ω|D
I< sh λt p2 −λ λ2 , J< ch λt p2 −p λ2 ,
Re p > | Re λ|D Re p > | Re λ|D
2
.
M< tneαt (p −n!α)n+1 ,
Re p > Re αD
, ;N< t sin ωt (p2 2pω + ω 2 )2 2
Re p > | Im ω|D
2
Re p > | Im ω|D ;;< t cos ωt (pp2 +−ωω2)2 , ;>< eλt sin ωt (p − λ)ω2 + ω2 , Re p > (Re λ + | Im ω|)D
;A< eλt cos ωt (p −pλ)−2λ+ ω2 ,
;C< sintωt π2 − arctg ωp , pπ , ;F< | sin ωt| p2 +ω ω2 cth 2ω ;G<
Re p > (Re λ + | Im ω|)D
Re p > | Im ω|D
1 π − arctg p , si t p 2
!
Re p > | Im ω|D Re p > 0D
#
2 5
B! 6 ( $, = 7 F (p) 6 %! 7 f (t) % 9 K $( 9 !7 |f (t)| < Meat , !"7 ! a 6 !(!$ %B( ! (! 9 69 %( !$ K! 9 f (t) 57 !(!"! H76# 7 ! ( * 9 $
,&(&(& 2 "
4 91 ) & # F (p)
" Re p > a ) " - - f (t) - - t& # ) ' a% 4 1 f (t) = 2πi
x+i∞
ept F (p)dp ,
x > a.
:FCG<
x−i∞
- * 7 ( 9 ϕ(t) = e−xt f (t), x > a 57 7 6 ) * !(# 9, ! ( # % ( 7 t !6 " # # " $ K * (, ! # 76 ( 7 $ t → ∞ ' !
' 3 4
%$ ( ! ! ! ( * 9 *! ! QH6 1 ϕ(t) = 2π
∞
∞ dξ
−∞
:FCI<
ϕ(η)eiξ(t−η)dη .
−∞
'$ $ !6 $!"7 ϕ(t) e−xt f (t) =
1 2π
1 = 2π
∞
∞
e−xη f (η)eiξ(t−η)dη =
dξ −∞ ∞
−∞
eiξt dξ
∞
−∞
:FCJ< e−(x+iξ)η f (η)dη ,
0
#$ f (η) = 0 $ η < 0 @ (, !"!9"$ p = x + iξ , $ %$( " $$:FC>< ! ext f (t) =
1 2π
∞ −∞
e(x+iξ)t dξ
∞
e−(x+iξ)η f (η)dη =
0
1 2πi
x+i∞
ept F (p)dp .
x−i∞
Q ! :FCG< ! ! ! !$ " 8 , #$ ! ( *!6 !$ $*! ( %! 9 - * 7 (7 ! !7 6 $
,&(&+& 91 f1(t) F1(p)& Re p > a1 f2(t) F2(p)& Re p > a2% 4 1 f (t) = f1 (t)f2(t) F (p) = 2πi
x+i∞
x−i∞
1 F1(q)F2(p−q)dq = 2πi
x+i∞
F1 (p−q)F2(q)dq , x−i∞
:FCM<
F (p) ) " Re p > a1 + a2 & & # ) ) ' a1 < Re q < Re p − a2& 5 a2 < Re q < Re p − a1%
2
.
∞
f (t) F (p) = 0
=
1 e−pt f1(t)f2(t)dt = 2πi
1 2πi
x+i∞
eqt F1(q)dq
∞
∞
e−ptf2 (t)dt
0
e−pt f2(t)dt =
0
x−i∞
x+i∞
eqt F1(q)dq =
x−i∞ x+i∞
1 2πi
F1(q)F2(p − q)dq . x−i∞
57 ! (7 %! ( ! $ F>J
$ ((& 91 f1(t) = cos ωt& f2(t) = t% $ " f (t) = t cos ωt%
E #$ cos ωt p2 +p ω2 , t p12 , ! ( :FCM< ! ! $ !$ 1 f (t) F (p) = 2πi x+ iR r
Im q 6
r
r
| Im ω|
Re p
r
x − iR
-$ FA
x+i∞
x−i∞
qdq , (q 2 + ω 2 )(p − q)2
-
Re q
Re p > | Im ω| ,
:FFN<
( * !7 ($# 7 ( ( ) # 7 , = 6 !! # 9 ( 7 ) $# ! K ! 7 Re q = | Im ω| B! 7 7! B ($# ) K ! " q = p :$ FA<, * 7) !$ Γ, = !(!6# 7 (! [x − iR, x + iR] 6 7 ! ! ! |q − x| = R ' ($ # * ! (*! #! 7 6 !! ) $" 9 9($ ! $7 "$ q = p, 7! 6 9 (* * 7( @ (
q 1 res F (p) = −2πi · ,q = p = 2 + ω 2 )(p − q)2 2πi (q q p2 − ω 2 d 2 (p − q) 2 = , = − dq (q + ω 2 )(p − q)2 q=p (p2 + ω 2 )2
= !(!6 #! ! IN @ * !7 (% !6# 7 $ * ($$ $, = !B ! ! \−\ E , p2 − ω 2 t cos ωt 2 . :FF;< (p + ω 2)2
' 3 4
0 1" "%& +* #%
% " " " !$ % "
%
6 # # ,
2 "! &
(
& ) * )
) #4
- * 7 *! $ $* 7(
F (λ) =
:I;<
ϕ(z)eλf (z) dz ,
C
( ϕ(z) ! f (z) & z, = 6 !! $"$$ (7 % ! G , 7 $# 7 $ ! C D λ & $ ( (! "$ ( ! !$, = *! 6, %6 $!$ ! $ $"$ !( F (λ) %$B 7B !!! λ ? $ !(, * 7 *! , = $!"!6 *!!)9 ! ∞ Γ(p + 1) =
:I><
xp e−xdx ,
0
$ * !% $ !"7 $ $$B !"7B p O 69 9 %$ (7 7 (*! # * $! ∞ Γ(p + 1) =
:IA<
ep ln x−x dx ,
0
5 4
= $ ($# K!$ *! ( $* 7( :I;< 4(*! #! 7 7) 6 ( 7 $ x → 0 ! x → ∞ @ !"7 *! ! $!"!6# 7 !"7 (*! # ! $! ! $! # !"7 7 f (x) = p ln x − x :IC< !% !6 " x = p, 6 "$ " 9 ( K$B " !( ! ! $ !$ f (x) p ln p − p −
1 (x − p)2 . 2p
:IF<
5 ( 76 * $!$ !"7 *! ! p+δ p ln p − p − 1 (x − p)2 p+δ − 1 (x − p)2 2p Γ(p + 1) e dx = ppe−p e 2p dx p−δ
p−δ
pp e−p
∞ − 1 (x − p)2 p p 2p e dx = 2πp . e
:IG<
−∞
4$% $ !9# ! ( * , = (*! #! 7 $ $ ! $ |x − p| > δ ! ! K $( 76 ( 7 Q ! :IG< (!6 !% $ !"7 *! ! :I>< $ $$B !") 7B p 59 "! !$ !9# 4$ $! 6 $ % % $ " % $B !% $ #, $ !( $# $K 9 !$ $ B!!, H7) "$ %$ ( #! ! ! #, = ( 7# *K $ (9 (! ! Q ! :IG< (!6 !% $ !"7 *! ! :I>< " !"7 (*! # " ! $! :ppe−p< (7 ( (! $ $, = ( (!6 ( $ (! * !7, ! 7 !"7 (*! # 6 ( !# % $#$ ( ! $! # * @ ! !*!(!$, = (*! #! !# 7( $ :IG< ( ) (!6 ! # :S! < ( ($ 69 p 4 # ( *! :I;<, 7 (*! #! 7 6 !! ) $" 9 % ! G =$$ z 5 *! ! %$
5 1 0
!% $ %"$ $ " ! $! # !"7 ( 7 (*! # ! 9 ! K $( # !(!7 ! * !7 ?= K 7B * !7, = H6(6 "$ z1 z2, 6 !$, = ! $ * ( 7 !% 9 !"7 (*! # ( 7*!6 !% #K * !) "7, ! K $( !(!6, $ ( $ $$, = !(! $"$! (! # ( % !% $ 7 E #$ 7 f (z) 6 !! $" 9 % ! G , ! ( $ K !"7 *! ! :I;< $!"!6# 7 !"7$ "! z1 z2 " K 7B * !7, ! $* 7( $ C O ( $ $ $ !6, = ( 7 !(! * *! :I;< $ # * !% $ * %"$ 7 ! ( ) * 9 (!, = * 7(!6# 7, H7!! $ 9 $% ! * ! * !7, = % !( #7 !! $= $ $ K!6 !"7 *! :I;< $ $$B ( (!$B !"7B !) !! λ, = # ! "$ $ @ 6 $ ($ " !$, = * $ !"7 *! (!(# ( 7$ K 7B * !) 7, ! 7$B 7 u(x, y) ( 7*!6 !% #K$B !"# @ u(x, y) 6 ( 9 "! $ 9 f (x, y) = u(x, y) + iv(x, y) 4$ # %B( %!$ ( !*$, = 7 u(x, y) 6 *! " 9 % ! G , ( 7*!$ !% ) 9 * ! $! KB "!B 6 % ! , % ($ % ! G !6 " , 7$B 7 u(x, y) !6 !% K6# 7 B ! 7!B 4 B7 u(x, y) !$ $K
u v !"
∂v ∂u = , ∂x ∂y
∂u ∂v =− , ∂y ∂x
x, y ∈ G ,
# $% u v &' ( ) & *
∂2u ∂2u + = 0, ∂x2 ∂y 2
∂2v ∂2v + = 0, ∂x2 ∂y 2
x, y ∈ G .
B! "! z0 = x0 + iy0 6 6($ 9 ( 9 " 9 B u(x, y) % !) G - * 7 $B !"# u(x, y) = u(x0, y0) = const u(x, y), = B (7# " 9 " '! ( $$ ! $! ( 7 *!) "$B , # 9 !$ !$B $ $B, % $ !% $!9# 7 *!$9 % ! G , !% 79# ! " # $ !( % % ! $ u(x, y) = u(x0, y0) %$ !9# % ! # G ! ) $, ($ 7$B !"7 u(x, y) ( ( !% K, !% % #K
5 4
!"7 u(x0, y0) 4K $ !$ !9# 7 0), ! (* & ?= *!$" "$ z1 z2 $ * !7 !# ( 7 u(x, y) $!6 $B "!B !"7, , " $( , ! !$ "$ ( !$ , = % ! # 7 u(x, y) 9 ! ! # 4$ # $ !"7 *! ! %( (! !$ 6 *!$" "$, 7 !"7 u(x, y) 6 !% #K$ @!! $!7 !6 $ !(, $ "$ z1 z2 !# (! ( (! , ! (*! (H6 !B ( ! ! 6# 7 $ !(, $ z1 z2 !# $B (H6$B !B, = (!6 $ # %!$ !$ * !7, $ %( B ($$ " ( " x0, y0, ! 7 !"7 u(x, y) 6 ! $! #$ " x0, y0 K $( !(!6 ! 7!B ( *!$"$B " E" $( , = # $ !( $ !"7 *! ! :I;< %( (! !$ ! ! ( 7! ( "$, $" ! ! %!$ $ K, "$ K $(K !(!9# !"7 u(x, y) ( $ * !7 ( ! $ !$ !9# '' 57 ! ! H7!! *!69 B u(x, y) ( "$ (
5 2
4$ ( 7( ( $B ( # :% ( (#<, = !# ) (! ! ! ! ! $ $" $ *! ( (
7 ( @ p > 0 A → ∞ ) A x
A e dx = Γ(p) + O e− 2 .
p−1 −x
0
!(! ! $ # %(# (*! !$ *! $ $* 7( a Φ(λ) =
2
ϕ(t)e−λt dt ,
0 < a < ∞,
−a
( 7 7$B !6 ! ! !
:II<
5 4 .
7 + 91 |t| ≤ δ ϕ(t) " :IJ<
ϕ(t) = c0 + c1 t + O(t2 )
λ0 > 0 ") a
:IM<
2
|ϕ(t)| e−λ0 t dt < M .
−a
4 λ > λ0 ) a
2
ϕ(t)e−λt dt = c0
Φ(λ) = −a
3 π + O(λ− 2 ). λ
:I;N<
< #! - + ! !!$, = $ !(, $ 9 ϕ(t)
! ! $ 7( @ ! ϕ(t) =
n−1
k
n
ck t + O(t ) ,
k=0
ϕ(k) (0) , ck = k!
!6 ! $ $"$ !( a Φ(λ) =
−λt2
ϕ(t)e −a
dt =
n−1 [ 2 ]
m=0
c2m
Γ(m + 12 ) m+ 12
λ
+ O(λ−
n+1 2
),
:I;;<
!"!6 !% #K "$ , = 6 K$ !% ( 96 n−1 ( $ n−1 2 2 O !, $ n = 1, $ !( ϕ(t) !6 $* 7( ϕ(t) = c0 + O(t), ! $K $ " :I;;< !6 7( λ−1 , #$ ( 7 $ ! $K * # (*!6 *! δ
2
O(t)e−λt dt < C
−δ
δ −δ
2
|t|e−λt dt = 2C
δ
2
te−λt dt .
0
n−1 O(tn ) ϕ(t) = ck tk + O(tn ) |t| ≤ δ k=0 n−1 k ck t < C|tn | C ϕ(t) −
½µ
k=0
5 4
< #! . + ! ! $K!6# 7 ! ( $ 9 $ !(, $ * !7 (% !6# 7 ( [a1, a2], ( a1 < 0, a2 > 0 −a1 = a2
7 , 91 |t| ≤ δ0 - ϕ(t) µ(t)
ϕ(t) = c0 + c1 t + O(t2 ) ,
µ(t) = c3 t3 + O(t4 ) ,
1 λ → ∞ δ(λ) ≤ δ0 ) λδ 2 (λ) → ∞ ,
:I;><
λδ 3 (λ) → 0 .
4 λ → ∞ ) δ(λ) I(λ) =
ϕ(t)eλ[−t
2
+µ(t)]
dt = c0
3 π + O(λ− 2 ). λ
:I;A<
−δ(λ)
5 $ ( 79# ( $ ! , = 6 9 ( ! ! ! ! $ $" * !( *! ( (
.&'&,& 91 f (t)& # ) [a, b]& )
" ' t0& f (t0) < 0& 1 ) δ0 > 0& # |t − t0 | < δ0 ) f (t) = f (t0) +
f (t0) (t − t0 )2 + µ(t) . 2
4& # - ϕ(t) µ(t) |t − t0| < δ0 A& " ϕ(t) = c0 + c1 t + O(t2 ) ,
µ(t) = c3 t3 + O(t4 ) ,
) b Ψ(λ) =
! ϕ(t)eλf (t) dt = eλf (t0 )
a
−
2π − 32 ϕ(t ) + O(λ ) 0 λf (t0 )
# * ¾µ
2
δ(λ) = λ− 5 ! "#
" ,
:I;C<
5 4 .
, δ0 ' f (t0) |µ(t)| < − (t − t0 )2 , 4 f (t0) − f (t) ≥ h > 0 ;
|t − t0| ≤ δ0 |t − t0| > δ0 ", λ0 > 0 ") b
|ϕ(t)|eλ0 f (t) dt ≤ M .
a
< #! ' .&'&,& @ ! ! $K!6# 7 ! ( $ 9 )
$ !(, $ (! !% %$( *!$ * !7 ( 99# ") < #! ( .&'&,& $!$ $K K$ " ! $) $" * !( *! ! :I;C< P! *"$ "$ ! $!$ $) !$ ( 7 ! $B " ! $ $" * !( < #! + .&'&,& $ !(, $ ! $! # !"7 ) f (t) ( 7*!6# 7 7)%(# *!$" " ( [a, b], :I;C< H7 76# 7 ( (! $ $ 12 < #! , .&'&,& $ !(, $ 7 f (t) ($ ( [a, b] !6 ( #! ! $ , = 6 $$ ! $"$ 9, ! $ ) $"$ !( *! :I;C< %$B 7B $ * !!! λ ! $!$ $ ( ($B *! δ) " ! $! ( 9"$ #!$ - * 7 $ !($ ! !7 6 $
$ (+& 6 - C ∞
Γ(p + 1) =
xp e−xdx .
0
4( ! $ (*! # 9 $* 7( xpe−x = ep ln x−x %$ !) $B x = pt @ ( ! ! $ !$ Γ(p + 1) = pp+1
∞ 0
ep(ln t−t) dt .
5 4
5 6 *! $ :I;C< ϕ ≡ 1 ! f (t) = ln t − t Q7 f (t) ( 7*!6 * ! $! # * !"7 $ t0 = 1, $" f (1) = −1 ,
f (t)
= 0,
t=1
f (t)
t=1
= −1 .
@ ( ( :I;C< ! $!$ Γ(p + 1) = e−p
p p
2π 1 3 1+O + O(p− 2 ) pp+1 = 2πp . p e p
@!$ "$ $ $! $ ! $ $" *!!) !, = %) ! $!! B$ !K ! ( % $ (7 ( #! K$B " $ 3 *! p p
1 1 139 1+ + Γ(p + 1) = 2πp − +... . e 12p 288p2 51840p3
(
5 #
4( ( * 7( ! ! * (! ! ( 7 $!7 ! $ $"$B !( *! $ :I;<
F (λ) = C
ϕ(z)eλf (z) dz .
4$ ( $ $$, = * !7 C 6 !$, = ! (7 ) $ * ( 7 ( ! "! $! u(x, y) f (z) ( 7*!6 !% #K * !"7, ! K $( !(!6 : $ # 7 ! "! $! ! $K!6# 7 !) $" ! 9, = % $$ %! !$B K $($B $ 7 (*! # < @ ( $ $"$ *! ! :I;< (!6 % (6 ) * !7 ( 7 ! C @ ( 7 !% $ * %"$ 7 # * *! %B( ( !$ C !$ "$ , = % (*! #! 7 ! # ! ! !! ! $ 4$ # %B(! ( !7 ! C $!"!6# 7 K "* *!69 B 7 u(x, y) O !, * !7 $ B ($$ " ( " B u(x, y) ! 7 !K $(K $ 6 O $$ # ( !(K ! *! B *! " u(x, y) ( "$ M0(x0, y0) ! 7$ !K $(K $ $) !"!9# 7 ! 7 ! grad u = ∇u B! ∇u = 0 E #$ ( 7
5' 4
!! $" ∇u · ∇v = 0, ! 7 ! ∇u $!"!6 $ v(x, y) = const ! ( $ $ % ( 7 7= ! (7 $ v(x, y) = const ! ∇u = 0, 7 u(x, y) ( 6 $ 96# 7 !K $(K$ "$ 4 ! ( " M0 ∇u = 0 - * 7 (! #K ( u(x, y) ! v(x, y) ( "$ E" $( , = " M0 B( ! 7 l ( $" ( $ v(x, y) = const ( 99# 9 ∂u (x0, y0) = 0 , ∂l
∂v (x0, y0) = 0 . ∂l
E #$ B(! ( !! $" ! $# ( ! 7, ( $ $ $ !6, = f (z0 ) = 0 .
@ % , !( f (z) "$ z0 !6 $* 7( f (z) = f (z0) + (z − z0 )p {c0 + c1 (z − z0 ) + . . .} ,
( p ≥ 2 c0 = 0 ( !"7 cn = rneiθ $!6 p
f (z) − f (z0) = ρ
#
i[pϕ+θ0 ]
r0 e
i[(p+1)ϕ+θ1 ]
+ ρr1e
n
, n = 0, 1, . . ., z − z0 = ρeiϕ ,
$ +... .
7 $ $B u(x, y) = const ! v(x, y) = const, = B (7# " " z0 !6 f (z) − f (z0) = 0 U (ρ, ϕ) = r0 cos[pϕ + θ0] + ρr1 cos[(p + 1)ϕ + θ1] + . . . = 0 , :I;F< V (ρ, ϕ) = r0 sin[pϕ + θ0] + ρr1 sin[(p + 1)ϕ + θ1 ] + . . . = 0 . :I;G< @ $ $ ! !"7 ( 7 ( 7 "! $ f (z) − f (z0), ! ! u(x, y) − u(x0, y0) = ρp U (ρ, ϕ) , v(x, y) − v(x0, y0) = ρp V (ρ, ϕ) .
E #$ 7 cos[pϕ+θ0] $ ϕ ( 0 ( 2π 96 ! 2p ! , $ :I;F< $ $ !6, = "$ z0 !(!6# 7 ! 2p $ $B , ( 7$B 7 U (ρ, ϕ) %*!6 ! S!$ $B ) $!"!9# 7 H7 77 :I;F< 3 $, 7$B U (ρ, ϕ) < 0
5 4
!$ !9# 7 (H6$$, ! $ 7$B U (ρ, ϕ) > 0 & ( (!$$ ! 7$ !K $(K * u(x, y) !# (H6$B !B $!"!) 9# 7 $$ !"7$ ! ϕ, ( 7 7$B "$ (x − 0, y0) 7 V (ρ, ϕ) = 0 U (ρ, ϕ) < 0, % cos(pϕ + θ0 ) = −1 5 !"7 ( 99# ϕm = −
θ0 2m + 1 + π, p p
:I;I<
m = 0, 1, . . . , p − 1 .
!(! %( * 7(!$ #$ $ !( p = 2, $ f (z0) = 0 4$ # c0 = 12 f (z0 ) θ0 = arg f (z0 ) ' # $ !( 6 $K ( ! (H6$B $, ($ 7$B B ($# 7 !K $(K * u(x, y) ! 7 ( $" ( 6 " z0 *( :I;I< $!"!6# 7 !$ ϕ0 =
−θ0 + π , 2
ϕ1 =
−θ0 + 3π = ϕ0 + π . 2
.&(&-& 91 - ϕ(z) f (z) = u(x, y) + iv(x, y) )
" G * +, @ 1 - u(x, y) ) G z0 = x0 + iy0 & f (z0 ) = 0% , < ) δ > 0& # - L - v(x, y) = v(x0, y0 )& # 1 z0 & "1 0)1 1 )-
u(x, y) δ z0 )
:I;J<
u(x0, y0) − u(x, y) ≥ h > 0 .
;, ! λ0 > 0 ")
|ϕ(z)|eλ0 u(x,y) ds < M ,
C
C " G & -- .z1, .z2, 1 1 0)1 1 z0 & # -1 0) L γ1 γ2 - & 1
u(x, y) ) :I;J<% 4 1 λ ≥ λ0 ) F (λ) = C
!
ϕ(z)eλf (z) dz = eλf (z0 )
2π iϕm − 32 ϕ(z )e + O(λ ) 0 λ|f (z0 )|
"
,
:I;M<
5' 4
π−θ
0 + mπ (m = 0, 1) θ0 = arg f (z0)% " ϕm ) ϕ = 2 .D%+E, C%
(7 $ $ ($
< #! % .&(&-& . $ $ $ !6, = 7= %$( *!)
$" "$ z1 ! z2 $ * !7 C !# ( (H6 ( "$ z0, !6 =! !
F (λ) =
ϕ(z)eλf (z) dz = eλf (z0 ) O(e−λh ) .
:I>N<
C
< #! ' .&(&-& ' ! !7B "! ( ($# 7 * 7)
(!$ *! $ $ :I;< % % ! , $ $ ! * !7 C 7) 6 ( " $B $ !(!B ( 7 % *! ! :I;< %B( , = % $ ! * !7 7 ! ! ( " (H6$B !B () "$ z0 4$ # ! IAG ! :I;M< %*!9# $
$ (,& $ " 8 Pn(cos θ) 1 1 n%
!*!(!6 (7 ( $ ' $ 99# 7 $ !( 6
∞
1 √ = Pn (x)tn , 1 − 2xt + t2 n=0
:I>;<
!( #79# ! ( K7 n n+1 Pn−1 (x) + Pn+1 (x) , 2n + 1 2n + 1 Pn (x) = Pn+1 (x) − 2xPn (x) + Pn−1 (x) , (2n + 1)Pn(x) = Pn+1 (x) − Pn−1 (x) . xPn (x) =
O! 9 - ($*! ! !$ 7 $ $* 7( $B 1 dn 2 n Pn (x) = n − 1) (x , 2 n! dxn
:I>>< :I>A< :I>C<
5 4
= ( $# $!$ ( 7 $B ! *! # ( ! 7 E( $B : $ ! x = cos θ< $ (! %( * 7(!$ 1 Pn (cos θ) = √ π 2
σ Dr
θ −θ
:I>F<
O %$ ! !, ! # ϕ $) $ !6 ζ
6
ih
-
1 ei(n+ 2 )ϕ √ dϕ . cos ϕ − cos θ
-
ϕ ⇒ ζ = s + iσ ,
Cr
* 7 !$ , = ) ? %! $ ! $ I; E #$ ($) 6 # * ! (*! #! 7 Ar !6 % $ , *! () rB −θ θ s ( [−θ, θ] ≡ AB ! !) $$ ! *! (!B -$ I; AD, DC ! CB - * 7 ( *! $ ! ( AD !6 ζ = −θ + iσ, dζ = idσ $ !( h → ∞ $!6
−i(n+ 21 )θ
∞
⇒ ie AD
1
0
e−(n+ 2 )σ cos(θ − iσ) − cos θ
:I>G<
dϕ .
7 !% $ $ # * *! ! $ $ !6 ( ! 7 $$B n 7 e−(n+ )σ !6 ! $ " σ = 0, $" !% #K $ 6 , ( 7 7 * σ $!6 ( !"7, ! = K 7B *) 1 ( e−(n+ ) !7 %! ( !$ !7 # $! √cos(θ−iσ)−cos θ $K $ 96 " σ = 0 , !6 1 2
1 2
iσ iσ cos(θ − iσ) − cos θ = −2 sin θ − ) sin , 2 2
( ( # $!
1 1 =% cos(θ − iσ) − cos θ 2 sh σ2 sin θ −
iσ 2
−→ ∞ σ→0
5' 4
! !(!6 K $( 9 !$ $ σ → ∞ @!$ "$ , * "! $ *! ! :I>G< $ $$B n ! %"$ ) $$, % 9"$ # $K ! #$ ! * !7 0 < σ < h, ! 7 π
1 e−i 4 √ ≈! = · σ− 2 . sin θ cos(θ − iσ) − cos θ iσ iσ 2 sin θ − 2 2
1
1
O 69 " 9 ! %*$ "! $ (h, ∞) ! ! * !7, = $ $$B n 7 e−(n+ )σ K $( !(!6 @ ( ( 7 K * ! ! $!6 1 2
∞ ≈
AD
(n+ 12 )θ+ π4
≈ ie−i[
A
]√ 1 sin θ
∞ σ
− 21
n+ 21
e−(
(n+ 21 )θ+ π4
)σ dσ = ie−i[
0
√
π ]% . 1 (n + 2 ) sin θ
P! *" $!6 ( 7 ( 7$ ! CB , ! !
B ≈
CB
(n+ 21 )θ+ π4
≈ −iei[
√ π
]%
(n +
∞
1 2 ) sin θ
.
! ( DC , ( ζ = s+ih, ( # (*! # ! $$ 1 ei(n+ 21 )ζ e−(n+ 2 )h √ ≤ cos ζ − cos θ | cos ζ| − | cos θ|
76 ( 7, $ h → ∞ @ %
−→ 0 . DC
h→∞
E%H6(9"$ !"7 *! ! (!B AD CB , 7 !() $B # $!6 !
Pn (cos θ) ≈
cos 2
π n + 12
n + 12 θ − π4 √ . cos θ
5 4
K9"$ " !$ ! !$$ n + 12 ! n $!6 8
Pn (cos θ) ≈
2 cos · πn
n + 12 θ − π4 √ . cos θ
:I>I<
' % ) !! ;; ;> ;A ;C ;F ;G ;I
"$ ! ( !( $$ E% ! # ! =$ S $" 77 Q $9 !7 ' ! $ !! $"$B
(
A F J J M ;A ;J
' ; ! *
%0
( ?! 6
'.
>; .*! K ;M
A; A> AA AC AF AG
-7($ 3 7($ -7( @ ! ($ # $!"7 !! $" -7( !! E % $ "$
+ ! 3 C; C> CA CC
$K !! $" # ! % $ " E ! ! $K P! $" ( 7 E%"$ 7 $!"$B *! CC; .*! $ $* 7(
2π
R(cos θ, sin θ)dθ 0
>I >M A; AA AC AI
+1
CN CA CF CF
CG
6
CC> .*! $ $* 7(
∞ f (x)dx
CJ
−∞
CCA ! W (!! F;
, $ ! 7
F; '$!"7 7 ! ! ! F;; O %! 7 ! ! ! !$B F> ' ! $ %! # F>; @!% $7 %! # FA Q ! !
,/
FJ GN G> GM IN
- !
.+
. "
.,
G; 4 ! ! !(!" !! * "$ 7 IC
I; ' ! ! 7 IF I> ( ! ! ! IJ IA ( ! J>
