Spherical Harmonics. Lecture Notes in Mathematics. No. 17

Lecture Notes in Mathematics An informal series of special lectures, seminars and reports on mathematical topics Edited by A. Dold, Heidelberg and B. Eckmann, Zerich

17 Claus Mailer Institut fur Reine und Angewandte Mathematik Technische Hochschule Aachen

Spherical Harmonics 1966 -",~!

Springer-Verlag. Berlin-Heidelberg. New York

All rights, especially that of translation into foreign languages, reserved. It is also forbidden to reproduce this book, either whole or in part, by photomechanical means (photostat, microfilm and/or microcard)or by other procedure without written permission from Springer Verlag. O by Springer-Verlag Berlin. Heidelberg 1966 Library of Congress Catalog Card Number 66-22467. Printed in Germany. Title No. 7537.

PREFACE

The subject regular

of these lecture notes is the theory of

spherical

harmonics

in any number of dimensions.

The approach is such that the two- or t h r e e - d i m e n s i o n a l problems

do not stand out separately.

regarded

as special

They are on the contrary

cases of a more general

seems that in this way it is possible standing of the basic

properties

thus appear as extensions elementary

functions.

of w e l l - k n o w n

One o u t s t a n d i n g result

coordinate

which goes back property

of the

the d i f f i c u l t i e s

which arise from the singularities

of

of the

system.

as possible

is to derive as many results

solely from the symmetry of the sphere,

prove the basic

properties

the r e p r e s e n t a t i o n

the completeness

which are, besides by a g e n e r a t i n g

and to

the addition

function,

and

of the entire system.

The r e p r e s e n t a t i o n

is self-contained.

This approach to the theory of spherical first p r e s e n t e d in a series of lectures Scientific

of

the use of a special

and thus avoids

The intent of these lectures

theorem,

which

is a proof of

harmonics,

does not require

system of coordinates representation,

properties

This proof of a fundamental

spherical harmonics

It

to get a better under-

of these functions,

the addition theorem of spherical to G. Herglotz.

structure.

R e s e a r c h Laboratories.

harmonics was

at the B o e i n g

It has since been slightly

modified.

I am grateful in p r e p a r i n g

to Dr. Theodore Higgins

these lecture notes

Dr. Ernest R o e t m a n

for a number

for his assistance

and I should like to thank of suggestions

to improve

the manuscript.

February

1966

Claus M U l l e r

C ON TEN TS

General Background Orthogonal

Representation Applications

Funk

...............................

I

....................................

5

..............................................

9

Transformations

Addition Theorem

Rodrigues

and N o t a t i o n

Theorem

........................................

of the A d d i t i o n

Formula

..........................................

Integral Representations Legendre

P r o p e r t i e s of the Differential Expansions

of S p h e r i c a l

Functions

Equations

Harmonic

14 16 18

................

21

.................................

22

Legendre Functions

in S p h e r i c a l

Bibliography

..........................

.............................................

- Hecke Formula

Associated

Theorem

11

..........................

29

........................................

37

Harmonics

.............................

..................................................

40 45

-

I

-

G E N E R A L B A C K G R O U N D AND NOTATION

Let

(Xl,...,Xq)

of q dimensions.

be Cartesian

coordinates

of a

Euclidean

space

Then we have w l t h Ixl

l

:

~-~

:

(x.)~+

....

+ ('x~) ~

the r e p r e s e n t a t i o n

where

represents

the system of coordinates

sphere in q dimensions. element d ~ 9

of the points on the unit

It will be called

and the total

surface

~9

~

, its surface , where this surface

is given by

By d e f i n i t i o n we set

If the vectors represent

~

~ . . .

~s ~s_~

X~9

system,

~'_.,

unit vector

,. --~_~ f_~4 ,. ~:~
in the space spanned by

of the unit sphere then can be w r i t t e n as ~-~

and we have from above

The integral

we may

by

__ .t.E< 1 + ~

is a

The surface element

are an orthonormal

,~

the points on

(1) where

=2. Then we have

A~#_I -i on the right hand side may be t r a n s f o r m e d 4

i


to

% %/

o

I) Here and in the following points of the unit denoted by greek letters.

sphere are

El,... , c ~ r

-

Which gives us for (2)

-

q = 2,3, ... ")

wI

Denote

2

=

CU,/

=

_-

.

"~ ( z ) z

by Z

(3) the Laplace Definition

operator. I :

We then introduce

Let Hn(X)

the

be a homogeneous

q dimensions,

polynomial

of degree n in

which satisfies

Then

is called a (regular)

spherical harmonic

of order n

in q dimensions. From this we get immediately Lemma 1 :

Let Hn(X)

~, (-~)

=

(-4)~

Sn (~)

and Hm(X ) be two homogeneous

degree n and m. Then by Green's

as the normal derivatives I 8~ H ~ ( ~ ) }

= ~ H~(~)

harmonic

of

theorem we have

of H m and H n on and

polynomials

[~

~9

are

H~(+~)] = n H~ (~)

respectively.

T= 4

From Definition

(I) we have therefore /

Lemma 2 :

Any homogeneous the form

.~ S ~ (~) S ~ ( ~ )

polynomial

~

= 0

in q variables

for

m #

can be represented

in

-3-

Z i:~

(4) where

(~) A._#(z,,

the An.j(Xl,...,Xq_1)

.

.

.

,~_.)

.

are homogeneous

(n-j) in Xl,...,Xq_ 1. A p p l i c a t i o n

H.(.)

:

polynomials

of the Laplace

of degree

operator

in the

form =

~-~

+

m-z

gives

nq

H (x) :

For a harmonic equating

--# ~

polynomial

coefficients

;(~-~)(~1~-~ A._~ + ~:s~)~ a~_~ A._~ this has to vanish identically.

By

we thus get

(5) Therefore

all the polynomials

An_ I 9 The number polynomials

of linearly

is thus equal

Aj are determined independent

to the number

if we know A n and

homogeneous

and harmonic

of coefficients

of A n and

An_ I 9 Denote

by M(q,n)

polynomial

the number

of coefficients

of degree n and q variables.

(4) that

It then follows

{ ~ ~(~-~,~)

(6)

Clearly

in a homogeneous

M (~,~)

M(1,n)

An(Xl,...,Xq_l)

(8) converges

:

MCq-s,n)

=

~(~q-~)

#'o (6) and

(7)

~o

nLO

9

available

in

is

~- kl (~-'1, n-s)

series

Ixl g 1. By

~

p

of coefficients

@~ for

0

and A n _ l ( X l , . . . , X q _ l )

N (~,n)

Then the power

~=a

= I, so that M(q,n)

Now the total number

(7)

=

,

from

:

(7(~ ~-z)

!

-4-

IVc9,.1

(9) Now

it follows

from

Z

=

,

(7)

N(I

=

,n)

1

for n = 0,1

0

for n > I,

o

so that Substituting

(9)

into

(8)

and interchanging

the

order

of

summation

we o b t a i n 4

and hence ~9 (x)

This

gives

Lemma

~ :

=

4Y" X

us

The n u m b e r harmonics

N(q,n)

of l i n e a r l y

of d e g r e e

n is g i v e n

(4 - x ) q -~

Specializing

independent

by the power

.=o

Oo

=

(4

*• -

2x"

=

X} z

:

>~

gives

.=o

(2.+~•

=

=o

for

4 + x

:

7~

the N(q,n)

NC3,,~)

•

=o

explicitly.

The

Ix l < I -,PC"+q-~}

(4~x)

( 4 - x ) I-~

n ( .+d

P(q-4l

x"

. :o

4 + 7" so

.Y__ N ( z , , , ) ~ <

.=~

F r o m L e m m a 3 we can d e t e r m i n e expansion

oo

4 + Z

-7 - x

binomial

series

to q = 2 and q = 3 we get 4+•

(10)

spherical

~,,,q-zJPC,~,f-z~

that

(11)

N (9,~)

I

/

"~

I

n~4

n=

0

-5If we set

N(%,m)

(12)

s,, c~)

:

Z ~--,f

rl

ci ,s,,,i(~)

we have Lemma 4: There exist N(q,n) linearly independent 5~,~(F) harmonic

of degree n in q dimensions

spherical harmonics and every spherical

of degree n can be regarded as a linear

combination of the

5~,~ (~) .

ORTHOGONAL TRANSFORMATIONS Suppose now that the functions S n , j ( ~ ) , an orthonormal

j = I,...,N constitute

set, i.e.,

If A is an orthogonal matrix, harmonic polynomial

then Hn(A~)

of degree

is a homogeneous

n in x if Hn(X) has this property,

so that Sn(A ~ ) is a spherical harmonic

of order n. In particular

(14)

~i~ s.,~. (~')

S.,~

( A ~)

;

Z

T--9

I,I

To every orthogonal matrix A there corresponds c ~T

9 We now have, because of (13) and I

(15)

therefore a matrix

(14),

Zo,~ ( A ~) So,~ ( A ~') ~

=

,_=~

C~?

~q The orthogonal

transformation A ~

transformation of unaltered.

O 9

may be regarded as a coordinate

which leaves the surface element d ~

This means that

-6-

~9 From

(15) we now get

~_~") ~=

(16)

kl c

so that the coefficients matrix. Besides

Ca

=

c~T

~,

are the elements of an orthogonal

(16) we therefore get also IV(q,.)

(17)

"r=4

Z

For any two points

~

c r''''~ C'r~ "

and

,?

-- ~: K on

/]9 we now form the function

Due to (17) we have for any orthogonal matrix A

F(A~',

A?) W(q,~)

Z[Z

I"= 9

The function F(

~, ~

is not changed if

y

W(f,.)

"

~:4

) thus has the important property that it and ~

undergo an orthogonal

transformation

simultaneously. To further studies of our function F( ~, ~ properties a)

of the group of orthogonal To every unit vector

~

formation such that A ~ b)

For any two vectors

c)

For any unit vector transformations,

~

~

) we use the following

transformations

there is an orthogonal = and

s9 ~

trans-

9 we have

there is a subgroup of orthogonal

which keeps

~

fixed and which trans-

-7forms

a given

unit

%

vector

in all those

vectors

for w h i c h

=Y'~o

~"~

LEGENDRE

FUNCTIONS

We n o w use

these

It follows

from

according

to

properties

to study

(a) that we may

(2),

~

would

our f u n c t i o n

transform

~

be r e p r e s e n t e d

F( ~, ~

into

E9

).

. Then,

in the form

(~8) From

(b) we k n o w and

~

that

of

~

it

can be seen that

before

the o r t h o g o n a l

We have

t is also carrying

group

in

transformation. fixpoint

(q-1)-dimensions

~q-4

~ ~_~z" ~ _ ~ )

is a f u n c t i o n

~

product From

(18)

is i s o m o r p h i c

to

~).

and

q q_~

Combining

Let Sn, j ( ~ ) , j = 1,...,N harmonics

(vectors)

depends

The o r t h o g o n a l formations

~

and

~

on

~q

on

~q

only on the scalar

group

on

does not d e p e n d

of t alone.

spherical

i)

with

of the scalar

therefore

F ( E~, t ~

5:

out the

the s u b g r o u p

for any two vectors

Lemma

the value

on

9 This

~q_~

this w i t h

implies

(18) we have

. Then

set of

for any two points

the f u n c t i o n

product

of

consists

~

that

. It t h e r e f o r e

be an o r t h o n o r m a l

in one d i m e n s i o n

x' I = • x I only.

~f~q_~

and

of the

~

9

two trans-

-8It is clear from the left hand side that this function is a spherical harmonic in ~ or ~ of degree n. From the right hand side it follows that it is symmetric with regard to all orthogonal transformations which leave

~

fixed. We are thus led to introduce

a special spherical harmonic which has this same symmetry. Definition 2:

Let Ln(X ) be a homogeneous,

harmonic polynomial of

degree n with the following properties: a)

Ln(AX ) = Ln(X ) for all orthogonal transformations A which leave the vector

b)

~9

unchanged.

Ln(~q) = I.

Then T 0

is called the Legendre function of degree n. By this definiton the function Ln( ~ ) is uniquely determined, for according to the representation (4), L ( x ) is uniquely determined by the homogeneous polynomials An(Xl,...,Xq_ I ) and An_1(xl,...,Xq.1). The condition (a) implies that these polynomials depend only on (xi)2 + (x2)2 + .... + (Xq_1)2. We thus get

A, =

c

[ (x,)Z+ .... +Cxtt-~)~j~ ; A,.=O

for

n = .Z~

'<;A,,:o

for

~= 2+<+4

and A,,_+

=

c/{x+l'+--.

+

Apart from a multiplicative constant,

the function Ln(X) is

therefore determined by condition (a). The value of the constant c is then fixed by condition (b). Using the parameter representation (2) we see that Ln( ~ ) depends on t only, as (•

+ (;
=. Tz

C.1_tz)

-9We now have:

Theorem

I:

The L e g e n d r e

where

The

last

Pn(t)

two r e l a t i o n s

function

is a p o l y n o m i a l

of this

As r = I, t = I, c o r r e s p o n d s condition Lemma

Ln( ~ ) may be w r i t t e n

(b) of D e f i n i t i o n

theorem

to

of degree

n with

csn be p r o v e d

~ = ~

easily:

, the first

2 and the second

as

statement

is

follows

from

equation

I.

ADDITION THEOREM We n o w can d e t e r m i n e we k n o w that

this

w i t h respect

to

by an o r t h o g o n a l

the f u n c t i o n

function ~

~

~ X.~)

is a s p h e r i c a l

9 It is m o r e o v e r

transformation

which

in L e m m a

harmonic

unchanged leaves

if

5,

of d e g r e e W

for n

is t r a n s f o r m e d

fixed,

so that

Z as the

function

To d e t e r m i n e

~ (~"~Z)

the constant

can only be p r o p o r t i o n a l = ~

c n we set

to Pn ( ~ - ~

and o b t a i n

N (
Integration

over

A~?

c.

c~

co~I

gives N [~,~)

and we get

_-

=

=

c.

.

.

-

Theorem

2 : (Addition of N(q,n)

10-

Theorem)

Let Sn,j( ~

spherical

harmonics

) be an orthonormal

set

of order n and dimension

Then

where

Pn(t)

dimension

Thls theorem addition

addition

according

theorem

for the function

case after introducing

In order

Polynomial

of degree n and

q.

is called

theorem

is the Legendre

to determine

polar

as it reduces

cos ~

in the two-dimensional

coordinates.

the spherical

harmonics

for the case q = 2

to this theory we first have to determine

independent

homogeneous

to the

and harmonic

polynomials

two linearly

of degree n.

We can take them as

We now introduce (19)

a system of polar •

:

"7- c,o-oy

!

Re

;

coordinates xz

--

in the usual way

-r ~ T

and get (x,+~xl)"

=

7r - ~o)

c,~ ~ ( ~ .

I"

s

]~

(x,+~•

:

sX,. n ( ~ - , f )

Th

From these two we get an orthonormal

set by

F- T)

q.

-11-

The L e g e n d r e polynomial

function now is o b t a i n e d

from a h o m o g e n e o u s

w h i c h is symmetric w i t h respect

w h i c h takes on the value

harmonic

to the x 2- axis,

and

I for x1= 0, x 2 = I. This gives us

L,, (xq,x~)

:

~rx~ t ' i x ~ )

Ee

'~

or

Now let t be the scalar product from

between

and

s

~

. We then have

(19) t

=

.~.:~

I,,

=

~,-~

c

-

w h i c h gives us

In two dimensions,

therefore,

the f u n c t i o n

k n o w n as the C h e b y c h e v

Polynomial.

If the points

~

respectively

y

and

the c o o r d i n a t e s

we get by o b s e r v i n g Y "~

the r e l a t i o n

have

:

( T - ~K) ;

~

P (t) is what n

9'

and

is otherwise

F

that IV E 2, ~ ) = 2 ,

,,'I

;

co z

:

2 ~

(for q = 2);

2 n"

~

Theorem cos ~

~c~-~)

2 therefore reduces

=

!

~(~(~-~)).

to the a d d i t i o n

in the t w o - d i m e n s i o n a l

case,

which explains why this result

is called the a d d i t i o n t h e o r e m of spherical

REPRESENTATION

harmonics.

THEOREM

As is well known,

all the t r i g o n o m e t r i c

d e r i v e d by simple a l g e b r a i c the q u e s t i o n

f o r m u l a for the function

arises

if there

processes

functions

can be

from a single one

is a c o r r e s p o n d i n g

result

(e.g.cosx), in the theory

-

of general spherical harmonics.

12-

The addition theorem suggests that it

might be possible to express all spherical harmonics in terms of the Legendre function. Theorem

~:

This is stated in

To every degree n, there is a system of N points W~,~,,

..........

Sn(~)

,

W~

such that every spherical harmonic

can be expressed in the form R=4

It is clear from the above that every spherical harmonic

can be

written as

so that it is only necessary to show that the functions Sn,j( ~ ) can be expressed

by the Legendre functions.

To this end we observe that it is certainly possible a point

~

such that Sn, 1( ~ ~ ) @ O. We then consider

As a function of

~

this cannot be identically O, because

Sn,1( ~ ) and Sn,2( ~ ) are linearly independent. point

~

to find

=

~z

Therefore there is a

such that this determinant

Discussing next the determinant

,%.,~C~]

S.,~ (~)

and using the same arguments we obtain by induction

does not vanish.

-

Lemma 6 :

13-

There is a system of points matrix

(Sn, j( ~ k)),

W~, ~

j = I,...,N;

such that the

.... , ~ N

k = I,...,N is

non-degenerate. From Theorem 2 we now have ~9

This is a n o n - d e g e n e r a t e as unknowns

system of linear equations with S

n,j

so that Theorem 3 follows by inversion.

In order to simplify the formulation

fo these relations we

introduce Definition ) :

A system of N points a fundamental

I ~

('~4''~K~

It can be seen readily that the matrix that the determinant minant is positive, since then det Theorem 4 :

A~Z 9 will be called

system of degree n, if

det

be obtained by multiplying

on

~,..-,~

~ 0 .

~

~ (~

~)

can

the matrix S n , j ( ~ i ) with its adjoint,

of Definition 3 is non-negative. the system has the properties

so

If the deter-

stated in Theorem 3,

(Sn,j( ~ k) ) @ O, which may also be formulated as

Every spherical harmonic of degree n may be represented in the form

if the points

~

form a fundamental

It is clear now that an orthonormal

system of spherical harmonics

can always be obtained by linear combinations Pn ( ~ k

' ~

represent

). Which fundamental the functions

system of this degree.

system

~K

of the functions

is best suited to

of degree n remains open at this stage

as it requires more information

on the polynomials

Pn(t).

-

APPLICATIONS

OF THE A D D I T I O N

Before

studying

obtain

several

which depend

(20)

THEOREM

polynomials

simple results

on the addition

If we remember represented

the Legendre

that every

I#-

in detail,

on spherical

we shall

harmonics

in general

theorem.

spherical

harmonic

of degree n can be

as

S,,

(~)

--

~, S,,,. ~(=,4

(g)

j

a,,

=

I

~'.('f)S,,,Kc2)d~ l

9O, 9

we get immediately Lemma 7 :

from Theorem

For every

spherical

("' 'I

Here

the letter

integration

Observing

~

2

harmonic

of degree n

90- 9

in connection

is carried

with

out with respect

d ~ to

means ~

that the

9

,vO,,,)

that

K=4

we g e t from ( 2 0 ) ,

u s i n g Schwa~z's i n e q u a l i t y

K=4

and Theorem 2,

~="1

I(=4

This gives us

Lemma 8 : Let S n ( ~ )

be a spherical

~- .,/~c,,.,

I s.c~)l

then we get from

Ncq,.,,

~ (~..,?)

~

--

(21) and Theorem

of degree n. Then

I I s~c~)l ~ ,.o.~

Put

S,, ('~) =

harmonic

.}

S.6 (Y)

S,,,~. (~/

,

2 N (~,.I

L

cvq which gives us

Z

Is.,,

= I ,v

].

-

Lemma 9

From

:

For

Theorem

15-

- I z_ t _z 1

2 we have m o r e o v e r IV(q,.)

[ N (,~,,,,, ] z.

This gives

[2

z

by i n t e g r a t i o n

over

El

(22)

=

As t h e on

S "4 (~) S.,~ (.z) ]

~

value

of

the

integral

, we may assume

representation

(2),

(23)

"~

~

on t h e

_-

left

hand side

to be ~I" Then,

using

does not

the c o o r d i n a t e

we g e t

at-' -' +4

It follows

from

(22)

and

(23)

-4

On the other

hand

, by L e m m a

2,

for

0 By t h e c o o r d i n a t e

representation

(2)

this

is

n

#

m,

equivalent

+4

~t

/ P~(~) P. (~) ( ~ - t z)

--

0

-4

which

gives

us,

combined

with

(24)

Lemma 10 : "

~(t

=

depend

for

n # m,

to

-

RODRIGUES'

16-

FORMULA

We shall now give

a representation

of the L e g e n d r e

polynomials

b a s e d on the f o l l o w i n g p r o p e r t i e s :

1.

P. (~)

2.

is a p o l y n o m i a l

I ?" (/''~ P,,, (~)

(~- L~)

z q-3

d~

of d e g r e e n in t.

: 0

for n # m.

-4

3-

The u s u a l p r o c e s s

P~ (,t) = W

.

of o r t h o g o n a l i z a t i o n

d e t e r m i n e d u p to a m u l t i p l i c a t i v e conditions.

This c o n s t a n t

Consider

the f u n c t i o n s

(25}

r

s h o w s that Pn (t)

c o n s t a n t by the f i r s t

c a n t h e n be f i x e d by the

--

is two

third condition.

9

T h e y are p o l y n o m i a l s

of d e g r e e n, and we see by p a r t i a l

integration

that §

9_ ~

-1 m ~- C q - 3 )

= (-4)" [(4-~')

,

(,l~,z)

-4

If n ~ m the r i g h t h a n d s i d e v a n i s h e s , (25) s a t i s f y

Put

the f i r s t

which proves

that

the f u n c t i o n s

two c o n d i t i o n s .

t = I - s, t h e n 3-fl

= (-~I ~ [r

~

so that we get

=

[_2) ~

Fc~,~-

q;~)

(~)~

/r

~e

C9-3]

-

17-

Thus we get

Theorem

5 :

(Rodrigues ' formula)

~ ctl ~

(~I ~

This has an immediate after

integrating

Lemma

11 :

Fc ~

and simple

application

n times by parts.

Let f(t) be n times

which we obtain

It is

continuously

differentiable,

then

,I-4

-1

+~r

F-' (,,-,

As an immediate

~-~)

application

coefficient

of the Legendre

coefficient

of the highest

1

-I

.+(q-3~

1

-

_.,

of Lemma

11 we determine

polynomial power

~ " (~) o4t

of order n. If c n is the

in Pn(t),

then

I

'"

-q

as the lower terms of the power series

for Pn ( t )

to the integral.

(26) is

to

the leading

The left hand side of

(24) and the right hand side equals

do not contribute

~ 9 4 ~q.~ N(r

according

-

18.-

§

I

= C. (~

P(.

_~ d~

~)

r(.+

, +~-3~

0

9

~)

Therefore n!

By (3)

SO that (27)

~. { t )

=

4 N {q,-I

r(,,~.} r(~)

_z" t " + ,!

.....

FUNK - HECKE F O R M U L A

B e f o r e going further into the details of the Legendre

functions

shall discuss a f o r m u l a w h i c h will prove to be the basis of a great many special results.

Let us consider an integral of the form

where

f(t)

is

a continuous

function

for

i n t e g r a t i o n is carried out w i t h respect orthogonal m a t r i x A

to

1 ~- t ~

_L 1 a n d t h e . Then w i t h any

we

-

19-

F(A=,AD)

(28)

where A ~ is

elements

the

~ ~9

adjoint

(transpose)

CA*R)

o f .&.. Now t h e

d~q~'~)

and

surface

are equal so that

(28)

becomes

"~'t This is equal to F( ~ , p variables.

) b e c a u se we may r e g a r d A * ~

U s i n g the same argument

see that F( ~,~

) is a f u n c t i o n

as the new

n ow w h i c h led to Lemma

of the scalar product

5, we

only,

which

gives us

Now a s

a function

As i t

depends

which

characterizes

of

/3

on the

this

scalar

is

product

P n ( o< ./3 ) .

I r In order to d e t e r m i n e

~

a spherical only,

Therefore

~(~~) set

~ = ~ =

q-3

we get ~4

ha-~ t h e

of degree

~

= ~ ~{~.D). and

n.

same s y m m e t r y

we g e t

d~q~

Then with

+4

it

harmonic

-

20

-

This leads to Lemma

12 :

Let

~

and

~

be any two points in

~

, and

suppose f(t) is continuous for - I ~ t ~ I. Then

/'z,! where d~. -*I

From Lemma 7 we now get by m u l t i p l i c a t i o n with Sn( ~ ) and integration with regard to Theorem 6 :

(Funk-Hecke formula)

Suppose f(t) is continuous

-I ~ t ~ I. Then for every spherical harmonic of degree n

n~ with +4

I ~(t) ~Ct) C~-~~) ~ -4

dt

for

-

INTEGRAL REPRESENTATIONS

To d i s t i n g u i s h spherical

21

-

OF S P H E R I C A L H A R M O N I C S

clearly we will designate

harmonic

in the following

a

of order n in q d i m e n s i o n s w i t h Sn(q; ~ )

and the Legendre polynomial

of degree n in q dimensions

with

Pn(q;t). It is obvious

that the integral

~ . q_.~

represents

a homogeneous

for any continous

harmonic

polynomial

function f ( ~ _ ~ ) , :

"~'1

=

~

~'1 +

of degree

n

if we set j~_~i"

,~_~

where

This enables us to get a new r e p r e s e n t a t i o n polynomials.

of the Legendre

To this end we now prove the identity

,1

Cx. q +,ix. Tr,)

d%_

=

L,, cq,•

63 q_~

X~ q-1

As this integral r e p r e s e n t s to

to all orthogonal

t r a n s f o r m a t i o n s which leave

Definition

assumes

, the integral

over all d i r e c t i o n s

are p e r p e n d i c u l a r

the integral

E9

the average

is symmetric with respect s

fixed.

the value one; hence the integral

2. We therefore have

Now

so that

and Theorem 6 with S o : I gives

which

For x = E~

satisfies

-

Theorem

7

: (Laplace's

22

-

representation)

?,

ds.

) -4

Similarly

we may get representations

functions

if we consider

For x =

~

this becomes

a spherical

dimensions which we may represent

According

to Hecke's

formula

for further

harmonic

spherical

of degree n in q

in the form

(Theorem 6)

thls

ls

+4

(29)

5~ (q-1,~,_,)

harmonic

9_ 4

~q-z

( ~ +4 41/'~-~-~z.S

"]~ (9-4, s) C4-S')

representation

os a system of

dS

-4

w h i c h can be w r i t t e n

ASSOCIATED

In order

LEGENDRE

as

FUNCTIONS

to get an explicit

orthonormal

Definition

spherical

4 :

harmonics

Suppose

Then,

we now introduce

the points

the function

associated

Legendre

of

~

are represented

An, j(q,t ) is called function

in the form

an

os degree n, order

J,

- 23 and dimension

q, if,

A,,~rq,~)

S~ ( ~ - ~ ; y q _ ~

is a spherical for every degree

spherical

J in q-1

The functions

As sperical

harmonics

have to determine The associated obtained,

which determined proportional,

From

degree

4 means

we only

zero are readily

in this

case that the properties

An,o(q,t ) and Pn(q,t)

F W('r''~ ~9-,

if

for the case n = m.

have the symmetry

Therefore

A,,,,, (q,~,~ :

(29) it is obvious

~ q_1 ) of

are orthogonal

of order

and we have to determine

(30)

Sj(q-1,

of n o r m a l i z a t i o n

harmonics

Pn(q,t).

n in q dimensions

dimensions.

functions

Definition

special

harmonic

of different

Legendre

of degree

An,j(q,t ) will be called normalized

the factor

because

corresponding

harmonic

%~ o , ~ , . . . , -

,

the constant.

are

This gives

P,, (q,~)

that

~

~-q

-4

is an associated

Legendre

dimension

q. From Lemma

plocative

constant

of degree n, order

11 we now see that apart

this is equal

is proportional

to Pn_j(2J

( 4 - s ~)

+ q,t)

from Theorem 7, so that

( ,r-

~')

j and

from a multi-

to

)(~,~-~.s)

(~-~') The integral

function

~,,_~ (2 i ,~, ~2

z

ds.

as follows

immediately

is an associated

Legendre

function

of degree n and order

j in

q dimensions.

Consider

now the function

(j+) which is a polynomial parts,

P. (+,~) = p.("(~,~)

of degree n-J.

Integrating

we see that for n> m, j = 0,...,

j times by

m; q ~- 3

+4

q,~.) (4- ~z)

(3~)

~

d~

+~

(_+)i I P-r+,~) (~)i l (+-~')

-_

because

the integrated

differentiated

"

m'~'~+,+~]~t

terms vanish for t = I and t = -I. The

term is of the form q-3

(32) where

(,I-(,) Pm(t)

p,.

'

is a polynomial

(~1

of degree m, which is best

seen by

using the formula +

with U = (I-t2) j+(q-3)/2

We thus obtain from

(33) On t h e

and V = Pm(J)(q,t).

(31) for m ~ n

I (~_ ~z) other

2

h a n d we h a v e

from

=~@~

P

Lemma 10 f o r

. m

~ n

(34) -4

As Pn(J)(q't) and P n - j ( q + 2 j ' t ) which satisfy be obtained

the same

intervall

conditions of orthogonality,

by a process

t n with the weight -lmtgl.

by a constant

are both polynomials of degree n-J

of o r t h o g o n a l i z a t l o n

function

over the

As they are not normalized

they differ

This gives

-

~)

from the powers

q - ~~, z ~

factor.

(~

they may

us

only

-

Lemma I~

:

25

-

The functions

~/z

and

?.(~) (~,~)

(4 t ~)

are associated Legendre functions of degree n, order j, and dimension q, which differ only by a factor of normalization,

which is given by

~-~ ( ~ ? ' ~ ) As Pn_j(2j+q,t)

~(~,~1

:

~(2~q,~.~)

and Pn (J)(q t) are proportional,

can be obtained by equating the coefficients given by

F (~)

2 ~ ~ ~#

~

this last result

of t n-j as

(27). This shows also that all Legendre polynomials

expressed either by Pn(3,t) or Pn(2,t) odd or even.

was not to find a

of different dimensions

but to give an explicit representation Legendre functions

of the normalized associated

An, j(q,t ).

Suppose now that the two unit vectors

~

and

~

are represented

in the form

"~

Then,

can be

according to whether q is

The purpose of the preceding study, however, relation between Legendre polynomials

(~'~).

if the function A

~

n,j

s-~, I

§

1/~-s

z'

,/~_~

are normalized

is a complete and normalized system of spherical harmonics of order n, as Sj,k( q-1 , ~ _ ~ dimensions because of

) has this property in (q-l)

-

26

-

V1

Thus we know t h a t A n , j ( q , t ) i s p r o p o r t i o n a l or by Lenuna (13) to (1-t2)j/2Pn(J)(q,t). ++

(35)

L (4-t') "~

"r,,_ ~ (2~+ ~, t ) ] z

++

(+- ~')

to (1-82)j/2s Now d~

z i + t-

To f i n d the n o r m a l i z i n g

factor

for

(I-t2)j/2P (j) (q,t),

we

observe that, for large t, Pn(j) (q,t) (I-t2) j+ ~q-3)/2 is a holomorphic function of t which may be written as

p(~l{q,+)

(<_+.}

~

c_4)~a + ~~ t2i*~ -J p.~i;{,,+) {+_~-2)

=

According to (27), the highest power of the Laurent expansion for Itl

>

1

is i+ {~-3)

(-4)

where we have set

b.i/.

z

{

b.,

(.-~)!

. + ~ + er_~

~.

for the leading coefficient in (27).

Thus by (32) q-

=

(-4)

6.

F' ( , + ~ + q - z )

("-~J!

Now with a constant c we get for

F" (

Itl >

I

,,,

+ 9 -z)

9

~

"++-~

+

-

.

.

-

.-

( .~-

~--~ ) ~

(_.~) ~

L C-~) "i .

b,,

-

c

F'[.+4.

~.-$H

As Pn(t) is a polynomial

27

t ".~ . . . .

.

+ ~-z)

]

/ : " + ....

P[..~q-zl

of degree n, we have

/4

-_

C.~_e~] 9

[ C..O ~

b.

F'C~+~r+~-z)

.

[""~}!

Substituting

this into

("-'i)!

§ .....

P ( ~+ q- zl

(31), we have for the value of that integral

P [n~-q-~l -4

From the analysis leading to formula

(27) we know that this last

integral is

f

2-"

,

which combined with the value of b n from Lemma 14

(27) yields

: +4

I [ [.-t'l ~/2 ?,,r

_

Thus from Lemma 15

~_

.~[

t ]z [~-t'~ ~ {" C ~ * ~, + q - z )

d~

4

(35) and Lemma 14 we get :

The functions

A .,~ cq,t)

or

Y

C-~- t ~) ~/~ ~ - i t2~, q, t~

-

A . , t (~,t) = /

~""

28

-

(.-i)! f'(.,~-z) . !

p(..i+q-z)

form a system of normalized

been obtained

from Lemma

formula itself. coefficient

associated

of An,j(q,t ) in terms of P~J)(q,t) (_

A representation

13 but Lemma

(~_~,)i/~p~i)(~.t)

N(,l,.)

Legendre

could have

14 is an interesting

The reader may find it interesting

to compare

above with that obtained by using Lemma

The addition theorem

the

13.

(Theorem 2) now can be written in the form

A ",i (*'t) ~'=0

A"'i ('/'~)

Si,~ ('-', ~',-,) Si,,~('t-~,'r/,-.) g= "I

N(el,.)

According

functions.

"/~_, ])

to Theorem 2, /v(q-,, i )

(]6)

Z

1,(=4

this may be written as "I ,

~.=o

(37)

The addition theorem is usually given in the literature An,j(q,t)

expressed

gives by Lemma

15

in terms of the derivatives

with

of Pn(q,t ) which

-

29

-

(38) t

PROPERTIES

OF THE LEGENDRE FUNCTIONS

Multiplying

(37) by PC (q-l, ~ q_1' ~

q-1 with respect

to

~ q-1

q_1) and integrating

, we get from 23 and Lemma

( 10 ) :

§

~_~

~,_~ I

A.,e {q,~) A.,e(q,s) = ~t,..~ ~

over

~cq'~"* ~-"/';:;~~~ ~-~,~c~-~'~

~ ~v

~q

From Lemma 15 we now get Lemma 16:

Ig(2.C,q, n-g.)

Wze+q_ ~

§

N(~,.}

,/-,r

wq_ z "4

In particular

it follows

for

4 = 0 ~4

't,O ,~ _ 4 t.,,O c/_

2

~,, c,,t~,,c~,~)

= I ~',,~q, t ' ~ ~ -4

We now prove

q-#

~-~zT"V)~

~ ~'~

-

L e m m a 17

30

-

For 0 _z x < I and -I __4 t _x I,

:

o~

4-- X z

1,1

(4 + x z - Z •

For q = 2 thls is a w e l l - k n o w n by s e t t i n g

t = cos ~

Z

N (e,,~)

identity

9/z

w h i c h we can best

obtain

. Then

x ~ "P, (2,t:)

~=o

x '"le~"9" -

=

4

.

4- x e ~

~=_~

4

+

4

4- xe-~

d --.Mz

~-2

~: ~ ,

4_X

Using

for the following,

the L a p l a c e

-

polynomials,

7.

4+x

that

9

we find for the left h a n d

~-2

N~q,~)

side

oo Z

(40)

9-~

x"(~ §

In L e m m a 3, we had p r o v e d

~ . s

I t + i ~

(41)

N(q,.)

x

.

4+

=

W

(4- x) q-~

is less t h a n one we m a y w r i t e s t a t e d in L e m m a 17

~-~

o~_~

ds.

sl 2 : t 2 + ( 1 - t 2 ) s 2 ~ t 2 + ( 1 - t 2) : I and h e n c e

Ix(t+i~s)l condition

( ~ - s 2)

the i d e n t i t y

.--o

As

a 3-

(Theorem 7) of the L e g e n d r e

,4

(39)

z - 2 ~

xf

therefore,

representation

=

z

4 + x z

We may a s s u m e

_4

+4 (

)

- the f o r m u l a

- under

(39) as q_~

~+ x ( ~ + ~ z ~ T ~ ' s ) l~- x ( ~ ~ ~r

(~-s'~

~-"

~

~s

the

9,

-

To prove equal

our Lemma,

we thus have

to the f u n c t i o n

In order

Using

31

given

-

to show that

this

on the r i g h t h a n d

to do this we i n t r o d u c e

the

integral

is

side of L e m m a 17"

substitution

s = t a n h u.

the a b b r e v i a t i o n s

(~2)

and o b s e r v i n g

( 4 - s~l z

ds

=

we o b t a i n for the i n t e g r a l

in

(41)

If

f(u)

stands

for

either

we have f r o m f''(u)

of

the

= +f(u)

(t.o',d,..J~-'t d~

two

functions

defined

in

(42),

for any c o m p l e x n u m b e r u O

N o w we i n t r o d u c e

(4.4)

the real n u m b e r

x, ,/,t-~,"

so that we can w r i t e

~

+ ,~'(1-~) f2(u

by

: "li,~+xZ-zxt"

e

~r

o~,-~

g

as

(4s) Apart

from a n u m e r i c a l

constant

the i n t e g r a l

(43)

therefore

equals

t~

.~ q - r

(r

Here

I

x ~-l~t)

the i n t e g r a l

r (-.,,r) ~

f,~+,;4") * r

--DO

reduces

to

f~

~e (-id')

(46)

where

is zero.

As

~

+oo

t e r m of this is g r e a t e r

sum v a n i s h e s ,

t h a n zero,

for all q ~- 3. It may be r e g a r d e d

(47)

C~,,-#,, ,~1 ~ - z -eo + r

[DV~A (~+4r) Jq-r o~

[s4;,~ (.a+4d~)]'q'z

the s e c o n d

.~,,.l,(~+q7

=

the i n t e g r a l

as a c o m p l e x

( s " ~ "~) q- ~ -~

because

+ 4"ll'/l"

the i n t e g r a l in

(46) e x i s t s

integral

-

where

this

last i d e n t i t y

integration expressing u = v + i ~

in

~- ~+]''+~ +"+"P~('i"+:)

(42)

=

by s h i f t i n g

the p a t h of

. Combining

~

these results

(47) b y use of the s u b s t i t u t i o n

:

4+" r

~

~+'-+ I ~'I-+

("~+xz-zx~') ~

~v (c,<~,~v)t-z

--CO

and

(44)

we get 4-

We have

and

, we n o w see that

~=o From

-

is o b t a i n e d

to the l l n e Im(u) the i n t e g r a l

32

thus p r o v e d

X

Z

that 9

_

~Z

I'1 ~ C,

In o r d e r

to d e t e r m i n e

the c o n s t a n t

C we set t = I, and o b t a i n

~ - ~ 1 q-+

.=o

As the r i g h t h a n d

s i d e of

we o b t a i n C = I and h a v e

(48) r e d u c e s thus proved

to this v a l u e

for t = I,

our i d e n t i t y .

Introducing

c,+,.++

S.,

:

~K=O

we h a v e @0

=

Z

4 +

19=4

I'1=0

x"

(s.,ct~j

o,o

=

Z

_ S,,-+c~,~=))

oO

x ~

S,,(',~,e)

- • Z

,~",.%,~o/,+~

l"l = 0

Where oo

(49)

~

x " ,5'. (0/, ~-) =

I"l = O

Set

for

n

= 0,1,...,

and

from

q

"t + ~(

( . I + x z - 2 • EJ r "z

-~ 3,

M

I"(,+§ p(n++). I-'(q-z)

(40)

-

33

-

so that oo

~"

I,i

C n (el) X

=

( 4 - x ) q-z

We then get

Lemma

18 :

For q -~ 3, 0 _L x < I, and -I z t -~ I

C4 + xz- ~>c~) ~

.:o

with c . t 9)

The

corresponding

result .

is well

Pn (2,c~

T

The proof

known

"p. (~ ~:) t

=

4- ,8, ( 4 + ~ z _ ~ ) Z

and can be proved

) = cosn T of L e m m a

i'~(.+q-z)

P(q-z). F'(.+4)

for q = 2 is

~'I=4

which

=

immediately

"

18 is quite

analogous

so that we can use the same n o t a t i o n s . of Pn(q,t)

by using

to the proof

Laplace's

of L e m m a

17

representation

gives +4

y_~

c~

.=o

I (4 -(4-sz} '~4 als x ( ~ I/TZ-P'.S)) ~-z

%-~

-'I

The s u b s t i t u t i o n transform

the integral

-4

so that Lemma

Lemma

(42),

(44),

(45)

to

I

"

18 may be proved

by the same

I

-00

arguments

that

led to

C n ( q ) P n (q't):C(q-2)/2(t)n

where

17.

4) This

CV(t) n

s = tanh u and the a b b r e v i a t i o n s

estabillshes

are

the r e l a t i o n

the Gegenbauer functions.

-

Suppose space,

now

that x and y are

any

-

two v e c t o r s

y=

~-.~

;

l}'l=

,I ,

I~I

= 4

for q ~ 3 and R> r

I~->'I ~-z

(R z + ~

- z

R.~" ~ ' . ~ t ) ~ -~

4

This

in q - d i m e n s i o n a l

with x = R.~" ,

Then

34

c a n be e x p r e s s e d

Lemma

19 :

If x

by L e m m a

R~

=

; y

,,<-•

Let xi,Ylbe

(50)

According be w r i t t e n

=

18 so t h a t we o b t a i n

R

Z

components

2-~ = [ ( x , - Z ~ +

to the T a y l o r

and R

'r.'~

=

the Cartesian

i•

4

, then

r

~c~,~.~).

c~c~(

of x a n d y. T h e n

C~-~,

expansion

>

.... , ( x ~ - ~ # ]

in s e v e r a l

Y

variables,

this

can

as oo

2-q

(51) (~

I'I=0

If ~7 x d e n o t e s

as u s u a l

the vector

operator

with

the

components

the

coefficients

we h a v e

y~ ~X~ -~ ~ . . . . .

a n d we g e t

from

(50)

xl

and

z-~

=

(51)

+ )'~ ~

for

:

IYl <

~o >-

~

(~

'V~)

Ix~ 2_ 9

VI=O

Comparing n of q~ ,

this with

Lemma

19 we h a v e

by equating

-

This gives with the explicit

Lemma

20

:

(Maxwell's

~

As

Ixl

2-q

value

1•

=

this shows

by repeated

in the direction

-

of Cn(q)

(_.f)',

F(~q-~

I xl

solution

~

polynomials

9 The potential

We know that every spherical

~

equation

in

may be

of the fundamental

may thus be regarded

a pole of order n with the axis

iXl~,*q-z

of the Laplace

that the Legendre

of the vector

P,,~:q,~'-~)

/'~(9-1)

differentiations

hand side of Lemma 20

and R =

representation)

is the fundamental

q dimensions, obtained

~

35

solution

on the right

as the potential

of

at the origin.

harmonic

can be expressed

in the form

I( _.- ,I

with a fundamental

system

~ k" Therefore

it is always

possible

to

write

which shows

that every

the potential

potential

of a combination

system of fundamental

points

of this type may be regarded

of multipoles

introduced

to a fundamental

system of multlpoles

off the spherical

harmonics.

A rather following

with

striking

interpretation

way. We first

observe

earlier

axis.

The

thus corresponds

in Maxwell's

of L e m m a 2 0 that

with real

as

interpretation

is obtained

in the

-

36

-

2,

~b

-1

P (., ~)

where Hn(q,x ) polynomial

rnSn(q, ~ ), which enables

=

us to express

the formal

Hn(q,~7~) as

q

Multiplication gration over Lemma 21 :

of both sides of Lemma 20 with Sn(q, ~ ) and intei-~

now gives

For every harmonic

polynomial

of degree n

r ( ~) Before leaving the special properties

i~/~"'~-2

of the spherical harmonics

it should be noted that many more can be derived from Lemmas to 21 of which the recursion the associated known.

functions,

formulas

for the Legendre polynomials 9

and their derivatives

They can be obtained by differentiating

formulated

in Lemma 18 with respect n coefficients of x .

18

are perhaps best the identity

to x or t and equating

As an example we take the formula h

(52)

{~-z). Z

N(~,~) ~(q,~)

-- c.(q) P.'(q,~). c~.(q)P.'.~(q,~).

K=O

From the Laplace representation P,' ( q , ~ ) =

~ ~- ~

i

(Theorem 7) we get

, ( ~ , ~ ~T:-~

which shows that for all t with

.s

).-s

O - ~ . V ~ ; -~"~

Itl -~ t o < 1

P'(q 9 9

1~. ' c ~ , ~ )

=

(~(.)

n

s) O-s')

z

q_q satisfies

ds,

-

uniformly. series

It is t h e r e f o r e

of L e m m a

-

permitted

18 t e r m w i s e .

7

37

to d i f f e r e n t i a t e

the p o w e r

We obtain

c,,(,~ •

P,,'cq,~)

=

C,t-2) (-,,-.+,-2~.~)~,',~

PI:O

w h i c h g i v e s us

rl=O

Comparing we get

this result with

(52). T h i s b e c o m e s

(49) and e q u a t i n g particularly

true of m a n y m o r e of t h e s e r e s u l t s . 3-

coefficients

simple

of x

n

f o r q = 3, as is

In this case we get

--

+-

.

K=O

DIFFERENTIAL

The basic

E~UATIONS

concept

and the s t a r t i n g p o i n t

to the t h e o r y of s p h e r i c a l polynomial. spherical

is the h a r m o n i c

and homogeneous

O n l y v e r y i n d i r e c t l y we m a d e use of the fact that

harmonics

are

shall now derive results special

harmonics

of our a p p r o a c h

differential

c o n n e c t e d w i t h the L a p l a c e which express

equations

In o r d e r to do this we h a v e

for the s p h e r i c a l

to e x p r e s s

t e r m s of the p o l a r c o o r d i n a t e s

this f a c t o r

the

~

equation. in t e r m s

the We

of

harmonics.

-operator

w h i c h we h a v e b e e n u s i n g .

in We wrote

(53) where s

is a u n i t v e c t o r s p a n n e d by the u n i t v e c t o r s

~9-~

.........

S u p p o s e n o w that we h a v e

s

representation

. ....... v~_,

of

some

coordinate

~-~q-4

9 We then

set

~I = ~

so that

~9

above notation

;

~q-1 = ~ ,"

is a f u n c t i o n of t and of

~,,. ...., ~ - I

~

-- v~

i-- ~,....,,t-2

for

v~, ........v~_~

, or in the

. W i t h the a b b r e v i a t i o n

- 38 -

~};,< _- a__&. ~

we may form the Beltrami

From

(53) it is clear

. ~= ~

Operator

~,~..,,~ ; ~, ~;;~ =

for

, ~-,,~ _- ~, ~,.., ~_ .~.

~

that for i,k = 1,2...,q-I,

_

9,a,i, 8__x

~},a/ . ax

-

#~.~ a.._x, a- -x

"rz~.,.,;

and we obtain by means

of the tensor

= o i

ax

~

=./

calculus

We had

~

=

{-~

9 ~_~

§ ~

so that for i,k = 1,...,q-2,

d'a,l,~" ~'~q-4

=

t: z

,I-

;

9,~.,

This gives us az

(54)

Lk I

=

It should be noted we

(,f_~=)

that

c~-~)~

atz for

get _

a 2

a § a-~

I -,I ~.z A~-4

-

39

We can thus define the operators the two -dimensional

-

~

successively,

starting with

case.

As rnSn(q, ~ ) is a harmonic

function we get

I'1 - 2 . _ _ ak

which gives us Lemma 22 :

Every spherical harmonic of degree n and dimension q satisfies ~

5~C~,~)

,

n(~+9-2)

S~q,~)

For the Legendre polynomials we thus get from Lemma 23 : The Legendre polynomial dt z

o .

(54)

satisfies

]P,,(.q,t:)~- n(n+ct-z)~C~,~)=o.

tq-~),~

-

Pn(q,t)

=

The associated Legendre functions satisfy

which gives us Lemma 24 : The associated Legendre functions An,j(q,t)

of degree n,

order j, and dimension q satisfy

Lc~-~ ~) d ~ _ ~ q - ~

~ ~ n~n+q-2~-

~t~-~]

A.,~(q,~ = O.

The extension of the concept of spherical harmonics for degrees and orders which are not integers, differential

equations,

Spherical Harmonics). the harmonic valued,

functions

may be started from these

as has been done previously

However,

(see Hobson,

if the condition is imposed that

thus obtained should be entire and nni-

the theory reduces to the functions discussed here, which

are therefore called the regular spherical harmonics.

-

EXPANSIONS

IN S P H E R I C A L

We shall n o w prove and c l o s e d

that

the s p h e r i c a l

harmonics

on the sphere.

as an e x t e n s i o n

the case of p r o b l e m s

-

HARMONICS

set of f u n c t i o n s

be r e g a r d e d

40

of the t h e o r y

with spherical

form a c o m p l e t e

This,

of course,

of F o u r i e r

symmetry

series

in any n u m b e r

may to of

dimensions.

Due

to the o r t h o g o n a l i t y

Lemma

17

of the L e g e n d r e

polynomials

we h a v e f r o m

(multiply by PO and i n t e g r a t e )

for all x w i t h 0 ~ x < I. We shall n o w prove

L e m m a 25

:

Suppose

f(t)

is c o n t i n u o u s

for -I m t m I. T h e n

+I

( 4 ~ ~ z - 2xd:) r

x..,4-o

~f~. =

~P(41.

.~~l - ,

I

We w r i t e

where

g(1)

= O. If f(t)

immediately,

is c o n s t a n t

so that L e m m a

the r e s u l t

25 is p r o v e d

follows

from

(55)

if we s h o w

4.4

)

)r

The c o n t i n u i t y

of g(t)

{ q + ~ z - Z ~< 4: ) ~II~

implies

that

there

is

a positive

function

m(s) w i t h S ...e O

s u c h that 4 ~'t I; "~/ "1- $

Moreover,

it f o l l o w s

f r o m the c o n t i n u i t y

that

there is a c o n s t a n t

with I ~I We n o w o b s e r v e

that

Z~6

for the same r a n g e 1-X

z

C

for

-~ -~ g 4.

for -1 g t g 1 - s and x -~ 0 4 ~- •

so that

~

=

[4-x)

z +

I of t and ~ x < 4- x z

2x{4-~)

>/ 2 x . S

I 4+2( . ~

(~ X i . z . . E ) ~

C

-

We now des

-

s by

s ~/~

(56) and divide

I

-/~-x'

=

the interval

I - s _z t -~ I . T h e n

(57)

#I

os i n t e g r a t i o n

into

-1 ~- t ~- 1 - s and

for x ~- 0

(~-'('~(,~§~(~) ( ~ - F ) ~-~ ~z _ Z x t ) ~1~

d~

_~ 2C ~-,()'/~

,a~

~-~

= ~'(~W;:-;-~)

-I

and

li as this

last

;

(4+ ~ - Z x t ) ~lz

integral

I

=

may be m a j o r i z e d

by

(4 9 ~z _ Z ~ t) q/~

According

to

(55),

our L e m m a

follows

We are n o w able Theorem

8 :

s tends from

towards

(57)

to prove

(Poisson's

and

zero

for x ~ I - O, so that

(58) w i t h

the f o l l o w i n g integral)

(55).

theorem:

Suppose

F( y ) is c o n t i n u o u s

on

A~q

Then

"l ~

-r~,1-o where

As

,D,~

thls

is compact,

the e x i s t e n c e

I ' (~-7 z) F ( ~ ) , (4+rz-zr~.~/)

limit

holds

we can deduce

of a p o s i t i v e

uniformly

from

function

m(s)

d~,l~

=

F(~)

~/~

w i t h regard

the c o n t i n u i t y

of F( ~ )

such that

(59) We now assume ~ =

s

#(~)

and define

=

Ir

(60)

~?q_, )

,9,?-I

so that

From

~.z?

=

(59) follows I ~(4)

-

~(t)

l

L_ w ' l - ~

" rn(s~

~-1

~',/)

to ~

,

.

- #2 for I ~ t ~ I - s. The integral in Theorem 8 can be written (~-~

SO t h a t

we g e t

for

9(,Jm

~ =

8~

ea~

{4,", -~- z','.r.~)

~/z

7"--" 4 - 0

~4

~--~ 4- o

(~

(q+~-Z~)

~'/z

-,,f

As any point of the sphere appropriately for all

~

of

~

may be chosen as

chosen system of coordinates, ~

. Moreover,

uniformly valld estimate

the estimate

E 9 of an

this argument holds (60) only involves the

(59) so that the limtis are approached

uniformly. From the identity oo

(61)

Z

r

.=,

=

,' (~+ 4T-~-2~y,.?)f/z

we may now deduce ,Theorem ~ : (Abel summation)

Every function F( ~ ) which is

continuous on A ~

can be approximated uniformly in the

sense of @o

I"--~4-0

by spherical harmonics Sn(9,~ ) which are given by

~q (c),~'/ = where

Ncq,.~

~. ~ , ~ . ? ) F ( ? ) ~(~)9c-~) =

c.,~ S.,~(~,}')

-

c.,~

This result

=

We may therefore representation

-

I

is an immmediate

which holds uniformly

43

consequence

with respect

integrate

to

termwise

of the spherical

of the identity

~

and W

and obtain

harmonics

(61)

for 0 g r < I. the last

from the addition

theorem. Using

the same notation,

we get from Parseval's oo

inequality

N(~,~)

oo

where we used the abbreviation

~'-

I c.,~ I ~

=

I I S. (~,~) I ~ d%c~

~ c~.

Set oo

so that (

) 'r .~, .t- o

I F('r,,'~)

12

~q

,D, ~/

as F(r, ~ ) approximates

F( ~ ) uniformly.

On the left hand side we may interchange summation

Theorem

because

10 :

of

I F ( ~ J l z d~,ir_, ?) conclusion

Theorem

11 :

the limit

have

and the

(62).

For every continuous

Another

We therefore

may b e d r a w n

--

function Z

F(~ )

(c.) z

from Theorem 9,

If the continuous

function

F( ~ ) satisfies

for all spherical

harmonics,

it vanishes

identically.

-

Our assumption

44

has the consequence

-

that F(r, ~ ) vanishes

for

all r ~ 1. Therefore

T~4-O

which proves

These

Theorem

last results

has the basic continuous

11.

show that the system of spherical

property

functions

to more general

on

classes

of being A"I~

complete

. Extensions

of functions

of the theory of approximations.

and closed

harmonics for the

of these results

may be obtained

by methods

-

45

-

BIBLIOGRAPHY

The following are either solely devoted to the subject of spherical harmonics or contain detailed information on this subject 9 E

rdelyi *

A.

,

,

W.Magnus,

F 9 0berhettinger,

transcendental Hobson,

E.W.

I ~id 2, New York,

1953.

1931.

Ku~elfunktionen~

Leipzig 1950.

Magnus, W. and F. 0berhettinger,

Formulas and theorems for the

functions of mathematical MUller,

Higher

The theory of spherical and ellipsoidal harmonics~

Cambridge, Lense, J.

functions, Vol.

and F. Tricomi,

C., Grundprobleme magnetischer

physics, New York,

der mathematischen

Schwingungen,

1954.

Theorie elektro-

Berlin, Heidelberg,

GSttingen, 1957. Morse,

P. M., and H. Feshbach, Methods of theoretical Vol.

I and 2, New York,

physics,

1953.

Sansone,

G. Orthogonal

Webster,

A.G. - SzegS, G. Partielle Differential~leichungen mathematischen

functions t New York,

1959.

Physik~ Leipzig, Berlin,

1930.

der