spherical harmonics

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 of these lecture notes is the theory of

regular spherical harmonics in any number of dimensions.

The approach is such that the two- or three-dimensional

problems do not stand out separately. They are on the contrary

regarded as special cases of a more general structure. It

seems that in this way it is possible to get a better under-

standing of the basic properties of these functions, which

thus appear as extensions of well-known properties of

elementary functions. One outstanding result is a proof of

the addition theorem of spherical harmonics, which goes back

to G. Herglotz. This proof of a fundamental property of the

spherical harmonics does not require the use of a special

system of coordinates and thus avoids the difficulties of

representation, which arise from the singularities of the

coordinate system.

The intent of these lectures is to derive as many results

as possible solely from the symmetry of the sphere, and to

prove the basic properties which are, besides the addition

theorem, the representation by a generating function, and

the completeness of the entire system.

The representation is self-contained.

This approach to the theory of spherical harmonics was

first presented in a series of lectures at the Boeing

Scientific Research Laboratories. It has since been slightly

modified.

I am grateful to Dr. Theodore Higgins for his assistance

in preparing these lecture notes and I should like to thank

Dr. Ernest Roetman for a number of suggestions to improve

the manuscript.

February 1966 Claus MUller

C ON TEN TS

General Background and Notation ............................... I

Orthogonal Transformations .................................... 5

Addition Theorem .............................................. 9

Representation Theorem ........................................ 11

Applications of the Addition Theorem .......................... 14

Rodrigues Formula ............................................. 16

Funk - Hecke Formula .......................................... 18

Integral Representations of Spherical Harmonic ................ 21

Associated Legendre Functions ................................. 22

Properties of the Legendre Functions .......................... 29

Differential Equations ........................................ 37

Expansions in Spherical Harmonics ............................. 40

Bibliography .................................................. 45

- I -

GENERAL BACKGROUND AND NOTATION

Let (Xl,...,Xq) be Cartesian coordinates of a Euclidean space

of q dimensions. Then we have wlth

the representation

I x l l : ~-~ : ( x . ) ~ + . . . . + ( 'x~) ~

where

represents the system of coordinates of the points on the unit

sphere in q dimensions. It will be called ~ , its surface

element d ~9 and the total surface ~9 , where this surface

is given by

By definition we set ~ =2. Then we have

If the vectors ~ . . . , ~ are an orthonormal system, we may

represent the points on X~9 by

(1 ) ~s __ .t.E< 1 + ~ ~ '_. , ,. --~_~ f_~4 ,. ~:~<r~<l

where ~s_~ is a unit vector in the space spanned by El,... ,c~ r

The surface element of the unit sphere then can be written as ~-~

and we have from above

A~#_ I -i

The integral on the right hand side may be transformed to

4

i <!-3 % %/

o

I) Here and in the following points of the unit sphere are

denoted by greek letters.

- 2 -

Which gives us for

( 2 ) w I =

D e n o t e b y

(3)

q = 2,3, ...

" ) =

Z

the Laplace operator. We then introduce the

Definition I :

"~ (z ) z C U , / _- .

Let Hn(X) be a homogeneous polynomial of degree n in

q dimensions, which satisfies

Then

is called a (regular) spherical harmonic of order n

in q dimensions.

From this we get immediately

Lemma 1 : ~, (-~) = (-4)~ Sn (~)

Let Hn(X) and Hm(X ) be two homogeneous harmonic polynomials of

degree n and m. Then by Green's theorem we have

as the normal derivatives of H m and H n on ~9 are

I 8~ H~(~)} = ~ H~(~) and [~ H~(+~)] = n H~ (~) T= 4

From Definition (I) we have therefore

respectively.

/

Lemma 2 : .~ S~ (~) S~(~) ~ = 0 for m #

Any homogeneous polynomial in q variables can be represented in

the form

-3-

Z (~) A._#(z,, . . . . ,~_.) : H.(.) (4) i :~

where the An.j(Xl,...,Xq_1) are homogeneous polynomials of degree

(n-j) in Xl,...,Xq_ 1. Application of the Laplace operator in the

form

= + ~ - ~

gives m-z

nq H (x) : --# ~ ;(~-~)(~1 ~-~ A._~ + ~:s ~)~ a~_~ A._~

For a harmonic polynomial this has to vanish identically. By

equating coefficients we thus get

(5)

Therefore all the polynomials Aj are determined if we know A n and

An_ I �9 The number of linearly independent homogeneous and harmonic

polynomials is thus equal to the number of coefficients of A n and

An_ I �9

Denote by M(q,n) the number of coefficients in a homogeneous

polynomial of degree n and q variables. It then follows from

(4) that

{ ~ ~(~-~,~) , ~ ~o

(6 ) M (~,~) = ~=a 0 p n L O

Clearly M(1,n) = I, so that M(q,n) = ~(~q-~) �9

Now the total number of coefficients available in

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

(7) N (~,n) : MCq-s,n) ~- kl (~-'1, n-s) : (7(~ ~-z) !

Then the power series

(8) @~ #'o

converges for Ixl g 1. By (6) and (7)

- 4 -

(9) IVc9,.1 = Z ,

Now it follows from (7)

N(I , n ) = 1 for n = 0,1

0 for n > I, o

so that

S u b s t i t u t i n g ( 9 ) i n t o ( 8 ) and i n t e r c h a n g i n g t h e o r d e r o f s u m m a t i o n

we obtain

and hence

This gives us

4

~9 (x) = 4Y" X

Lemma ~ : The number N(q,n) of linearly independent spherical

harmonics of degree n is given by the power series

(4 -x )q -~ .=o

Specializing to q = 2 and q = 3 we get Oo oo

4 + • = 4 + Z 2 x " = .Y__ N ( z , , , ) ~ < ( 1 0 ) -7 - x . = ~ . = o

*• : >- ( 2 . + ~ • = 7 - NC3,,~) • (4 - X} z ~ = o ~ = o

From Lemma 3 we can determine the N(q,n) explicitly. The

binomial expansion gives for Ix l < I

- , P C " + q - ~ } x" 4 + x : ( 4 ~ x ) n ( .+d P(q-4l

( 4 - x) I-~ . : o

s o t h a t

4 + 7" ~,,,q-zJPC,~,f-z~

(11) N (9 ,~) I n ~ 4 /

"~ n = 0 I

-5-

If we set

N(%,m) r l

(12) s,, c~) : Z c i ,s,,, i (~) ~--,f

we have

Lemma 4: There exist N(q,n) linearly independent spherical harmonics

5~,~(F) of degree n in q dimensions and every spherical

harmonic of degree n can be regarded as a linear

combination of the 5~,~ (~) .

ORTHOGONAL TRANSFORMATIONS

Suppose now that the functions Sn,j(~), j = I,...,N constitute

an orthonormal set, i.e.,

If A is an orthogonal matrix, then Hn(A~) is a homogeneous

harmonic polynomial of degree n in x if Hn(X) has this property,

so that Sn(A ~ ) is a spherical harmonic of order n. In particular

I,I

(14) S. ,~ ( A ~) ; Z ~i~ s.,~. (~') T - - 9

To every orthogonal matrix A there corresponds therefore a matrix

�9 We now have, because of (13) and (14), c ~T

I = C~? (15) Zo,~ ( A ~) So,~ ( A ~') ~ ,_=~ ~q

The orthogonal transformation A ~ may be regarded as a coordinate

transformation of O 9 which leaves the surface element d ~

unaltered. This means that

-6-

~9

From (15) we now get

~_~") C a ~, kl

(16) ~= c =

so that the coefficients c~T are the elements of an orthogonal

matrix. Besides (16) we therefore get also

IV(q,.)

(17) Z c r''' " -- ~: "r=4 '~ C'r~ K

For any two points ~ and ,? on /]9 we now form the function

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

F ( A ~ ' , A ? )

W(q,~) W(f,.)

Z[Z "

I"= 9 ~ : 4

The function F( ~, ~ ) thus has the important property that it

is not changed if y and ~ undergo an orthogonal transformation

simultaneously.

To further studies of our function F( ~, ~ ) we use the following

properties of the group of orthogonal transformations

a) To every unit vector ~ there is an orthogonal trans-

formation such that A ~ = s 9 �9

b) For any two vectors ~ and ~ we have

c) For any unit vector ~ there is a subgroup of orthogonal

transformations, which keeps ~ fixed and which trans-

-7-

forms a given unit vector

for which

%

~"~ =Y'~o

in all those vectors

LEGENDRE FUNCTIONS

We now use these properties to study our function F( ~, ~ ).

It follows from (a) that we may transform ~ into E 9 . Then,

according to (2), ~ would be represented in the form

(~8)

From (b) we know that t is also the value of the scalar product

of ~ and ~ before carrying out the transformation. From (18)

it can be seen that the subgroup with fixpoint ~ is isomorphic to

the orthogonal group in (q-1)-dimensions ~).

We have therefore

for any two vectors ~q-4 and q q_~ on ~f~q_~ �9 This implies that

F ( E~, t~ ~ ~_~z" ~_~ ) does not depend on ~q_~ . It therefore

is a function of t alone. Combining this with (18) we have

Lemma 5: Let Sn, j(~ ) , j = 1,...,N be an orthonormal set of

spherical harmonics on ~q . Then for any two points

(vectors) ~ and ~ on ~q the function

depends only on the scalar product of ~ and ~ �9

i) The orthogonal group in one dimension consists of the two trans-

formations x' I = • x I only.

-8-

It 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 ~9 unchanged.

b) 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 n d

A , , _ + = c/{x+l'+--. + '<;A,,:o for ~= 2+<+4

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

(• + (;<z)~+ .... § C,,<,l.:+ ) l =. Tz C.1_tz)

-9-

We now have:

Theorem I: The Legendre function Ln( ~ ) may be written as

where Pn(t) is a polynomial of degree n with

The last two relations of this theorem csn be proved easily:

As r = I, t = I, corresponds to ~ = ~ , the first statement is

condition (b) of Definition 2 and the second equation follows from

Lemma I.

ADDITION THEOREM

We now can determine the function ~ ~ X.~) in Lemma 5, for

we know that this function is a spherical harmonic of degree n

with respect to ~ �9 It is moreover unchanged if W is transformed

by an orthogonal transformation which leaves

Z

as the function ~ (~"~Z)

To determine the constant c n we set

N (<z,,~)

fixed, so that

can only be proportional to Pn ( ~-~

= ~ and obtain

.

_- c . = c . .

Integration over A~? gives

N [ ~ , ~ ) = c~ co~ I

and we get

- 1 0 -

Theorem 2 : (Addition Theorem) Let Sn,j( ~ ) be an orthonormal set

of N(q,n) spherical harmonics of order n and dimension q.

Then

where Pn(t) is the Legendre Polynomial of degree n and

dimension q.

Thls theorem is called addition theorem as it reduces to the

addition theorem for the function cos ~ in the two-dimensional

case after introducing polar coordinates.

In order to determine the spherical harmonics for the case q = 2

according to this theory we first have to determine two linearly

independent homogeneous and harmonic polynomials of degree n.

We can take them as

We now introduce a system of polar coordinates in the usual way

( 1 9 ) • : "7- c,o-oy ; x z -- -r ~ T

and get

7r ~o) ! Re ( x , + ~ x l ) " = c,~ ~ (~ . - I"

s ] ~ ( x ,+~• : sX,. n ( ~ - , f ) T h

From these two we get an orthonormal set by

F- T)

-11-

The Legendre function now is obtained from a homogeneous harmonic

polynomial which is symmetric with respect to the x 2- axis, and

which takes on the value I for x1= 0, x 2 = I. This gives us

o r

L,, (xq,x~) : Ee ~rx~ t ' i x ~ ) '~

Now let t be the scalar product between

from (19)

s

t = .~. :~ I , , = ~, -~

and ~ . We then have

c -

which gives us

In two dimensions, therefore, the function P (t) is what is otherwise n

known as the Chebychev Polynomial.

If the points y and ~ have the coordinates

respectively we get by observing that

Y "~ : ~ ( T - ~K ) ; IV E 2, ~ ) = 2 ,

the relation (for q = 2);

2

9' and F

,,'I ; co z : 2 ~

n"

~ ~c~-~) = ! ~(~(~-~)).

Theorem 2 therefore reduces to the addition formula for the function

cos ~ in the two-dimensional case, which explains why this result

is called the addition theorem of spherical harmonics.

REPRESENTATION THEOREM

As is well known, all the trigonometric functions can be

derived by simple algebraic processes from a single one (e.g.cosx),

the question arises if there is a corresponding result in the theory

- 12-

of general spherical harmonics. The addition theorem suggests that it

might be possible to express all spherical harmonics in terms of the

Legendre function. This is stated in

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

W ~ , ~ , , . . . . . . . . . . , W ~ such that every spherical harmonic

Sn(~) 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 to find

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. Therefore there is a

point ~ = ~z such that this determinant does not vanish.

Discussing next the determinant

,%.,~ C~] S.,~ (~)

and using the same arguments we obtain by induction

- 13-

Lemma 6 : There is a system of points W~, ~ . . . . , ~ N such that the

matrix (Sn, j( ~ k)), j = I,...,N; k = I,...,N is

non-degenerate.

From Theorem 2 we now have

~ 9

This is a non-degenerate system of linear equations with S n,j

as unknowns so that Theorem 3 follows by inversion.

In order to simplify the formulation fo these relations we

introduce

Definition ) : A system of N points ~ , . . - , ~ on A ~Z

a fundamental system of degree n, if

det I ~ ( ' ~ 4 ' ' ~ K ~ ~ 0 .

9 will be called

It can be seen readily that the matrix ~ ~ (~ ~) can

be obtained by multiplying the matrix Sn,j(~ i ) with its adjoint, so

that the determinant of Definition 3 is non-negative. If the deter-

minant is positive, the system has the properties stated in Theorem 3,

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

Theorem 4 : Every spherical harmonic of degree n may be represented

in the form

if the points ~ form a fundamental system of this degree.

It is clear now that an orthonormal system of spherical harmonics

can always be obtained by linear combinations of the functions

Pn (~k ' ~ ). Which fundamental system ~K is best suited to

represent the functions of degree n remains open at this stage

as it requires more information on the polynomials Pn(t).

- I#-

APPLICATIONS OF THE ADDITION THEOREM

Before studying the Legendre polynomials in detail, we shall

obtain several simple results on spherical harmonics in general

which depend on the addition theorem.

If we remember that every spherical harmonic of degree n can be

represented as

( 2 0 ) S , , (~) -- ~ , S , , , . (g) ~(=,4

we get immediately from Theorem 2

j a , , = I ~ ' . ( ' f ) S , , , K c 2 ) d ~ l �9 O, 9

Lemma 7 : For every spherical harmonic of degree n

("' 'I �9 0- 9

Here the letter ~ in connection with d ~ means that the

integration is carried out with respect to ~ �9

Observing that ,vO,,,)

K=4

we get f rom (20) , us ing Schwa~z's i n e q u a l i t y and Theorem 2,

K=4 ~="1 I (=4

This gives us

Lemma 8 : Let Sn(~) be a spherical harmonic of degree n. Then

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

Put ~ .} S,, ('~) = Ncq,.,, ~ (~. . ,?) -- S.6 (Y) S,,,~. (~/ ,

then we get from (21) and Theorem 2

cvq L

N (~,. I

Z Is.,, = I ,v ].

which gives us

- 15-

Lemma 9 : For - I z_ t _z 1

From Theorem 2 we have moreover

[ N (,~,,,,, ] z. z IV(q,.)

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

This gives by integration over El

( 2 2 ) = _-

As t h e v a l u e o f t h e i n t e g r a l on t h e l e f t hand s i d e d o e s n o t d e p e n d

on ~ , we may assume ~ to be ~I" Then, using the coordinate

r e p r e s e n t a t i o n ( 2 ) , we g e t

(23) "~ a t - ' - ' +4

It follows from (22) and (23)

-4

On the other hand , by Lemma 2,

0 f o r n # m ,

By t he c o o r d i n a t e r e p r e s e n t a t i o n (2) t h i s i s e q u i v a l e n t t o +4

/ P~(~) P. (~) (~ - t z) -- 0 f o r n # m, ~ t

-4

which gives us, combined with (24)

Lemma 10 :

" ~(t =

- 16-

RODRIGUES' FORMULA

We shall now give a representation of the Legendre polynomials

based on the following properties:

1. P. (~) is a polynomial of degree n in t.

2. I ?" (/''~ P,,, (~) (~- L~) q-3 z d~ : 0 for n # m. - 4

3- P~ (,t) = W .

The usual process of orthogonalization shows that Pn (t) is

determined up to a multiplicative constant by the first two

conditions. This constant can then be fixed by the third condition.

Consider the functions

( 2 5 } r -- �9

They are polynomials of degree n, and we see by partial integration

t h a t § 9_ ~

- 1

= (-4)" [(4-~') , (,l~,z) - 4

m ~- C q - 3 )

If n ~ m the right hand side vanishes, which proves that the functions

(25) satisfy the first two conditions.

Put t = I - s, then

so that we get

3- f l ~ e C9-3] = (-~I ~ [ r ~ ( ~ ) ~ /r

= [ _ 2 ) ~ F c ~ , ~ - q;~)

- 17-

Thus we get

Theorem 5 : (Rodrigues ' formula)

~ ctl ~ (~I ~ Fc ~

This has an immediate and simple application which we obtain

after integrating n times by parts. It is

Lemma 11 : Let f(t) be n times continuously differentiable, then

,I-4

-1

+~r . + ( q - 3 ~

1 - ~ " (~) o4t F-' (,,-, ~-~) _.,

As an immediate application of Lemma 11 we determine the leading

coefficient of the Legendre polynomial of order n. If c n is the

coefficient of the highest power in Pn(t), then

1 I '" -I -q

as the lower terms of the power series for Pn (t) do not contribute

to the integral. The left hand side of (26) is ~ �9 4 according ~q.~ N(r

to (24) and the right hand side equals

- 1 8 . -

§

= C. (~ r ( . + ~)

I

0

, +~-3~ _ ~ d~

P ( . �9 ~ )

Therefore

By (3)

SO that

n!

( 2 7 ) ~. { t ) = 4 N {q,-I

r ( , , ~ . } _z" t " + . . . . . r ( ~ ) , !

FUNK - HECKE FORMULA

Before going further into the details of the Legendre functions we

shall discuss a formula which will prove to be the basis of a

great many special results.

Let us consider an integral of the form

w h e r e f ( t ) i s a c o n t i n u o u s f u n c t i o n f o r - 1 ~- t _L 1 and t h e

integration is carried out with respect to ~ . Then with any

orthogonal matrix A

- 1 9 -

(28) F(A=,AD)

where A ~ i s t h e a d j o i n t ( t r a n s p o s e ) o f .&.. Now t h e s u r f a c e

elements ~ ~9 CA*R) and d~q~'~) are equal so that (28)

becomes

"~'t

This is equal to F( ~,p ) because we may regard A*~ as the new

variables. Using the same argument now which led to Lemma 5, we

see that F( ~,~ ) is a function of the scalar product only, which

gives us

Now a s a f u n c t i o n o f /3 t h i s i s a s p h e r i c a l h a r m o n i c o f d e g r e e n .

As i t d e p e n d s on t h e s c a l a r p r o d u c t o n l y , i t ha-~ t h e same s y m m e t r y

w h i c h c h a r a c t e r i z e s Pn( o< ./3 ) . T h e r e f o r e we g e t

I r ~ ( ~ ~ ) d ~ q ~ = ~ ~{~.D).

In order to determine ~ set ~ = ~ = ~ and

Then with

we get

q-3

~4

+4

- 20 -

This leads to

Lemma 12 : Let ~ and ~ be any two points in ~ , and

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

where

/'z,!

-*I

d~.

From Lemma 7 we now get by multiplication with Sn( ~ ) and

integration with regard to

Theorem 6 : (Funk-Hecke formula) Suppose f(t) is continuous for

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

degree n

with

n~

+4

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

- 21 -

INTEGRAL REPRESENTATIONS OF SPHERICAL HARMONICS

To distinguish clearly we will designate in the following a

spherical harmonic of order n in q dimensions with 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 harmonic polynomial of degree

for any continous function f(~_~), if we set

: "~'1 = ~ ~'1 + j ~ _ ~ i " , ~ _ ~

where

n

This enables us to get a new representation of the Legendre

polynomials. To this end we now prove the identity

,1 Cx. q +,ix. Tr,) d%_ = L,, cq,• 63 q_~

X~ q-1

As this integral represents the average over all directions which

are perpendicular to E 9 , the integral is symmetric with respect

to all orthogonal transformations which leave s fixed. For x = E~

the integral assumes the value one; hence the integral satisfies

Definition 2. We therefore have

Now

so that

and Theorem 6 with S o : I gives

- 2 2 -

Theorem 7 : (Laplace's representation)

?, )

- 4

ds.

Similarly we may get representations for further spherical harmonic

functions if we consider

For x = ~ this becomes a spherical harmonic of degree n in q

dimensions which we may represent in the form

A c c o r d i n g t o Hecke ' s f o r m u l a (Theorem 6) t h l s l s

+ 4 9_ 4

(29) 5~ (q-1,~,_,) ~q-z ( ~ +4 41/'~-~-~z .S "]~ (9-4, s) C4-S') dS - 4

wh ich can be w r i t t e n as

ASSOCIATED LEGENDRE FUNCTIONS

In order to get an explicit representation os a system of

orthonormal spherical harmonics we now introduce

Definition 4 : Suppose the points of ~ are represented in the form

Then, the function An, j(q,t ) is called an

associated Legendre function os degree n, order J,

- 23 -

and dimension q, if,

A, ,~ rq ,~ ) S~ ( ~ - ~ ; y q _ ~ , %~ o , ~ , . . . , -

is a spherical harmonic of degree n in q dimensions

for every spherical harmonic Sj(q-1, ~ q_1 ) of

degree J in q-1 dimensions.

The functions An,j(q,t ) will be called normalized if

As sperical harmonics of different degree are orthogonal we only

have to determine the factor of normalization for the case n = m.

The associated Legendre functions of order zero are readily

obtained, because Definition 4 means in this case that the

corresponding special harmonics have the symmetry properties

which determined Pn(q,t). Therefore An,o(q,t ) and Pn(q,t) are

proportional, and we have to determine the constant. This gives

A,,,,, (q,~,~ : F W('r''~ ~9- , P,, (30) (q,~)

From (29) it is obvious that

~ ~ -q

- 4

is an associated Legendre function of degree n, order j and

dimension q. From Lemma 11 we now see that apart from a multi-

plocative constant this is equal to

(~-~') )(~,~-~.s) ( 4 - s ~) z ds.

The integral is proportional to Pn_j(2J + q,t) as follows immediately

from Theorem 7, so that

( ,r- ~') ~,,_~ (2 i ,~, ~2

is an associated Legendre function of degree n and order j in

q dimensions.

Consider now the function

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

which is a polynomial of degree n-J. Integrating j times by

parts, we see that for n> m, j = 0,..., m; q ~- 3

+4

q,~.) (4- ~z) ~ d~

(3~) +~

-_ (_+)i I P-r+,~) (~)i l (+-~') " m'~'~+,+~] ~t

because the integrated terms vanish for t = I and t = -I. The

differentiated term is of the form

q-3

(32) ( , I - ( , ) ' p,. (~1

where Pm(t) is a polynomial of degree m, which is best seen by

u s i n g the f o r m u l a

+

with U = (I-t2) j+(q-3)/2 and V = Pm(J)(q,t).

We thus obtain from (31) for m ~ n

(33) I (~_ ~z) 2 =~@~ P .

On t h e o t h e r hand we h a v e f r o m Lemma 10 f o r m ~ n

(34) -4

As Pn(J)(q't) and P n - j ( q + 2 j ' t ) a r e bo th p o l y n o m i a l s o f deg ree n-J

which satisfy the same conditions of orthogonality, they may

be obtained by a process of orthogonalizatlon from the powers

t n with the weight function (~ ~) q-~,z~ - ~ over the

intervall -lmtgl. As they are not normalized they differ only

by a constant factor. This gives us

- 25 -

Lemma I~ : 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

~(~,~1 F (~) 2 ~ ~ ~# ~-~ (~? '~ ) : ~(2~q,~.~) ~ (~'~).

(J)(q t) are proportional, this last result As Pn_j(2j+q,t) and Pn

can be obtained by equating the coefficients of t n-j as

given by (27). This shows also that all Legendre polynomials can be

expressed either by Pn(3,t) or Pn(2,t) according to whether q is

odd or even.

The purpose of the preceding study, however, was not to find a

relation between Legendre polynomials of different dimensions

but to give an explicit representation of the normalized associated

Legendre functions An, j (q , t ).

Suppose now that the two unit vectors ~ and ~ are represented

in the form

"~ ~ s - ~ , I § 1/~-s z' ,/~_~

Then, if the function A are normalized n,j

is a complete and normalized system of spherical harmonics of

order n, as Sj,k( q-1 , ~_~ ) has this property in (q-l)

dimensions because of

- 2 6 -

V1

Thus we know tha t A n , j ( q , t ) i s p r o p o r t i o n a l to ( 1 - 8 2 ) j / 2 s or by Lenuna (13) to ( 1 - t 2 ) j / 2 P n ( J ) ( q , t ) . Now

++ t') "~ _ (2~+ t ) ] z L (4- "r,, ~ ~, (+- ~') d~

(35) ++ z i + t -

To f i n d the no rma l i z i ng f a c t o r f o r ( I - t 2 ) j / 2 P ( 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 + ~~ t 2 i * ~ -J p.~i;{,,+) {+_~-2)

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

I t l > 1 is i+ {~-3) .+~+ er_ ~

( - 4 ) z b., ~. (.-~)!

where we have set

Thus by (32) b.i/. { for the leading coefficient in (27).

q -

6. F' ( , + ~ + q - z ) "++-~ = (-4) �9 ~ + - . .

("-~J! F" ( ,,, + 9 - z )

Now with a constant c we get for Itl > I

- 27 -

.- ( . ~ - ~--~ ) ~ L C - ~ ) "i . c . t ".~ . . . . ]

(_.~) ~ b,, F ' [ . + 4 . + ~ - z ) / : " + ....

~.-$H P [ . . ~ q - z l

As Pn(t) is a polynomial of degree n, we have

/4

-_ C.~_e~] �9 [ C..O ~ b. . F'C~+~r+~-z) § . . . . . [""~}! P ( ~+ q- zl

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

("- ' i ) ! 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 (27) yields

Lemma 14 : + 4

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

_ ~_ .~[ {" C ~ * ~, + q - z ) 4

Thus from (35) and Lemma 14 we get

Lemma 15 : The functions

A .,~ cq,t) Y C-~- t ~) ~/~ ~ - i t2~, q, t~

or

- 2 8 -

A. , t (~,t) = / ~"" (.-i)! f ' ( . ,~-z) N(,l,.) (~_~,)i/~p~i)(~.t) . ! p( . . i+q-z)

form a system of normalized associated Legendre functions.

A representation of An,j(q,t ) in terms of P~J)(q,t) ( _ could have

been obtained from Lemma 13 but Lemma 14 is an interesting

formula itself. The reader may find it interesting to compare the

coefficient above with that obtained by using Lemma 13.

The addition theorem (Theorem 2) now can be written in the form

A " , i (* ' t ) A" ' i ('/'~) Si,~ ( ' - ' , ~',-,) Si,,~('t-~,'r/,-.) ~'=0 g= "I

N(el,.) "/~_, ] )

According to Theorem 2,

(]6) / v ( q - , , i )

Z 1,(=4

The addition theorem is usually given in the literature with

An,j(q,t) expressed in terms of the derivatives of Pn(q,t ) which

gives by Lemma 15

,

this may be written as

"I

(37) ~.= o

- 29 -

(38) t

PROPERTIES OF THE LEGENDRE FUNCTIONS

Multiplying (37) by PC (q-l, ~ q_1' ~ q_1) and integrating over

q-1 with respect to ~ q-1 , we get from 23 and Lemma ( 10 ) :

§ ~_~

~t,..~ ~,_~ I ~cq'~"* ~-"/ ' ;:;~~~ ~-~,~c~-~ '~ ~ ~v A.,e {q,~) A. ,e (q , s ) = ~ ~ q

From Lemma 15 we now get

Lemma 16:

Ig(2.C,q, n-g.) Wze+q_ ~

N ( ~ , . } wq_ z § , / - , r

" 4

In particular it follows for 4 = 0

' t,O ,~ _ 4

t.,,O c / _ 2

~4 q-#

~,, c , , t~, ,c~,~) = I ~',,~q, t ' ~ ~ ~-~zT"V)~ ~ ~ '~ - 4

We now prove

- 30 -

Lemma 17 : For 0 _z x < I and -I __4 t _x I,

o~ 1,1 4-- X z

(4 + x z - Z • 9/z

For q = 2 thls is a well-known identity which we can best obtain

by setting t = cos ~ . Then

Z ~ '"le~"9" 4 4 N (e , , ~ ) x "P, (2,t : ) = x - . + ~=o ~=_~ 4- x e ~ 4- xe-~

4

~-2 ~: ~ , _ 4 = d --.M z

4+x z - 2 ~ 9,

4_X z

4 + x z - 7. xf

We may assume for the following, therefore, that 9 a 3-

Using the Laplace representation (Theorem 7) of the Legendre

polynomials, we find for the left hand side

,4 9-~

(39) ~ - 2 N~q,~) x " ( ~ § ~ . s ( ~ - s 2) d s .

In Lemma 3, we had proved the identity

oo Z . 4+ W

(40) N(q,.) x = .--o (4- x) q-~

As I t + i ~ sl 2 : t2+(1-t2)s 2 ~ t2+(1-t 2) : I and hence

Ix(t+i~s)l is less than one we may write - under the

condition stated in Lemma 17 - the formula (39) as

+4 q_~

(41) ~ - ~ ( ~+ x ( ~ + ~ z ~ T ~ ' s ) (~-s'~ ~ ~s o~_~ ) l~- x ( ~ ~ ~r ~-"

- 31 -

To prove our Lemma, we thus have to show that this integral is

equal to the function given on the right hand side of Lemma 17"

In order to do this we introduce the substitution s = tanh u.

Using the abbreviations

( ~ 2 )

and observing

( 4 - s~l z ds = (t .o' ,d, . .J~- ' t d~

we obtain for the integral in (41)

I f f ( u ) s t a n d s f o r e i t h e r o f t h e t w o f u n c t i o n s d e f i n e d i n ( 4 2 ) ,

we have from f''(u) = +f(u) for any complex number u O

Now we introduce the real number ~ by

~r (4.4) x, , / , t -~ , " + ,~'(1-~) : " l i , ~+xZ-zx t " e o ~ , - ~ g

so that we can write f2(u as

(4s)

Apart from a numerical constant the integral (43) therefore equals

.~ q - r

x ~ ( r - l ~ t )

t~

I r (-.,,r) ~ f,~+,;4") * r .~,,.l,(~+q7

--DO

Here the integral reduces to

f ~ +oo

( 4 6 ) ~e (-id') [s4;,~ (.a+4d~)]'q'z [DV~A (~+4r) Jq-r o~

where the second term of this sum vanishes, because the integral

is zero. As ~ is greater than zero, the integral in (46) exists

for all q ~- 3. It may be regarded as a complex integral

( 4 7 ) C ~,,-#,, ,~1 ~ - z = ( s " ~ "~) q- ~ -eo + r - ~ + 4"ll'/l"

- 3 2 -

where this last identity is obtained by shifting the path of

integration to the llne Im(u) = ~ Combining these results and .

expressing the integral in (47) by use of the substitution

u = v + i ~ , we now see that

~- ~+]''+~ +"+"P~('i"+:) : ~ 4+" r ~+'-+ I ~v ~=o ( "~+xz-zx~ ' ) ~ ~'I-+ (c,<~,~v)t-z

--CO

From (42) and (44) we get

Z 4 - X

We have thus proved that

�9 _ ~Z

I'1 ~ C,

In order to determine the constant C we set t = I, and obtain from (40)

. = o ~ - ~ 1 q-+

As the right hand side of (48) reduces to this value for t = I,

we obtain C = I and have thus proved our identity.

Introducing

we have

@0

S., c,+,.++ : ~ - K = O

I'1=0

= 4 + Z x " ( s . , c t ~ j _ S , , -+c~ ,~=) ) 19=4

Where

o,o

= Z x ~ S,,(',~,e) l"l = 0

o o

(49) ~ x " ,5'. (0/, ~-) = I"l = O

o O

- • Z ,~",.%,~o/,+~

"t + ~(

( . I+ x z - 2 • E J r "z

Set for n = 0,1,..., and q -~ 3,

M

I"(,+§ p(n++). I-'(q-z)

- 33 -

so that o o

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

. : o C 4 + xz- ~>c~) ~

with

c . t 9) = i ' ~ ( . + q - z ) P(q-z). F'(.+4)

The corresponding result for q = 2 is

. "p. (~ ~:) = 4- ,8, ( 4 + ~ z _ ~ ) t Z

~'I=4

which is well known and can be proved immediately by using

Pn (2,c~ T ) = cosn T "

The proof of Lemma 18 is quite analogous to the proof of Lemma 17

so that we can use the same notations. Laplace's representation

of Pn(q,t) gives

+4

y_~ c~ I (4-sz} '~4 als .=o %-~ (4 - x ( ~ I/TZ-P'.S)) ~-z

- ' I

The substitution s = tanh u and the abbreviations (42), (44), (45)

transform the integral to

I " I - 4 - 0 0

so that Lemma 18 may be proved by the same arguments that led to

Lemma 17.

4) This estabillshes the relation Cn(q)Pn (q't):C(q-2)/2(t)n where

CV(t) a r e t h e G e g e n b a u e r f u n c t i o n s . n

- 34 -

Suppose now that x and y are any two vectors in q-dimensional

space, with

x = R.~" , y = ~-.~ ; l } ' l = ,I , I ~ I = 4

Then for q ~ 3 and R> r

I~->'I ~-z (R z + ~ - z R.~" ~ ' .~ t )~ -~

4 4

This can be expressed by Lemma 18 so that we obtain

Lemma 19 : If x = R~ ; y = ' r . ' ~ and R > r , then

,,<-• = R Z c ~ c ~ ( ~c~,~.~).

Let xi,Ylbe the Cartesian components of x and y. Then

(50) i • 2-~ = [ ( x , - Z ~ + C ~ - ~ , . . . . , ( x ~ - ~ # ] Y

According to the Taylor expansion in several variables, this can

be written as

(51) oo 2-q

I ' I=0 (~

If ~7 x denotes as usual the vector operator with the components

we have

y~ -~ ~ . . . . . + ~ (~ V ~ ) ~X~ )'~ : ~ '

and we get from (50) and (51) for IYl < Ix~

z - ~ ~o 2_ 9 x l = >- VI=O

Comparing this with Lemma 19 we have by equating the coefficients n

of q~ ,

- 3 5 -

This gives with the explicit value of Cn(q) and R = I xl

Lemma 20 : (Maxwell's representation)

~ ~ 1• = (_.f)', F ( ~ q - ~ P , , ~ : q , ~ ' - ~ ) /'~(9-1) iXl~,*q-z

As I x l 2-q is the fundamental solution of the Laplace equation in

q dimensions, this shows that the Legendre polynomials may be

obtained by repeated differentiations of the fundamental solution

in the direction of the vector ~ �9 The potential on the right

hand side of Lemma 20 may thus be regarded as the potential of

a pole of order n with the axis ~ at the origin.

We know that every spherical harmonic can be expressed in the form

I ( _.- , I

with a fundamental system ~ k"

write

Therefore it is always possible to

which shows that every potential of this type may be regarded as

the potential of a combination of multipoles with real axis. The

system of fundamental points introduced earlier thus corresponds

to a fundamental system of multlpoles in Maxwell's interpretation

off the spherical harmonics.

A rather striking interpretation of Lemma20 is obtained in the

following way. We first observe that

with

- 3 6 -

- 1

2, ~b

P (., ~)

where Hn(q,x ) = rnSn(q, ~ ), which enables us to express the formal

polynomial Hn(q,~7~) as

q

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

gration over i-~ now gives

Lemma 21 : For every harmonic polynomial of degree n

r ( ~ ) i~/~"'~-2

Before leaving the special properties of the spherical harmonics

it should be noted that many more can be derived from Lemmas 18

to 21 of which the recursion formulas for the Legendre polynomials�9

the associated functions, and their derivatives are perhaps best

known. They can be obtained by differentiating the identity

formulated in Lemma 18 with respect to x or t and equating n

coefficients of x .

As an example we take the formula

(52) h

{~-z ) . Z N(~,~) ~(q,~) -- c.(q) P.'(q,~). c~.(q)P.'.~(q,~). K=O

From the Laplace representation (Theorem 7) we get

i q_q ).-s ~" s) O - s ' ) z ds, P,' ( q , ~ ) = ~ ~ - ~ , ( ~ , ~ ~ T : - ~ . s O - ~ . V ~ ; - ~

which shows that for all t with I tl -~ to< 1 P'(q�9 satisfies �9 n

1~. ' c ~ , ~ ) = ( ~ ( . )

- 3 7 -

uniformly. It is therefore permitted to differentiate the power

series of Lemma 18 termwise. We obtain

7 c , , ( , ~ • P,, 'cq,~) = C, t -2 ) (-,,-.+,-2~.~)~,',~ PI:O

which gives us

rl=O n

Comparing this result with (49) and equating coefficients of x

we get (52). This becomes particularly simple for q = 3, as is

true of many more of these results. In this case we get

3 - -- +- . K=O

DIFFERENTIAL E~UATIONS

The basic concept and the starting point of our approach

to the theory of spherical harmonics is the harmonic and homogeneous

polynomial. Only very indirectly we made use of the fact that the

spherical harmonics are connected with the Laplace equation. We

shall now derive results which express this factor in terms of

special differential equations for the spherical harmonics.

In order to do this we have to express the ~ -operator in

terms of the polar coordinates which we have been using. We wrote

(53)

where ~9-~

s . . . . . . . . . s representat ion s e t

is a unit vector spanned by the unit vectors

Suppose now that we have some coordinate

. ....... v~_, of ~-~q-4 �9 We then

~I = ~ ; ~q-1 = ~ ," ~ -- v~ for i-- ~,....,,t-2

so that ~9 is a function of t and v~, ........ v~_~ , or in the

above notation of ~,,. .... , ~-I . With the abbreviation

- 3 8 -

~};,< _- a__&. ~ . ~ = ~ ~,~..,,~ ; ~, ~;;~ = , ~-,,~ _- ~, ~,. . , ~_ .~.

we may form the Beltrami Operator for ~

From (53) it is clear that for i,k = 1,2...,q-I,

_

9,a,i, ~},a/ #~.~

__ a.._x, a x o a x ~ = . / 8x . a x - " r z ~ . , . , ; - - = i

and we obtain by means of the tensor calculus

We had

~ = { - ~ § ~ �9 ~ _ ~

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

d'a,l,~" ~'~q-4 = , I - t: z ; 9 , ~ . ,

This gives us

a z ( 5 4 ) Lk I = ( , f _ ~ = ) c ~ - ~ ) ~ a § ,I a t z a-~ I - ~.z A~-4

It should be noted that for

w e get

_ a 2

- 39 -

We can thus define the operators ~

the two -dimensional case.

successively, starting with

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 , ~ ) = o .

For the Legendre polynomials we thus get from (54)

Lemma 23 : The Legendre polynomial Pn(q,t) satisfies

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

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, may be started from these

differential equations, as has been done previously (see Hobson,

Spherical Harmonics). However, if the condition is imposed that

the harmonic functions thus obtained should be entire and nni-

valued, the theory reduces to the functions discussed here, which

are therefore called the regular spherical harmonics.

- 40 -

EXPANSIONS IN SPHERICAL HARMONICS

We shall now prove that the spherical harmonics form a complete

and closed set of functions on the sphere. This, of course, may

be regarded as an extension of the theory of Fourier series to

the case of problems with spherical symmetry in any number of

dimensions.

Due to the orthogonality of the Legendre polynomials we have from

Lemma 17 (multiply by PO and integrate)

for all x with 0 ~ x < I. We shall now prove

Lemma 25 : Suppose f(t) is continuous for -I m t m I. Then

+ I

x . . , 4 - o ( 4 ~ ~ z 2xd:) r ~f~. = ~P(41. . ~ - ~ l - , I

We write

where g(1) = O. If f(t) is constant the result follows from (55)

immediately, so that Lemma 25 is proved if we show 4.4

) { q + ~ z - Z ~< 4: ) ~I I~ ) r

The c o n t i n u i t y o f g ( t ) i m p l i e s t h a t t h e r e i s a p o s i t i v e f u n c t i o n

m(s) wi th

S ...e O

such that

4 ~'t I; "~/ "1- $

Moreover, it follows from the continuity that there is a constant C

with I ~I ~ C for -~ -~ g 4.

We now observe that for -1 g t g 1 - s and x -~ 0

4 ~- • Z ~ 6 = [ 4 - x ) z + 2 x { 4 - ~ ) >/ 2 x . S

I so that for the same range of t and ~x< I

1 - X z 4 - x z 4+2( . ~ (~ X i . z . . E ) ~

- #I -

We now des s by

(56) s ~/~ = - / ~ - x '

and divide the interval os integration into -1 ~- t ~- 1 - s and

I - s _z t -~ I . Then for x ~- 0

(57) I (~-'('~ ~(~) ( ~ - F ) ~-~ d~ _~ 2C ~- , ( ) ' /~ ,a~ = ~'(~W;:-;-~) (,~§ ~z _ Z x t ) ~1~ ~ - ~ - I

and

l i ; I (4+ ~ - Z x t ) ~lz =

as this last integral may be majorized by

(4 �9 ~z _ Z ~ t) q/~

According to (55), s tends towards zero for x~ I - O, so that

our Lemma follows from (57) and (58) with (55).

We are now able to prove the following theorem:

Theorem 8 : (Poisson's integral) Suppose F( y ) is continuous on A~q

Then

"l I ' ( ~ - 7 z) F ( ~ ) d ~ , l ~ = F ( ~ ) -r~,1-o ~ , ( 4 + r z - z r ~ . ~ / ) ~/~

where thls limit holds uniformly with regard to ~ .

As ,D,~ is compact, we can deduce from the continuity of F( ~ )

the existence of a positive function m(s) such that

(59)

We now assume ~ = s

#(~) =

so that

Ir =

From (59) follows

and define

~.z? ~?q_, ) ~-1 ,9,?-I

~',/) ,

( 6 0 ) I ~(4) - ~ ( t ) l L_ w ' l - ~ " r n ( s ~

- #2 -

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

(~ -~

SO t h a t we g e t f o r ~ = 8~

�9 (,Jm ea~ { 4 , " , -~ - z ' , ' . r . ~ ) ~/z 7"--" 4 - 0

~ 4

~--~ 4- o ( ~ ( q + ~ - Z ~ ) ~'/z -,,f

As any point of the sphere ~ may be chosen as E 9 of an

appropriately chosen system of coordinates, this argument holds

for all ~ of ~ . Moreover, the estimate (60) only involves the

uniformly valld estimate (59) so that the limtis are approached

uniformly.

From the identity

oo

(61) Z r = 4 - , ' .=, (~+ T ~-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),~'/ = Ncq,.~ ~. ~ , ~ . ? ) F ( ? ) ~(~)9c-~) = c.,~ S. ,~(~,} ' )

where

- 4 3 -

c . , ~ = I

This result is an immmediate consequence of the identity (61)

which holds uniformly with respect to ~ and W for 0 g r < I.

We may therefore integrate termwise and obtain the last

representation of the spherical harmonics from the addition

theorem.

Using the same notation, we get from Parseval's inequality

where we used the abbreviation

~'- I c . , ~ I ~

Set

so that

o o

oo N (~ ,~ ) oo

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

(

) I F ( ' r , , ' ~ ) 12 ' r .~, . t - o ,D, ~/ ~ q

as F(r, ~ ) approximates F( ~ ) uniformly. We therefore have

On the left hand side we may interchange the limit and the

summation because of (62).

Theorem 10 : For every continuous function F(~ )

I F ( ~ J l z d~,ir_, ?) -- Z ( c . ) z

A n o t h e r c o n c l u s i o n may be d r a w n f r o m T h e o r e m 9 ,

Theorem 11 : If the continuous function F( ~ ) satisfies

for all spherical harmonics, it vanishes identically.

- 44 -

Our assumption has the consequence that F(r, ~ ) vanishes for

all r ~ 1. Therefore

T ~ 4 - O

which proves Theorem 11.

These last results show that the system of spherical harmonics

has the basic property of being complete and closed for the

continuous functions on A"I~ . Extensions of these results

to more general classes of functions may be obtained by methods

of the theory of approximations.

- 45 -

BIBLIOGRAPHY

The following are either solely devoted to the subject of

spherical harmonics or contain detailed information on this

subject�9

E * , , �9 rdelyi A. W.Magnus, F 0berhettinger, and F. Tricomi, Higher

transcendental functions, Vol. I ~id 2, New York, 1953.

Hobson, E.W. The theory of spherical and ellipsoidal harmonics~

Cambridge, 1931.

Lense, J. Ku~elfunktionen~ Leipzig 1950.

Magnus, W. and F. 0berhettinger, Formulas and theorems for the

functions of mathematical physics, New York, 1954.

MUller, C., Grundprobleme der mathematischen Theorie elektro-

magnetischer Schwingungen, Berlin, Heidelberg,

GSttingen, 1957.

Morse, P. M., and H. Feshbach, Methods of theoretical physics,

Vol. I and 2, New York, 1953.

Sansone, G. Orthogonal functions t New York, 1959.

Webster, A.G. - SzegS, G. Partielle Differential~leichungen der

mathematischen Physik~ Leipzig, Berlin, 1930.

spherical harmonics

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