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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 ,. ~:~
% %/
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 , , , ) ~