Mathematical Methods for Physicists: A concise introduction - Site Map

SOLUTIONS OF LAPLACE'S EQUATION
the series diverges for jxj ˆ1. A solution which converges for all x can be
obtained if either the even or odd series is terminated at the term in x j . This
may be done by setting equal to
ˆ…j ‡ m†… j ‡ m ‡ 1† ˆl…l ‡ 1†:
On substituting this into Eq. (10.32a), the resulting equation is
" #
…1 x 2 † d2 P dP
m2
2x ‡ l…l ‡ 1†
2
dx dx 1 x 2 P ˆ 0;
which is identical to Eq. (7.25). Special solutions were studied there: they were
written in the form P m l …x† and are known as the associated Legendre functions of
the ®rst kind of degree l and order m, where l and m, take on the values
l ˆ 0; 1; 2; ...; and m ˆ 0; 1; 2; ...; l. The general solution of Eq. (10.32) for
m 0 is therefore
P…x† ˆ…† ˆa l P m l …x†:
…10:33†
The second solution of Eq. (10.32) is given by the associated Legendre function of
the second kind of degree l and order m: Q m l …x†. However, only the associated
Legendre function of the ®rst kind remains ®nite over the range 1 x 1 (or
0 2†.
Equation (10.31) for R…r† becomes
When l 6ˆ 0, its solution is
d
dr
r 2 dR
dr
l…l ‡ 1†R ˆ 0:
…10:31a†
R…r† ˆb…l†r l ‡ b 0 …l†r l1 ;
…10:34†
and when l ˆ 0, its solution is
R…r† ˆcr 1 ‡ d:
The solution of Eq. (10.30) is
(
ˆ f …m†eim' ‡ f 0 …l†e im' ; m 6ˆ 0; positive integer;
g; m ˆ 0:
The solution of Laplace's equation (10.28) is therefore given by
8
>< ‰br l ‡ b 0 r l1 ŠP m l …cos †‰ fe im' ‡ f 0 e im' Š; l 6ˆ 0; m 6ˆ 0;
…r;;'†ˆ ‰br l ‡ b 0 r l1 ŠP l …cos †; l 6ˆ 0; m ˆ 0;
>:
‰cr 1 ‡ dŠP 0 …cos †; l ˆ 0; m ˆ 0;
…10:35†
…10:36†
…10:37†
where P l ˆ P 0 l .
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