Mathematical Methods for Physicists: A concise introduction - Site Map

SOLUTIONS OF LAPLACE'S EQUATION
Again, assume that
…r;;'†ˆR…r†…†…'†:
…10:29†
Substituting into Eq. (10.28) and dividing by we obtain
sin 2
d
r 2 dR
‡ sin
d d
sin ˆ 1 d 2
R dr dr d d d' 2 :
For a solution, both sides of this last equation must be equal to a constant, say
m 2 . Then we have two equations
d 2
d' 2 ‡ m2 ˆ 0;
…10:30†
sin 2
R
d
dr
r 2 dR
dr
‡ sin
d
d
d
sin
d
ˆ m 2 ;
the last equation can be rewritten as
1 d d
sin
m2
sin d d sin 2 ˆ1 d
r 2 dR
:
R dr dr
Again, both sides of the last equation must be equal to a constant, say . This
yields two equations
1 d
r 2 dR
ˆ ;
…10:31†
R dr dr
1 d d
sin
m2
sin d d sin 2 ˆ:
By a simple substitution: x ˆ cos , we can put the last equation in a more familiar
form:
!
d
dx …1 x2 † dP ‡ m2
dx 1 x 2 P ˆ 0
…10:32†
or
" #
…1 x 2 † d2 P dP
2x
2
dx dx ‡ m2
1 x 2 P ˆ 0;
…10:32a†
where we have set P…x† ˆ…†.
You may have already noticed that Eq. (10.32) is very similar to Eq. (10.25),
the associated Legendre equation. Let us take a close look at this resemblance.
In Eq. (10.32), the points x ˆ1 are regular singular points of the equation. Let
us ®rst study the behavior of the solution near point x ˆ 1; it is convenient to
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