Page 83 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

SERIES SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS 73
We shall see later that on separating variables in the case where u is a function of r,θ, ,
and t, we find the following differential equation in the µ variable:

d df m 2
2
(1 − µ ) + n(n + 1) − f = 0 (4.153)
dµ dµ 1 − µ 2
m
We state without proof that the solution is the associated Legendre function P (µ). The
n
associated Legendre polynomial is given by
d m
2 1/2m
m
P = (1 − µ ) P n (µ) (4.154)
n
dµ m
The orthogonality condition is
1
2(n + m)!

m 2
[P (µ)] dµ = (2n + 1)(n − m)! (4.155)
n
−1
and
1

m
m
P P dµ = 0 n = n (4.156)
n n
−1
The associated Legendre function of the second kind is singular at x =±1and maybe
computed by the formula

m
d Q n (x)
m
2 m/2
Q (x) = (1 − x ) (4.157)
n
dx m
Problems
1. Find and carefully plot P 6 and P 7 .
2. Perform the integral above and show that
x
dξ C 1 + x

Q 0 (x) = CP 0 (x) = ln
2
(1 − ξ )P 0 (ξ) 2 1 − x
ξ=0
and that
x
dξ Cx 1 + x

Q 1 (x) = Cx = ln − 1
2
2
ξ (1 − ξ ) 2 1 − x
ξ=0
0
1
3. Using the equation above find Q (x)and Q (x)
0 1