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SERIES SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS 69
4.3 LEGENDRE FUNCTIONS
We now consider another second-order linear differential that is common for problems involv-
ing the Laplacian in spherical coordinates. It is called Legendre’s equation,

2

(1 − x )u − 2xu + ku = 0 (4.132)
This is clearly a Sturm–Liouville equation and we will seek a series solution near the origin,
which is a regular point. We therefore assume a solution in the form of (4.3).

∞
j
u = c j x (4.133)
j=0

Differentiating (4.133) and substituting into (4.132) we find

∞
j−2 2 j j
[ j( j − 1)c j x (1 − x ) − 2 jc j x + n(n + 1)c j x ] (4.134)
j=0

or

∞
j j−2
{[k − j( j + 1)]c j x + j( j − 1)c j x }= 0 (4.135)
j=0

On shifting the last term,

∞
j
{( j + 2)( j + 1)c j+2 + [k − j( j + 1)]c j }x = 0 (4.136)
j=0

The recurrence relation is

j( j + 1) − k
c j+2 =− c j (4.137)
( j + 1)( j + 2)

There are thus two independent Frobenius series. It can be shown that they both diverge at
x = 1 unless they terminate at some point. It is easy to see from (4.137) that they do in fact
terminate if k = n(n + 1).
Since n and j are integers it follows that c n+2 = 0 and consequently c n+4 , c n+6 , etc. are
all zero. Therefore the solutions, which depend on n (i.e., the eigenfunctions) are polynomials,
series that terminate at j = n. For example, if n = 0, c 2 = 0 and the solution is a constant. If