Page 36 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 2: Legendre’s Polynomials

Notes
2. Recurrence relations derived in this unit help us in finding unknown P n ( ) in terms of
x
two or three known Legendre polynomial
x
x
 The Legendre’s polynomials P n ( ) have zeroes at some x x i ,i 1, 2, ... i.e. P 2 ( )
has two zeroes, P 3 ( ) has three and so on.
x
 Legendre polynomials are quite suited in numerical evaluations of certain integrals.
2.1 Legendre’s Differential Equation from Laplace’s Equation

Laplace’s equation in spherical polar coordinates is

2 V 1 V 1 2 V
r sin 2 2 0 ...(A)
r r sin sin
Let us put

n
V = r F n ( , ) ...(B)
Here F n ( , ) is a function of and . So

V
= n r n 1 F
r n
V n F n
= r

2 2
V n F n
2 = r 2
Substituting in Laplace equation, we get

1 F r n 2 F
n r n 1 F n r n sin n n = 0
r sin sin 2 2

r n F n r n F n
n
or ( n n 1)r F n sin 2 2 = 0
sin sin
n
Dividing by r , we have
1 F 1 2 F
( n n 1)F n sin n n = 0 ...(C)
sin sin 2 2

Next consider the case when F n ( , ) is independent of , so

1 F
( n n 1) F n sin n = 0 ...(D)
sin
Let us put the independent variable in terms of x given by
x = cos

d F n x F n
F n = sin
d x x

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