Page 35 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation Richa Nandra, Lovely Professional University

Notes Unit 2: Legendre’s Polynomials

CONTENTS
Objectives

Introduction
2.1 Legendre’s Differential Equation from Laplace’s Equation

2.1.1 Power Series Solution of Legendre’s Equation in Ascending Powers of x
2.1.2 Solution of Legendre’s Equation in Descending Powers of x.
2.2 Rodrigue’s Formula for Legendre Polynomials

2.3 Generating Function for Legendre Polynomials
2.4 Recurrence Relations for Legendre Polynomials

2.5 Orthogonal Properties of Legendre Polynomials
2.6 Expansion of a f(x) in terms of Legendre’s Polynomials
2.7 Summary

2.8 Keywords
2.9 Review Questions

2.10 Further Readings

Objectives

After studying this unit, you should be able to:
 Observe that Legendre’s differential equation is obtained from the Laplace differential
equation
n
 Obtain the Legendre’s polynomial P (x) as a power series having x as a maximum
n
powerterm for n > 0 integer
 See recurrence relations of P (x) help in finding all P (x) in terms of two or three lower
n n
P (x).
n
 See that a generating function is found by which various P (x) are found.
n
 See that orthogonal properties of P (x) help in expressing any function f(x) in terms of
n
various P (x).
n
Introduction

x
The Legendre’s polynomials P n ( ) play an important role in potential problems i.e. in
x
electrostatics and gravitational field. It is therefore important to study the properties of P n ( ) .
1. First of it is important to study the solution of Legendre’s equations so that more insight
x
to P n ( ) can be seen.

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