Explanation Approach: The Legendre polynomial P n ( x ) is given by, P n ( x ) = 2 − n ∑ m = 0 [ n / 2 ] ( − 1 ) m ( 2 n − 2 m ) ! ( n − m ) ! m ! ( n − 2 m ) ! x n − 2 m For n ≥ 0 . Calculation: When n is even, n = 2 k Here, k ≥ 0 The Legendre polynomial can be written as, P n ( x ) = 2 − 2 k ∑ m = 0 k ( − 1 ) m ( 4 k − 2 m ) ! ( 2 k − m ) ! m ! ( 2 k − 2 m ) ! x 2 ( k − m ) It can be observed from the above equation that p n ( x ) contains all even degrees of x

Fundamentals of Differential Equations and Boundary Value Problems

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ISBN: 9780321977106

Author: Nagle, R. Kent

Publisher: Pearson Education, Limited

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Chapter 8.8, Problem 30E

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The Legendre polynomials of even degree are even functions of x, while those of odd degree are odd functions.

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Chapter 8.8, Problem 29E

Chapter 8.8, Problem 31E

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Fundamentals of Differential Equations and Boundary Value Problems

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The Legendre polynomials of even degree are even functions of x , while those of odd degree are odd functions.

Solution Summary: The author explains that the Legendre polynomials of even degree are even functions of x, while those of odd degrees are odd functions.

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The Legendre polynomials of even degree are even functions of x, while those of odd degree are odd functions.

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Chapter 8 Solutions

The Legendre polynomials of even degree are even functions of x , while those of odd degree are odd functions.

Solution Summary: The author explains that the Legendre polynomials of even degree are even functions of x, while those of odd degrees are odd functions.

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To show: The Legendre polynomials of even degree are even functions of x, while those of odd degree are odd functions.

Want to see the full answer?

Chapter 8 Solutions

To show:

The Legendre polynomials of even degree are even functions of x, while those of odd degree are odd functions.