Explanation Given: From problem 17 the values are as follows: P 0 ( cos θ ) = 1 P 1 ( cos θ ) = cos θ P 2 ( cos θ ) = 1 4 ( 3 cos 2 θ + 1 ) Calculation: The Fourier-Legendre series of a function F ( θ ) defined as follows: F ( θ ) = ∑ n = 0 ∞ c n P n ( cos θ ) ...... ( 1 ) Here, c n is defined as follows: c n = 2 n + 1 2 ∫ 0 π F ( θ ) P n ( cos θ ) sin θ d θ ...... ( 2 ) Substitute 0 for n in equation (2) and solve. c 0 = 2 ( 0 ) + 1 2 ∫ 0 π F ( θ ) P 0 ( cos θ ) sin θ d θ = 1 2 ∫ 0 π ( 1 − cos 2 θ ) ⋅ 1 ⋅ sin θ d θ ( P 0 ( cos θ ) = 1 ) = 1 2 ∫ 0 π ( 1 − cos 2 θ + 1 ) sin θ d θ Further solve the above equation. c 0 = 1 2 ∫ 0 π 2 sin θ d θ − ∫ 0 π cos 2 θ sin θ d θ = [ − cos θ ] 0 π + [ 1 3 cos 3 θ ] 0 π = 2 − 2 3 = 4 3 Substitute 1 for n in equation (2) and solve

WEB ASSIGN FOR ZILL'S DIFFERENTIAL EQUAT

9th Edition

ISBN: 9780357539545

Author: ZILL

Publisher: CENGAGE L

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Chapter 11.5, Problem 18E

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The Fourier-Legendre expansion of the function F(θ)=1−cos2θ.

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Chapter 11.5, Problem 17E

Chapter 11.5, Problem 19E

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Chapter 11 Solutions

WEB ASSIGN FOR ZILL'S DIFFERENTIAL EQUAT

Ch. 11.1 - In problem 16 show that the given functions are...Ch. 11.1 - In problem 16 show that the given functions are...Ch. 11.1 - In problem 16 show that the given functions are...Ch. 11.1 - In problem 16 show that the given functions are...Ch. 11.1 - In problem 16 show that the given functions are...Ch. 11.1 - Prob. 6E Ch. 11.1 - Prob. 7E Ch. 11.1 - In Problems 712 show that the given set of...Ch. 11.1 - In Problems 712 show that the given set of...Ch. 11.1 - In Problems 712 show that the given set of...

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The Fourier-Legendre expansion of the function F ( θ ) = 1 − cos 2 θ .

Solution Summary: The author explains the Fourier-Legendre expansion of a function F(theta )=1-mathrmcos2tha .

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The Fourier-Legendre expansion of the function F(θ)=1−cos2θ.

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