Explanation Definition used: Fourier series with period 2 L : The Fourier series of the function with period 2 L is given as: f ( x ) = a 0 + a 1 cos ( π x L ) + a 2 cos ( 2 π x L ) + ⋅ ⋅ ⋅ + a n cos ( n π x L ) + ⋅ ⋅ ⋅ + b 1 sin ( π x L ) + b 2 sin ( 2 π x L ) + ⋅ ⋅ ⋅ + b n sin ( n π x L ) + ⋅ ⋅ ⋅ The n th coefficients are given as a 0 = 1 2 L ∫ − L L f ( x ) d x a n = 1 L ∫ − L L f ( x ) cos ( n π x L ) d x b n = 1 L ∫ − L L f ( x ) sin ( n π x L ) d x Result used: For an even function, the Fourier series contains only cosine terms and possibly constant term. For an odd function, the Fourier series contains only sine terms and no constant term. Calculation: It is given that the periodic force function is f ( x ) = x 2 + 2 , − 1 < x < 1 . Obtain the value of a 0 as follows. a 0 = 1 2 ∫ − 1 1 ( x 2 + 2 ) d x = 1 2 [ x 3 3 + 2 x ] − 1 1 = 1 2 [ ( 1 ) 3 3 + 2 ( 1 ) − ( − 1 ) 3 3 − 2 ( − 1 ) ] = 1 2 [ 1 3 + 2 + 1 3 + 2 ] That is, a 0 = 7 3 . Obtain the value of a 1 as follows

11th Edition

ISBN: 9780134437705

Author: Washington

Publisher: PEARSON

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Chapter 30.7, Problem 3PE

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To find: The Fourier series of the given function.

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Chapter 30.7, Problem 2PE

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Chapter 30 Solutions

Basic Technical Mathematics

Ch. 30.1 - Prob. 1PE Ch. 30.1 - Prob. 2PE Ch. 30.1 - Prob. 1E Ch. 30.1 - Prob. 2E Ch. 30.1 - Prob. 3E Ch. 30.1 - Prob. 4E Ch. 30.1 - Prob. 5E Ch. 30.1 - Prob. 6E Ch. 30.1 - Prob. 7E Ch. 30.1 - Prob. 8E

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