Explanation Definition used: Fourier series with period 2 L : The Fourier series of a function with period 2 L is obtained 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 obtained 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 function is f ( x ) = { 0 − 2 ≤ x < 0 , 1 ≤ x < 2 1 0 ≤ x < 1 . This function is neither even nor odd; hence all the coefficients are to be calculated. The period of the function is 2 L = 4 ⇒ L = 2 . Obtain the value of a 0 as follows. a 0 = 1 2 ( 2 ) ∫ − 2 0 0 d x + 1 2 ( 2 ) ∫ 0 1 d x + 1 2 ( 2 ) ∫ 1 2 0 d x = 1 4 [ x ] 0 1 = 1 4 [ 1 − 0 ] = 1 4 Obtain the value of a n as follows. a n = 1 ( 2 ) ∫ − 2 0 0 cos n π x 2 d x + 1 ( 2 ) ∫ 0 1 cos n π x 2 d x + 1 ( 2 ) ∫ 1 2 0 cos n π x 2 d x = 1 2 [ 2 n π sin n π x 2 ] 0 1 = 1 n π [ sin n π 2 − 0 ] = 1 n π sin n π 2 However, for all even values of n , the values of a n are 0. Thus, for n = 1 , the value of b 1 becomes 1 ( 1 ) π sin ( 1 ) π 2 = 1 π ; for n = 3 , the value of b 3 becomes 1 ( 3 ) π sin ( 3 ) π 2 = − 1 3 π

Pearson eText for Basic Technical Mathematics with Calculus -- Instant Access (Pearson+)

11th Edition

ISBN: 9780137554843

Author: Allyn Washington, Richard Evans

Publisher: PEARSON+

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Chapter 30.7, Problem 20E

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To find: The Fourier series of the given function.

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Chapter 30.7, Problem 19E

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Pearson eText for Basic Technical Mathematics with Calculus -- Instant Access (Pearson+)

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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