Explanation Definition used: Fourier series with period 2 π : An infinite sum of sine and cosine waves with period 2 π is, f ( x ) = { a 0 + a 1 cos x + a 2 cos 2 x + ⋅ ⋅ ⋅ + a n cos n x + ⋅ ⋅ ⋅ + b 1 sin x + b 2 sin 2 x + ⋅ ⋅ ⋅ + b n sin n x + ⋅ ⋅ ⋅ } . Here, the values of a 0 = 1 2 π ∫ − π π f ( x ) d x , a n = 1 π ∫ − π π f ( x ) cos n x d x and b n = 1 π ∫ − π π f ( x ) sin n x d x . Calculation: It is given that the function is f ( x ) = { − 1 − π ≤ x < 0 1 0 ≤ x < π in which –1 is replaced by 0. Thus, the function becomes f ( x ) = { 0 − π ≤ x < 0 1 0 ≤ x < π . Obtain the value of a 0 as follows. a 0 = 1 2 π ∫ − π 0 ( 0 ) d x + 1 2 π ∫ 0 π ( 1 ) d x = 0 + 1 2 π [ x ] 0 π = 1 2 π [ ( π ) − ( 0 ) ] = 1 2 π [ π ] That is, a 0 = 1 2 . Obtain the value of a n as follows. a n = 1 π ∫ − π 0 ( 0 ) cos n x d x + 1 π ∫ 0 π ( 1 ) cos n x d x = 0 + 2 π [ sin n x ] 0 π = 0 + 0 = 0 That is, a n = 0 . Obtain the value of b 1 as follows. b 1 = 1 π ∫ − π 0 ( 0 ) sin x d x + 1 π ∫ 0 π ( 1 ) sin x d x = 0 + 1 π [ − cos x ] 0 π = − 1 π [ cos ( π ) − cos 0 ] = − 1 π [ ( − 1 ) − 1 ] That is, b 1 = − 1 π [ − 2 ] = 2 π Obtain the value of b 2 as follows

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 30.6, Problem 2E

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To find: The Fourier series of the function.

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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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