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 ) = { 0 − π ≤ x < 0 x 2 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 ( x 2 ) d x = 0 + 1 2 π [ x 3 3 ] 0 π = 1 6 π [ ( π ) 3 − ( 0 ) 3 ] = 1 6 [ π 2 ] That is, a 0 = 1 6 π 2 . Obtain the value of a 1 as follows. a 1 = 1 π ∫ − π 0 ( 0 ) cos x d x + 1 π ∫ 0 π ( x 2 ) cos x d x = 0 + 1 π [ x 2 sin x + 2 x cos x − 2 sin x ] 0 π = 1 π [ 2 π cos π ] = − 2 Obtain the value of a 2 as follows. a 2 = 1 π ∫ − π 0 ( 0 ) cos 2 x d x + 1 π ∫ 0 π ( x 2 ) cos 2 x d x = 0 + 1 π [ 1 2 x 2 sin 2 x + 1 2 x cos 2 x − 1 4 sin 2 x ] 0 π = 1 π [ 1 2 π cos 2 π ] = 1 2 Obtain the value of b 1 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 12E

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To find: The Fourier series of the function.

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Chapter 30 Solutions

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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