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 < − π 2 , π 2 ≤ x < π 1 − π 2 ≤ x < π 2 . Obtain the value of a 0 as follows. a 0 = 1 2 π ∫ − π − π 2 ( 0 ) d x + 1 2 π ∫ − π 2 π 2 ( 2 ) d x + 1 2 π ∫ π 2 π ( 0 ) d x = 0 + 1 2 π [ 2 x ] − π 2 π 2 + 0 = 1 2 π [ 2 ( π 2 ) − 2 ( − π 2 ) ] = 1 2 π [ π + π ] That is, a 0 = 1 2 π ⋅ 2 π = 1 . Obtain the value of a n as follows. a n = 1 π ∫ − π − π 2 ( 0 ) cos n x d x + 1 π ∫ − π 2 π 2 ( 2 ) cos n x d x + 1 π ∫ π 2 π ( 0 ) cos n x d x = 0 + 2 π [ sin n x n ] − π 2 π 2 + 0 = 2 π [ sin n ( π 2 ) − sin n ( − π 2 ) ] = 2 π n [ sin n π 2 − sin ( − n π 2 ) ] That is, when n = 1 , the value of a 1 becomes a 1 = 2 π [ sin π 2 − sin ( − π 2 ) ] = 2 π [ 1 − ( − 1 ) ] = 2 π [ 2 ] = 4 π When n = 2 , the value of

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ISBN: 9780134437705

Author: Washington

Publisher: PEARSON

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Chapter 30.6, Problem 4E

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To find: The Fourier series of the function.

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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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