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

Student Solutions Manual For Basic Technical Mathematics And Basic Technical Mathematics With Calculus

11th Edition

ISBN: 9780134434636

Author: Allyn J. Washington, Richard Evans

Publisher: PEARSON

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Chapter 30.6, Problem 14E

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To find: The Fourier series of the function.

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Chapter 30.6, Problem 13E

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Student Solutions Manual For Basic Technical Mathematics And Basic Technical Mathematics With Calculus

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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