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 ) = e x , − π ≤ x < π . Obtain the value of a 0 as follows. a 0 = 1 2 π ∫ − π π ( e x ) d x = 1 2 π [ e x ] − π π = 1 2 π [ e π − e − π ] = e π − e − π 2 π Obtain the value of a 1 as follows. a 1 = 1 π ∫ − π π ( e x ) cos x d x = 1 π [ e x sin x + e x cos x 2 ] − π π = 1 2 π [ e π sin π + e π cos π − e ( − π ) sin ( − π ) − e ( − π ) cos ( − π ) ] = 1 2 π [ 0 + e π ( − 1 ) − 0 − e ( − π ) ( − 1 ) ] a 1 = − 1 2 π [ e π − e − π ] = − e π − e − π 2 π Obtain the value of a 2 as follows. a 2 = 1 π ∫ − π π ( e x ) cos 2 x d x = 1 π [ 2 e x sin 2 x + e x cos 2 x 5 ] − π π = 1 5 π [ 2 e π sin 2 π + e π cos 2 π − 2 e ( − π ) sin 2 ( − π ) + e ( − π ) cos 2 ( − π ) ] = 1 5 π [ 0 + e π ( 1 ) − 0 − e ( − π ) ( 1 ) ] a 2 = 1 5 π [ e π − e − π ] = e π − e − π 5 π Obtain the value of b 1 as follows

11th Edition

ISBN: 9780134437705

Author: Washington

Publisher: PEARSON

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Chapter 30.6, Problem 13E

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To find: The Fourier series of the function.

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Chapter 30.6, Problem 12E

Chapter 30.6, Problem 14E

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