Explanation The given function is f ( x ) = { 1 , − 5 < x < 0 1 + x , 0 ≤ x < 5 The Fourier series expansion of a function defined on the interval ( − p , p ) is given by, f ( x ) = a 0 2 + ∑ n = 1 ∞ ( a n cos n π x p + b n sin n π x p ) The values of coefficients of the expansion is as follows. a 0 = 1 p ∫ − p p f ( x ) d x a n = 1 p ∫ − p p f ( x ) cos n π x p d x b n = 1 p ∫ − p p f ( x ) sin n π x p d x Here, the value of p = 5 . Now, the value of the constant a 0 is obtained as, a 0 = 1 5 ∫ − 5 5 f ( x ) d x = 1 5 { ∫ − 5 0 1 d x + ∫ 0 5 ( 1 + x ) d x } = 1 5 { [ x ] − 5 0 + [ x + x 2 2 ] 0 5 } = 1 5 ( 5 + 5 + 25 2 ) Thus, the value of a 0 is a 0 = 9 2 . Substitute p = 5 and obtain the value of a n as follows. a n = 1 5 ∫ − 5 5 f ( x ) cos n π x 5 d x = 1 5 { ∫ − 5 0 ( 1 ) ⋅ cos n π x 5 d x + ∫ 0 5 ( 1 + x ) cos n π x 5 d x } = 1 5 { [ sin n π x 5 n π 5 ] − 5 0 + [ sin n π x 5 n π 5 ] 0 5 + [ x sin n π x 5 n π 5 ] 0 5 − ∫ 0 5 sin n π x 5 n π 5 d x } = 1 5 { 0 + 0 + 0 + [ cos n π x 5 ( n π 5 ) 2 ] 0 5 } Simplify further as, a n = 1 5 { [ ( − 1 ) n − 1 ( n π 5 ) 2 ] } = 5 n 2 π 2 [ ( − 1 ) n − 1 ] Thus, the value of a n is a n = 5 n 2 π 2 [ ( − 1 ) n − 1 ] . Substitute p = 5 and obtain the value of b n as follows

WEB ASSIGN FOR ZILL'S DIFFERENTIAL EQUAT

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Chapter 11.2, Problem 13E

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To find: The Fourier series expansion of the given function on the given interval and also find the number to which it converges at a point of discontinuity of function.

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Chapter 11.2, Problem 12E

Chapter 11.2, Problem 14E

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WEB ASSIGN FOR ZILL'S DIFFERENTIAL EQUAT

Ch. 11.1 - In problem 16 show that the given functions are...Ch. 11.1 - In problem 16 show that the given functions are...Ch. 11.1 - In problem 16 show that the given functions are...Ch. 11.1 - In problem 16 show that the given functions are...Ch. 11.1 - In problem 16 show that the given functions are...Ch. 11.1 - Prob. 6E Ch. 11.1 - Prob. 7E Ch. 11.1 - In Problems 712 show that the given set of...Ch. 11.1 - In Problems 712 show that the given set of...Ch. 11.1 - In Problems 712 show that the given set of...

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The Fourier series expansion of the given function on the given interval and also find the number to which it converges at a point of discontinuity of function.

Solution Summary: The author explains the Fourier series expansion of a function defined on the interval (-p,p) and the number to which it converges.

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To find: The Fourier series expansion of the given function on the given interval and also find the number to which it converges at a point of discontinuity of function.

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Chapter 11 Solutions