(a) Explanation Procedure used: The Fourier series of an even function on the interval ( − P , P ) is the cosine series given as follows: f ( x ) = a 0 2 + ∑ n = 1 ∞ a n cos n π p x Here, a 0 and a n are defined as follows: a 0 = 2 p ∫ 0 p f ( x ) d x a n = 2 p ∫ 0 p cos ( n π p x ) d x Calculation: Use the above definition to obtain the cosine expansion of the function f ( x ) = e − x , 0 < x < 1 as follows: f ( x ) = a 0 2 + ∑ n = 1 ∞ a n cos n π x ( 1 ) Substitute f ( x ) = e − x and p = 1 in expression a 0 = 2 p ∫ 0 p f ( x ) d x to obtain the value of a 0 as follows: a 0 = ∫ 0 1 f ( x ) d x = 2 ∫ 0 1 e − x d x = − 2 e − x | 0 1 = 2 ( 1 − e − 1 (b) To determine The expansion of the function f ( x ) = e − x , 0 < x < 1 using sine series.

Differential Equations with Boundary-Value Problems (MindTap Course List)

9th Edition

ISBN: 9781305965799

Author: Dennis G. Zill

Publisher: Cengage Learning

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Chapter 11, Problem 15RE

(a)

To determine

The expansion of the function f(x)=e−x, 0<x<1 using cosine series.

(b)

To determine

The expansion of the function f(x)=e−x, 0<x<1 using sine series.

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Chapter 11, Problem 14RE

Chapter 11, Problem 16RE

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Differential Equations with Boundary-Value Problems (MindTap Course List)

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 expansion of the function f ( x ) = e − x , 0 < x < 1 using cosine series.

Solution Summary: The author explains the Fourier series of an even function on the interval (-P,P).

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The expansion of the function f(x)=e−x, 0<x<1 using cosine series.

The expansion of the function f(x)=e−x, 0<x<1 using sine series.

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Chapter 11 Solutions

The expansion of the function f ( x ) = e − x , 0 &lt; x &lt; 1 using cosine series.

Solution Summary: The author explains the Fourier series of an even function on the interval (-P,P).

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The expansion of the function f(x)=e−x, 0<x<1 using cosine series.

The expansion of the function f(x)=e−x, 0<x<1 using sine series.

Want to see the full answer?

Chapter 11 Solutions

The expansion of the function f ( x ) = e − x , 0 < x < 1 using cosine series.