Explanation Given: The complex form is f ( x ) = 1 2 π ∫ − ∞ ∞ C ( α ) e − i α x d α . Calculation: The equation is given below: f ( x ) = 1 2 π ∫ − ∞ ∞ ∫ − ∞ ∞ f ( t ) cos α ( t − x ) d t d α ...... ( 1 ) The given function is f ( t ) = e − | t | , − ∞ < t < ∞ . { f ( t ) = e t − ∞ < t < ∞ f ( t ) = e − t − ∞ < t < 0 Therefore, the expression ∫ − ∞ ∞ f ( t ) cos α ( t − x ) d t is evaluated as follows: ∫ − ∞ ∞ f ( t ) cos α ( t − x ) d t = ∫ − ∞ 0 f ( t ) cos α ( t − x ) d t + ∫ 0 ∞ f ( t ) cos α ( t − x ) d t = ∫ − ∞ 0 e t cos α ( t − x ) d t + ∫ 0 ∞ e − t cos α ( t − x ) d t ...... ( 2 ) Consider t = − u and the derivative of t is given below. t = − u d t = − d u When the value of t is − ∞ then the value of u is ∞ and when t = 0 then u = 0 . Substitute − u for t in equation (2) as follows: ∫ − ∞ ∞ f ( t ) cos α ( t − x ) d t = − ∫ ∞ 0 e − u cos α ( − u − x ) d u + ∫ 0 ∞ e − t cos α ( t − x ) d t = ∫ 0 ∞ e − u cos α ( u + x ) d u + ∫ 0 ∞ e − t cos α ( t − x ) d t = ∫ 0 ∞ e − t cos α ( t + x ) d t + ∫ 0 ∞ e − t cos α ( t − x ) d t = ∫ 0 ∞ e − t [ cos α ( t + x ) + cos α ( t − x ) d t ] Further simplify the above equation

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

9th Edition

ISBN: 9781305965799

Author: Dennis G. Zill

Publisher: Cengage Learning

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Question

Chapter 14.3, Problem 20E

To determine

The Fourier integral representation of $f(x)=e−|x|$ using complex form $f(x)=12π∫−∞∞C(α)e−iαxdα$ and show that the obtained solution is $f(x)=2π∫0∞cosαx1+α2 dα$ .

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Chapter 14.3, Problem 19E

Chapter 14.4, Problem 1E

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Chapter 14 Solutions

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

Ch. 14.1 - (a) Show that erf(t)=10ted. (b) Use part (a), the...Ch. 14.1 - Prob. 2E Ch. 14.1 - Prob. 3E Ch. 14.1 - Prob. 4E Ch. 14.1 - Prob. 5E Ch. 14.1 - Prob. 6E Ch. 14.1 - Prob. 7E Ch. 14.1 - Prob. 8E Ch. 14.1 - Prob. 9E Ch. 14.1 - Use the third and fifth entries in Table 14.1.1 to...

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The Fourier integral representation of f ( x ) = e − | x | using complex form f ( x ) = 1 2 π ∫ − ∞ ∞ C ( α ) e − i α x d α and show that the obtained solution is f ( x ) = 2 π ∫ 0 ∞ cos α x 1 + α 2 d α .

Solution Summary: The author explains the Fourier integral representation of f(x)=e- left|x-right| using complex form.

The Fourier integral representation of $f(x)=e−|x|$ using complex form $f(x)=12π∫−∞∞C(α)e−iαxdα$ and show that the obtained solution is $f(x)=2π∫0∞cosαx1+α2 dα$ .

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Chapter 14 Solutions