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Chapter 8: Problem 6

For \(a>0\), find the Fourier transform, \(\hat{f}(k)\), of \(f(x)=e^{-a|x|}\).

Short Answer

Expert verified
The Fourier transform of the given function \(f(x)=e^{-a|x|}\) is \(\hat{f}(k) = \frac{a}{a^2 + (2\pi k)^2}\)

Step by step solution

01

Write f(x) for x>0 and x

Because of the absolute value, we have to consider the function \(f(x)\) for \(x>0\) and \(x<0\) separately. For \(x>0\), \(f(x)=e^{-ax}\) and for \(x<0\), \(f(x)=e^{ax}\).
02

Implement the definition of Fourier transform

The Fourier Transform is given by \(\hat{f}(k) = \int_{-\infty}^{\infty} f(x) e^{-2\pi i k x} dx\). The two cases for \(f(x)\) need to be incorporated into the formula. The Fourier transform, therefore, splits into two integrals: \(\hat{f}(k) = \int_{0}^{\infty} e^{-ax} e^{-2\pi i k x} dx + \int_{-\infty}^{0} e^{ax} e^{-2\pi i k x} dx\)
03

Simplify the integrals

We should combine the exponents in each integral and simplify as much as possible. We get \(\hat{f}(k) = \int_{0}^{\infty} e^{-(a+2\pi ik)x} dx + \int_{-\infty}^{0} e^{(a-2\pi ik)x} dx\)
04

Perform the integrations

On integrating each term, and then combining the results, the final expression for the Fourier transform of \(f(x)\) is: \(\hat{f}(k) = \frac{a}{a^2 + (2\pi k)^2}\)

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Most popular questions from this chapter

Compute the convolution \((f * g)(t)\) (in the Laplace transform sense) and its corresponding Laplace transform \(\mathcal{L}[f * g]\) for the following functions: a. \(f(t)=t^{2}, g(t)=t^{3}\). b. \(f(t)=t^{2}, g(t)=\cos 2 t\). c. \(f(t)=3 t^{2}-2 t+1, g(t)=e^{-3 t}\). d. \(f(t)=\delta\left(t-\frac{\pi}{4}\right), g(t)=\sin 5 t\)

In this problem you will show that the sequence of functions $$ f_{n}(x)=\frac{n}{\pi}\left(\frac{1}{1+n^{2} x^{2}}\right) $$ approaches \(\delta(x)\) as \(n \rightarrow \infty\). Use the following to support your argument: a. Show that \(\lim _{n \rightarrow \infty} f_{n}(x)=0\) for \(x \neq 0\). b. Show that the area under each function is one.

Find the inverse Laplace transform of the following functions using the properties of Laplace transforms and the table of Laplace transform pairs. a. \(F(s)=\frac{18}{s^{3}}+\frac{7}{s} .\) b. \(F(s)=\frac{1}{s-5}-\frac{2}{s^{2}+4}\). c. \(F(s)=\frac{s+1}{s^{2}+1} .\) d. \(F(s)=\frac{3}{s^{2}+2 s+2}\) e. \(F(s)=\frac{1}{(s-1)^{2}}\) f. \(F(s)=\frac{e^{-3 s}}{s^{2}-1} .\) g. \(F(s)=\frac{1}{s^{2}+4 s-5}\) h. \(F(s)=\frac{s+3}{s^{2}+8 s+17} .\)

Use the result from Problem 6 plus properties of the Fourier transform to find the Fourier transform, of \(f(x)=x^{2} e^{-a|x|}\) for \(a>0\).

Find the inverse Laplace transform in two different ways: (i) Use tables. (ii) Use the Bromwich Integral. a. \(F(s)=\frac{1}{s^{3}(s+4)^{2}}\) b. \(F(s)=\frac{1}{s^{2}-4 s-5}\) c. \(F(s)=\frac{s+3}{s^{2}+8 s+17}\) d. \(F(s)=\frac{s+1}{(s-2)^{2}(s+4)}\)
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