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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

A damped harmonic oscillator is given by $$ f(t)=\left\\{\begin{array}{cl} A e^{-a t} e^{i \omega_{0} t}, & t \geq 0 \\ 0, & t<0 \end{array}\right. $$ a. Find \(\hat{f}(\omega)\) and b. the frequency distribution \(|\hat{f}(\omega)|^{2}\). c. Sketch the frequency distribution.

Find a Fourier series representation of the Dirac delta function, \(\delta(x)\), on \([-L, L]\)

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.

Use Laplace transforms to solve the following initial value problems. Where possible, describe the solution behavior in terms of oscillation and decay. a. \(y^{\prime \prime}-5 y^{\prime}+6 y=0, y(0)=2, y^{\prime}(0)=0\). b. \(y^{\prime \prime}-y=t e^{2 t}, y(0)=0, y^{\prime}(0)=1\). c. \(y^{\prime \prime}+4 y=\delta(t-1), y(0)=3, y^{\prime}(0)=0\). d. \(y^{\prime \prime}+6 y^{\prime}+18 y=2 H(\pi-t), y(0)=0, y^{\prime}(0)=0\).

Verify that the sequence of functions \(\left\\{f_{n}(x)\right\\}_{n=1}^{\infty}\), defined by \(f_{n}(x)=\) \(\frac{n}{2} e^{-n|x|}\), approaches a delta function.
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