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Chapter 8: Problem 8

Find the Fourier transform, \(\hat{f}(k)\), of \(f(x)=e^{-2 x^{2}+x}\).

Short Answer

Expert verified
The Fourier transform, \( \hat{f}(k) \), of the function \(f(x)=e^{-2 x^{2}+x}\) is given by \(\hat{f}(k) = e^{\pi^2 k^2 - \frac{1}{16}}\sqrt{\pi}\)

Step by step solution

01

Identify the function

Identify the function which is to be transformed which in this case is \(f(x)=e^{-2 x^{2}+x}\).
02

Recall the formula for Fourier Transform

The formula for the Fourier Transform is \(\hat{f}(k) = \int_{-\infty}^{\infty} f(x) e^{-2 \pi i k x} dx\). Here, \( \hat{f}(k) \) represents the Fourier transform of \( f(x) \), and \( k \) is the frequency variable.
03

Substitute the function into the Fourier Transform formula

Now, substitute \( f(x) = e^{-2 x^{2}+x} \) in the Fourier transform formula. It gives \(\hat{f}(k) = \int_{-\infty}^{\infty} e^{-2 x^{2}+x} e^{-2 \pi i k x} dx\). This creates a manageable integral.
04

Consolidate the exponentials

Combine the terms \(-2x^{2} + x\) and \(-2\pi ikx\) under one exponential to get an equation in the form of a quadratic identity which can be later simplified for the integral. It gives \(\hat{f}(k) = \int_{-\infty}^{\infty} e^{-2 x^{2}+(1-2\pi ik)x} dx\).
05

Identify the quadratic form

To integrate this exponential, it is ideal to identify a perfect square. This can be factored into \((x-\pi ik)^{2}\). After solving, the integrand matches the formula of integral of a Gaussian, \(\int_{-\infty}^{\infty} e^{-x^{2}} dx = \sqrt{\pi}\) and the Fourier Transform is calculated.
06

Final evaluation of Fourier Transform

After evaluating and simplifying, we get the Fourier Transform. \(\hat{f}(k) = e^{\pi^2 k^2 - \frac{1}{16}}\sqrt{\pi}\)

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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.

Use Laplace transforms to sum the following series or write as a single integral. a. \(\sum_{n=0}^{\infty} \frac{(-1)^{n}}{1+2 n}\) b. \(\sum_{n=1}^{\infty} \frac{1}{n(n+3)} .\) c. \(\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n(n+3)}\) d. \(\sum_{n=0}^{\infty} \frac{(-1)^{n}}{n^{2}-a^{2}}\) e. \(\sum_{n=0}^{\infty} \frac{1}{(2 n+1)^{2}-a^{2}}\) f. \(\sum_{n=1}^{\infty} \frac{1}{n} e^{-a n}\)

Show that the convolution operation is associative: \((f *(g * h))(t)=\) \(((f * g) * h)(t)\)

Consider the initial boundary value problem for the heat equation: $$ \begin{array}{rr} u_{t}=2 u_{x x}, & 00 \\ u(1, t)=0, & t>0 \end{array} $$ Use the finite transform method to solve this problem. Namely, assume that the solution takes the form \(u(x, t)=\sum_{n=1}^{\infty} b_{n}(t) \sin n \pi x\) and obtain an ordinary differential equation for \(b_{n}\) and solve for the \(b_{n}\) 's for each \(n\).

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