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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

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 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. \(F(s)=\frac{s+1}{(s-2)^{2}(s+4)}\)

Find the Laplace transform of the following functions: a. \(f(t)=9 t^{2}-7\) b. \(f(t)=e^{5 t-3}\). c. \(f(t)=\cos 7 t\). d. \(f(t)=e^{4 l} \sin 2 t\). e. \(f(t)=e^{2 t}(t+\cosh t)\). f. \(f(t)=t^{2} H(t-1)\). g. \(f(t)=\left\\{\begin{array}{cl}\sin t, & t<4 \pi \\ \sin t+\cos t, & t>4 \pi\end{array}\right.\) h. \(f(t)=\int_{0}^{t}(t-u)^{2} \sin u d u\). i. \(f(t)=(t+5)^{2}+t e^{2 t} \cos 3 t\) and write the answer in the simplest form.

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. \(f(t)=\delta\left(t-\frac{\pi}{4}\right), g(t)=\sin 5 t\)

Use Laplace transforms to convert the following system of differential equations into an algebraic system, and find the solution of the differential equations. $$ \begin{array}{ll} x^{\prime \prime}=3 x-6 y, & x(0)=1, & x^{\prime}(0)=0 \\ y^{\prime \prime}=x+y, & y(0)=0, & y^{\prime}(0)=0 \end{array} $$
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