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Chapter 12: Problem 12

Evaluate Fourier transforms of the following distributional functions: (a) \(\delta(x-a)\). (b) \(\delta^{\prime}(x-a)\). (c) \(\delta^{(n)}(x-a)\). (d) \(\delta\left(x^{2}-a^{2}\right)\) (c) \(\delta^{\prime}\left(x^{2}-a^{2}\right)\)

Short Answer

Expert verified
(a) \( e^{-j\omega a} \), (b) \( j\omega e^{-j\omega a} \), (c) \((j\omega)^n e^{-j\omega a} \), (d) undefined, (e) undefined.

Step by step solution

01

Fourier Transform of Delta function

Fourier transform \(F(\omega)\) of a function f(t) is given by \(F(\omega) = \int_{-\infty}^{+\infty}f(t) e^{-j\omega t} dt \). If function f(t) = \(\delta (t-a)\) ,we get \(F(\omega) = e^{-j\omega a} \) .
02

Fourier Transform of Delta function derivatives

The Fourier Transform of the nth derivative of the delta function \(\delta^{(n)}(t-a)\) is given by \( (j\omega)^n e^{-j\omega a} \). This is obtained by differentiating the Fourier transform of the delta function n times.
03

Fourier Transform of Zero function delta

The Fourier transform of a zero function delta, \(\delta(x^2-a^2)\) is zero as it's undefined. Delta function is only defined for linear functions, not for quadratic or higher degree functions.
04

Fourier Transform of Zero function delta derivative

The derivative of a zero function delta, \(\delta'(x^2-a^2)\) is also not defined, hence its Fourier Transform is also zero.

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Most popular questions from this chapter

Show that the Green ' function for the one-dimensional diffusion equation, $$ \left.\frac{\partial^{2} G(x, t)}{\partial x^{2}}-\frac{1}{\kappa} \frac{\partial G(x, t)}{\partial t}=\varepsilon\left(x-x^{\prime}\right)\right\\}\left(t-t^{\prime}\right) $$ is given by $$ G\left(x-x^{\prime}, t-t^{\prime}\right)=-\theta\left(t-t^{\prime}\right) \sqrt{\frac{K}{4 \pi\left(t-t^{\prime}\right)}} e^{-\left(x-r^{\prime}\right)^{2} / 4(t-h)} $$ and write out the corresponding solution of the inhomogencous equation $$ \frac{\partial^{2} \psi(x, t)}{\partial x^{2}}-\frac{1}{x^{\prime}} \frac{\partial \psi(x, t)}{\partial t}=F(x, t) $$ Do the same for the two- and thrce-dimensional diffusion equations $$ \nabla^{2} G(x, t)-\frac{1}{\kappa} \frac{\partial G(x, t)}{\partial t}=\delta^{n}\left(x-x^{\prime}\right) \delta\left(t-t^{\prime}\right) \quad(n=2,3) $$

Show the following identities: (a) \(\delta((x-a)(x-b))=\frac{1}{b-a}(\delta(x-a)+\delta(x-b))\) (b) \(\frac{\mathrm{d}}{\mathrm{dx}} \theta\left(x^{2}-1\right)=\delta(x-1)-\delta(x+1)=2 x \delta\left(x^{2}-1\right)\). (c) \(\frac{\mathrm{d}}{\mathrm{d} x} \delta\left(x^{2}-1\right)=\frac{1}{2}\left(\delta^{\prime}(x-1)+\delta^{\prime}(x+1)\right)\) (d) \(\delta^{\prime}\left(x^{2}-1\right)=\frac{1}{4}\left(\delta^{\prime}(x-1)-\delta^{\prime}(x+1)+\delta(x-1)+\delta(x+1)\right)\)

Prove that $$ x^{m} \delta^{(n)}(x)=(-1)^{m} \frac{n !}{(n-m) !} \delta^{(n-m)}(x) \quad \text { for } n \geq m $$ Hence show that the Fourier trarsform of the distrobution $$ \sqrt{2 \pi} \frac{k !}{(m+k) !} x^{m} \delta^{(m+k)}(-x) \quad(m, k \geq 0) $$ is \((-2 y)^{k}\)

Which of the following is a distribution? (a) \(T(\phi)=\sum_{n=1}^{m} \lambda_{n} \phi^{(n)}(0) \quad\left(\lambda_{n} \in \mathbb{R}\right)\) (b) \(T(\phi)=\sum_{n=1}^{m} \lambda_{n} \phi\left(x_{n}\right) \quad\left(\lambda_{n}, x_{n} \in \mathbb{R}\right)\). (c) \(T(\phi)=(\phi(0))^{2}\). (d) \(T(\phi)=\sup \phi\) (c) \(T(\phi)=\int_{-\infty}^{\infty}|\phi(x)| \mathrm{d} x\).

Evaluate (a) \(\int_{-\infty}^{\infty} \mathrm{e}^{a t} \sin b t \delta^{(n)}(t) \mathrm{d} t \quad\) for \(n=0,1,2\). (b) \(\int_{-\infty}^{\infty}(\cos t+\sin t) \delta^{(n)}\left(t^{3}+t^{2}+t\right) d t \quad\) for \(n=0,1\).
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