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Chapter 12: Problem 14

Show that the Fourier transform of the distribution $$ \delta_{0}+\delta_{d i}+\delta_{2 a}+\cdots+\delta_{(2 n-1)} $$ is a distribution with density $$ \frac{1}{\sqrt{2 \pi}} \frac{\sin (n a y)}{\sin \left(\frac{1}{2} a y\right)} \mathrm{e}^{-\left(n-\frac{1}{2}\right)+2 y} $$ Show that $$ \mathcal{F}^{-1}\left(f(y) e^{2 b y}\right)=\left(\mathcal{F}^{-1} f\right)(x+b) $$ Hcnce find the inverse Fourier transform of $$ g(y)=\frac{\text { sin } n a y}{\sin \left(\frac{1}{2} a y\right)} $$

Short Answer

Expert verified
The Fourier transform of the given distribution is \( \frac{1}{\sqrt{2 \pi}} \frac{\sin (n a y)}{\sin \left(\frac{1}{2} a y\right)} e^{-\left(n-\frac{1}{2}\right)+2 y} \). Moreover, the Fourier transform of \( f(y)e^{2by} \) is \( \mathcal{F}^{-1}(f)(x+b) \). Thus, the inverse Fourier transform of \( g(y) = \frac{\sin(nay)}{\sin(ay/2)} \) is \( \mathcal{F}^{-1}(f)(x+b) \).

Step by step solution

01

Fourier Transform of Distributions

To find the Fourier transform of the given distribution, observe that the Fourier transform of \( \delta (x-a) \) is \( e^{iay} / \sqrt{2 \pi} \). Then, the Fourier transform of the given distribution is the sum for k = 0 to n, of \( e^{-ia(2k-1)y} / \sqrt{2 \pi} \). Now, this sum can be rewritten as \( \frac{1}{\sqrt{2 \pi}} \frac{\sin (n a y)}{\sin \left(\frac{1}{2} a y\right)} e^{-\left(n-\frac{1}{2}\right)+2 y} \) by using the formula for sum of geometric series and trigonometric identities.
02

Fourier Transform of Function times Exponential

This requires proving that the Fourier transform of \( f(y)e^{2by} \) is \( \mathcal{F}^{-1}(f)(x+b) \). Given that the Fourier transform of \( f(y) \) is \( \mathcal{F}(f)(x) \), we can apply the shift theorem to \( f(y)e^{2by} \). The shift theorem states that the Fourier transform of \( e^{bx}f(x) \) is \( \mathcal{F}(f)(x-b) \). Applying this result with a shift -b in frequency domain, we get \( \mathcal{F}^{-1}(f)(x+b) \) as required.
03

Inverse Fourier Transform of a Function

We have to find the inverse Fourier transform of \( g(y) = \frac{\sin(n a y)}{\sin\left(a y /2 \right)} \). Instead of taking the inverse Fourier transform directly, we find it easier to write g(y) as \( g(y) = f(y)e^{2by} \), with \( f(y) = \frac{1}{\sqrt{2 \pi}} \frac{\sin (n a y)}{\sin \left(\frac{1}{2} a y\right)} \) and \( b = -\left(n-\frac{1}{2}\right)+2 \). Now, we can use the result in Step 2 to find that \( \mathcal{F}^{-1}(g)(x) = \mathcal{F}^{-1}(f)(x+b) \). This yields the inverse Fourier transform of g(y).

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Most popular questions from this chapter

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 \((-2 y)^{k}\)

Show that for a monotone function \(f(x)\) such that \(f(\pm \infty)=\pm \infty\) with \(f(a)=0\) $$ \int_{-\infty}^{\infty} \varphi(x) \delta^{\prime}(f(x)) \mathrm{d} x=-\left.\frac{1}{f^{\prime}(x)} \frac{\mathrm{d}}{\mathrm{d} x}\left(\frac{\varphi(x)}{\left|f^{\prime}(x)\right|}\right)\right|_{x=6} $$ For a general function \(f(x)\) that is monotone on a neighbourhood of all its zeros, find a general formula for the distribution \(\delta^{\prime} \circ f .\)

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) \(\delta^{\prime}\left(x^{2}-a^{2}\right)\)

Show that the Green's function for the time-independent Klein-Gordon equation $$ \left(\nabla^{2}-m^{2}\right) \phi=\rho(r) $$ can be expressed as the Fourier integral $$ G\left(x-x^{\prime}\right)=-\frac{1}{(2 \pi)^{3}} \iiint d^{3} k \frac{e^{u k\left(x-y^{\prime}\right)}}{k^{2}+m^{2}} $$ Evaluate this integral and show that it results in $$ G(\mathbf{R})=-\frac{\mathrm{e}^{-\operatorname{m} k}}{4 \pi R} \quad \text { where } \quad \mathbf{R}=\mathbf{x}-\mathbf{x}^{\prime}, \quad R=|\mathbf{R}| $$ Find the solution \(\phi\) correspondmg to a point source $$ \rho(\mathbf{r})=q \delta^{3}(r) $$

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 $$ \nabla^{2} G(x, t)-\frac{1}{\kappa} \frac{\partial G(x, t)}{\partial t}=\delta^{n}\left(x-x^{\prime}\right) \delta\left(t-t^{\prime}\right) \quad(n=2,3) $$
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