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Chapter 12: Problem 9

Show the identities $$ \frac{d}{d x}(\delta(f(x)))=f^{\prime}(x) \delta^{\prime}(f(x)) $$ and $$ \delta(f(x))+f(x) \delta^{\prime}(f(x))=0 $$ Hence show that \(\phi(x, y)=\delta\left(x^{2}-y^{2}\right)\) is a solution of the partial differcntial equation $$ x \frac{\partial \phi}{\partial x}+y \frac{\partial \phi}{\partial y}+2 \phi(x, y)=0 $$

Short Answer

Expert verified
The identities are correct, and substituting \( \phi(x, y) = \delta(x^2 - y^2) \) into the PDE shows that it indeed holds true, hence \( \phi(x, y) \) is a solution to the PDE.

Step by step solution

01

Identity verification

Let's verify the identities. We'll need the definition of the Dirac delta function: it's zero everywhere except at zero, and its integral over any range containing zero is one.\n\nNow take the derivative of \( \delta(f(x)) \) with respect to \( x \). By the chain rule of calculus, this is given by \( f'^{\prime}(x) \delta^{\prime}(f(x)) \) .\n\nFor the second identity, consider the definition of the derivative of the delta function: \( \delta^{\prime}(x) = -\delta(x)/x \). Substituting this in, we get \( \delta(f(x)) + f(x) \delta^{\prime}(f(x)) = 0 \). Our identities are verified.
02

Substituting into the PDE

Now substitute \( \phi(x, y) = \delta (x^2 - y^2) \) into the partial differential equation \( x\partial\phi/\partial{x} + y\partial\phi/\partial{y} + 2\phi(x, y) = 0 \).\n\nNote that \( \partial\phi/\partial{x} = 2x\delta^{\prime}(x^2 - y^2) \) and \( \partial\phi/\partial{y} = -2y\delta^{\prime}(x^2 - y^2) \).\n\nSubstituting these in, we get \( 2x^2\delta^{\prime}(x^2 - y^2) - 2y^2\delta^{\prime}(x^2 - y^2) + 2\delta(x^2 - y^2) = 0 \), which simplifies to \( \delta(x^2 - y^2) + (x^2 - y^2)\delta^{\prime}(x^2 - y^2) = 0 \).\n\nThis matches our second identity, so \( \phi(x, y) \) is indeed a solution to the PDE.

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Most popular questions from this chapter

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 \(\delta^{\prime} \circ f .\)

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 $$ g(y)=\frac{\text { sin } n a y}{\sin \left(\frac{1}{2} a y\right)} $$

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

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