Articles of integral transforms

Is there a way to compute $\int_0^\infty \frac{\cos (qt) J_1 (qr)}{1+q^2} \, \mathrm{d} q$ provided that $0<t<r$?

In a dual integral situation, the following integral has to be involved $$ \int_0^\infty \frac{\cos (qt) J_1 (qr)}{1+q^2} \, \mathrm{d} q \quad\quad (0<t<r) \, . $$ Visibly this integral is convergent. I was wondering whether an amenable analytical expression is possible? This will be useful for my further analysis. Any help is highly appreciated. Thanks. […]

What is the difference between resolvent kernel and iterative kernel of an integral equation?

As we have different methods to find resolvent kernel, which is more suitable among all those methods? And what is the difference between resolvent and iterative kernels?

Stueckelberg Feynman propagator computation

On page 35 of Itzykson-Zuber’s textbook on quantum field theory, I am having trouble deriving equation (1-180): $\displaystyle G_F(0,r) = \frac{i}{(2\pi)^2 r} \int_m^\infty dp \frac{p}{\sqrt{p^2-m^2}} e^{-pr}$ Here $G_F(0,r)$ is the Stueckelberg Feynman propogator $G_F(x) = \frac{-1}{(2\pi)^4} \int d^4 p e^{-i p\cdot x} \frac{1}{p^2 – m^2 + i\epsilon}$, when $x$ is of the form $(0,\vec{x})$ and […]

Fourier transform of $te^{-t^2}$?

How can I find the Fourier transform of: $$f(t) = te^{-t^2}$$

$\int_{-\infty}^{+\infty}(1+\frac{1}{v^2})\exp(-\frac{u^2}{2\sigma_1^2\sigma_2^2}(\sigma_2v+\frac{\sigma_1}{v})^2)\,dv$

$\int_{-\infty}^{+\infty}(1+\frac{1}{v^2})e^{-\frac{u^2}{2\sigma_1^2\sigma_2^2}(\sigma_2v+\frac{\sigma_1}{v})^2}\,dv$ Does this integral has a close form solution? What if $\sigma_1=\sigma_2=1, u=\sqrt2, i.e,\int_{-\infty}^{+\infty}(1+\frac{1}{v^2})e^{-(v+\frac{1}{v})^2}dv$.

Why does it seem I can't apply the Radon transform to the Helmholtz equation?

Say we a function $u$ and a bounded region $\Omega \subset \mathbb{R}^2$, such that $(\Delta+\lambda)u = 0$ everywhere, and $u=0$ on the boundary. We extend it to the entire plane by defining $u=0$ everywhere outside of $\Omega$. Wikipedia states that, for the Radon transform, $R\Delta = \partial^2 / \partial s^2 R$, hence we can derive […]

Fourier transform convention: $\frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(x)e^{\pm ikx}dx $?

I’ve come across the Fourier transform being defined as: $$\tilde{f}(k)=\frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(x)e^{ikx}dx$$ But this convention is not present in the Wikipedia article. The one given there, under “Fourier transform: unitary, angular frequency” has a minus sign in the exponent. Are the two equivalent? Switching variables from $x$ to $-x$ wouldn’t work because I would get […]

Bilateral Laplace transform

My knowledge of Bilateral Laplace transform is less. Here are the few questions I need answer. What is the condition for existence of bilateral Laplace transform? How is the condition for existence related to ROC ? What’s the bilateral Laplace transform of $\sin \omega t$ ? How is the bilateral Laplace related to Fourier transform […]

Smoothness of Fourier transform of $\frac{1}{|x|^p}$

Consider the “function” (more precisely it is a tempered distribution) given by $f : \mathbb{R}^n \to \mathbb{R}$, $f(x) = \frac{1}{|x|^p}$, where $0 < p < n$. It can be calculated that the Fourier transform of $f$ is given (upto a constant) by $\hat{f}(\xi) = \frac{1}{|\xi|^{n – p}}$. Now, I am trying to prove that the […]

Consistency/range conditions for (integral) transform mapping into higher-dimensional space

I am interested in learning more about what the (formal) implications are when transforming a $n$-dimensional function space (e.g., the space of all $\mathbb{R}^n\to\mathbb{R}$) to a higherdimensional one (say, $\mathbb{R}^m\to\mathbb{R}$, $m>n$); in particular, in terms of consistency or range conditions (is there a difference?). A real example could be the relationship between the result of […]