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. […]

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?

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 […]

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

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

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 […]

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 […]

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 […]

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 […]

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 […]

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