Articles of integral transforms

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 […]

An inequality involving arctan of complex argument

I have the following conjecture: \begin{equation} \text{Re}\left[(1+\text{i}y)\arctan\left(\frac{t}{1+\text{i}y}\right)\right] \ge \arctan(t), \qquad \forall y,t\ge0. \end{equation} Which seems to be true numerically. Can anyone offer some advice on how to approach proving (or disproving) this? It originates from a question involving the (complex) Hilbert transform of a symmetric non-increasing probability distribution: \begin{equation} h(y) = (1+\text{i}y)\int_{-\infty}^\infty \frac{1}{1 + \text{i}(y-t)}\text{d}G(t) […]

Evaluating improper integrals using laplace transform

I want to calculate the following improper integral using Laplace and transforms (and laplace transforms only). $$\int_0^{\infty} x e^{-3x} \sin{x}\, dx$$ I propose the following method. I plan to use $\mathcal{L}(\int_0^x f(t) dt)=\frac{\mathcal{L}(f)(s)}{s}$. So, first I need to find out the laplace transform of the integrand, which can be done using $\mathcal{L}({xf})=-\frac{dL(f)}{ds}=-\frac{d}{ds}(e^{-3x} \sin{x})=-\frac{d}{ds}(\frac{1}{(s+3)^2+1})=\frac{2(s+3)}{((s+3)^2)+1)^2}$ So, my […]

What is the easiest known expression for inverse of Laplace transform?

There is a very simple expression for the inverse of Fourier transform. What is the easiest known expression for the inverse Laplace transform? Moreover, what is the easiest way to prove it?

About Mellin transform and harmonic series

Let $\mathfrak{M}\left(*\right)$ the Mellin transform. We know that holds this identity$$\mathfrak{M}\left(\underset{k\geq1}{\sum}\lambda_{k}g\left(\mu_{k}x\right),\, s\right)=\underset{k\geq1}{\sum}\frac{\lambda_{k}}{\mu_{k}^{s}}\mathfrak{M}\left(g\left(x\right),s\right)$$ so for example if we want a closed form of$$\underset{k\geq1}{\sum}\frac{\sin\left(nx\right)}{n^{3}}=x^{3}\underset{k\geq1}{\sum}\frac{\sin\left(nx\right)}{n^{3}x^{3}}\,\,(1)$$ we are in the case $$\lambda_{k}=1,\thinspace\mu_{k}=k,\, g\left(x\right)=\frac{\sin\left(x\right)}{x^{3}}$$ and so our Mellin transform is $$\zeta\left(s\right)\mathfrak{M}\left(g\left(x\right),s\right)$$ and then we find the sum of $(1)$ by the residue theorem. I tried to use this argument for […]

solution for integral $\int_0^{\infty} \frac{k}{k^3-a}J_0\left(k \, r\right) dk $ involving Bessel function (Hankel transform)

during my research I am facing for the first time integrals involving Bessel functions. In particular i need to evaluate the following integral: $\int_0^{\infty} \frac{k}{k^3-a}J_0\left(k \, r\right) dk $ with $a$ and $r$ being two real positive numbers. $J_0$ is Bessel function of the first kind and order zero. I know this can be seen […]

How fast does the function $\displaystyle f(x)=\lim_{\epsilon\to0}\int_\epsilon^{\infty} \dfrac{e^{xt}}{t^t} \, dt $ grow?

Let $x$ be a positive real number and $f(x):=\lim_{\epsilon\to0}\int_\epsilon^{\infty} \dfrac{e^{xt}}{t^t} \, dt $. How fast does this function grow ? In other words can we find a good asymptote for $f(x)$ as $x$ goes to $+\infty$ ? Can we show one of these two limits converges to a constant : A) $\lim_{x\to+\infty} \dfrac{\ln(f(x))}{P(x)} $ B) […]

Fourier transform of Bessel functions

I’m curious as to how the Fourier transform of the various types of Bessel functions would be calculated. The Wikipedia page on the Fourier transform gives the transform of $J_o(x)$ as being $\frac{2rect(\pi\zeta)}{\sqrt{1-4\pi^2\zeta^2}}$. I’ve searched the web some but I seem to be unable to find a derivation for that value. Since $J_o$ only “damps […]

An alternating series identity with a hidden hyperbolic tangent

How does one use the inverse Mellin transform to prove that the following identity holds? $$\sum_{n=1}^{\infty}\frac{(-1)^n}{n(e^{n\pi} + 1)} = \frac{1}{8}(\pi – 5\log(2))$$ The identity follows from MO189199 and the penultimate identity on this page (for $x=\frac{1}{2}$). Scratch-work: I computed the Mellin transform of $$f(x) = \frac{(-1)^x}{x(e^{x\pi} + 1)}$$ and re-wrote the function in terms of […]

Relationship Between The Z-Transform And The Laplace Transform

Below I’ve quoted Wikipedia’s entry that relates the Z-Transform to the Laplace Transform. The part I don’t understand is $z \ \stackrel{\mathrm{def}}{=}\ e^{s T}$; I thought $z$ was actually an element of $\mathbb{C}$ and thus would be $z \ \stackrel{\mathrm{def}}{=}\ Ae^{s T}$ (but then it would be different to the Laplace Transform…). I don’t understand […]