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 […]

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) […]

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 […]

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?

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 […]

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 […]

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) […]

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 […]

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 […]

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 […]

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