Questions tagged [mellin-transform]

Question 1

In Apostol's book "Modular Functions and Dirichlet Series in Number Theory", it states the inverse Mellin transform of the gamma function: $$e^{-x} = \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\...

Question 2

I'm struggling to provide a proper conceptual reason for what is going on here. For background, I was taught in school the three major transforms, Laplace first, Fourier second and Mellin last (but ...

Question 3

Start with $$ H_M(x)=x\sum_{k=1}^{M} a_k\,u^k,\qquad u=e^{1/\ln x}-1 $$ where $a_k$ are Stirling coefficients, and $H_M(x)$ matches the first $M$ terms in the asymptotic expansion of $\mathrm{Li}(x)$. ...

Question 4

Let $$f_n(x) = {}_3F_2 \left(\left.\begin{matrix} -n, n+2k+2a,k+1 \\ 2k+1,k+a+1+i x\end{matrix}\right| 1 \right),$$ and also $$w(x) = \left| \frac{\Gamma(a+ix)}{\Gamma(a+k+1+ix)} \right|^2.$$ For $k \...

Question 5

I have been having trouble with the definitions of the inverse Laplace Transform (LT). LT being: $$F\left(s\right)=\intop_{0}^{\infty}f\left(t\right)e^{-st}dt$$ And the Inverse transform: $$f\left(t\...

Question 6

I am trying to work out the following exercise on the Mellin's tranformation: “Let $f:[0,+\infty)\to \mathbb{R}$ be a function such that $f(x)x^{\sigma-1}\in L^1([0,+\infty))$ for $\sigma\in [a,b]$. ...

Question 7

When looking into the derivation of Perron's formula, I found that it seems to come from using the inverse Mellin transform of the equation $$\mathcal{M}_x\left[\sum_{n \le x} a_n\right](s) = \frac{1}{...

Question 8

In this article, the condition for the Mellin inversion theorem (MIT) is that the inverse Mellin transform $$ f(x) = \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty} F(s) x^{-s} \, ds $$ converges ...

Question 9

Define shifted Legendre polynomials with Gauss hypergeometric representation, $$ P_n(2x-1)=(-1)^n\,_2F_1(-n,n+1;1;x). $$ It can be shown that, for $f\in L(0,1)^2$, such that $$ f(x)\sim\sum_{n=0}^{\...

Question 10

The Mellin transform of $(1+t)^{-1}$ is given by $$ \int_0^\infty \frac{t^{s-1}}{1+t} \, d t, $$ whenever this integral exists. I have used contour integration to evaluate the integral, but I have ...

Question 11

I have a CDF F(x) which is non-differentiable. However, I have managed to write its Mellin transform as a meromorphic function, with simple poles at $s_m$, $m \in \mathbb{Z}$. I get something like; $$ ...

Question 12

Imagine that I want to evaluate the Mellin transform of the function $f$ defined by: $$ \exp(-x)= \sum_{n=1}^{\infty}f\left(\frac{x}{n}\right). $$ Now if I take the Mellin transform to both sides I am ...

Question 13

The (unitary, continuous) characters of $(\mathbb{R}_{> 0},\cdot)$ are given by $\{ \chi_{x} \colon t \mapsto t^{ix} \mid x \in \mathbb{R} \}$, with group structure given by addition. So $\widehat{(...

Question 14

I am solving the following eigenfunction problem for Fredholm operator: $$ \lambda f(x) = \int_0^{\infty} ds \, x^{5/3} s^{-8/3} \sin^2 \pi \left(\frac{s}{x}\right)^2 f(s). $$ Particular form of the ...

Question 15

I originally discovered the formula $$f(x)=\mathcal{M}_s^{-1}[\omega(s)](x)=\int\limits_{\alpha -i \infty}^{\alpha +i \infty} \omega(s)\, x^{-s}\, ds\,,\quad\alpha>0\\=2 x \sum\limits_{n=0}^{\infty}...

Stack Exchange Network

Questions tagged [mellin-transform]

Inverse Mellin Transform of $\Gamma(s)$

Why is the Reciprocal Log Transform so "un-creative"? Why does it seem to "interpolate" between Fourier, Mellin and Laplace?

Can we get modest bounds on $\pi(x)$ by inverting $F_M(s)$?

A hypergeometric orthogonal rational function

Inverse Laplace Transform and the ROC

Exercise on Mellin's tranform

Proof that $\mathcal{M}_x\left[\displaystyle\sum_{n \le x} a_n\right](s) = \displaystyle\sum_{n=1}^\infty \frac{a_n}{n^s}$?

Sufficient conditions for the Fourier transform to be integrable

Mellin Inversions of Terminate Multiple Hypergeometric Summations

How to show that the Mellin transform of $(1+t)^{-1}$ diverges if $\Re (s) \leq 0$ or $\Re (s) \geq 1$?

Inverse Mellin Transform with Singular Measure

Mellin transform calculation of a function

How is the Mellin transform the Fourier transform for $(\mathbb{R}_{> 0},\cdot)$?

Fredholm operator eigenfunctions: functional equation and generalized functions

Question on closed form for $f(x)=2 x \sum\limits_{n=0}^{\infty} (-1)^n\, (2 n+1)\, e^{-\frac{\pi}{4} (2 n+1)^2 x^2}$

Hot Network Questions