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Questions tagged [mellin-transform]

The Mellin transform is an integral transform similar to Laplace and Fourier transforms.

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Mellin transform of $e^{iat}$

When doing the change of variables $v=-iat$, shouldn't the limits be reversed? Or is it because its the same as $v=\frac{at}{i}$ I cant see why this is not the case
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Question about Dirichlet Series Related to Formula for $\frac{1}{e}$

This question is related to the three functions defined in (1) to (3) below where $\coth(z)$ gives the hyperbolic cotangent of $z$. (1) $\quad M(x)=\sum\limits_{n=1}^x\mu(n)\quad\text{(Mertens ...
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61 views

What is the Dirichlet Transform of $a(n)=\frac{\mu(n)}{n^2}\,\log\left(\frac{2\,\pi}{n}\right)$?

This question is related to my previous question at the following link. Do the Prime Number Theorem and/or Riemann Hypothesis predict a limit on the accuracy of this formula for $\gamma$? This ...
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Mellin Transform $M[Re (f(re^{i\theta}));s] = cos(s\theta)M[f(r);s]$

I want to derive the equations \begin{align} M\left[\mathrm{Re}\, (f(re^{i\theta}));s\right] &= \phantom{-}\cos(s\theta)M\left[f(r);s\right]\\ M\left[\mathrm{Im} \, (f(re^{i\theta}));s\right] &...
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Mellin Transform $M[\log(x)f(x)] = \frac{d}{ds}F(s)$

My solutions say: $$ \frac{\mathrm d}{\mathrm ds}F(s) = \frac{\mathrm {d}}{\mathrm {d}s}\int_0^\infty \! x^{s-1}f(x) \, \mathrm {d}x = \int_0^\infty \! \log(x)x^{s-1}f(x)\,\mathrm {d}x = M\left[(\log(...
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Is $\int_0^1 \Psi(x)\Psi(1-x)\,dx$ related to any transform?

Is this related to any integral transform? $$\int_0^1 \Psi(x)\Psi(1-x)\,dx=\int_{0}^{1} e^{{\frac{1}{\log(x)}}+{\frac{1}{\log(1-x)}}} \, dx.$$ The integral, where $K$ is the modified Bessel function ...
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Mellin Transform problem

The question is : if \begin{equation} \mathcal{M}[F(x)]=f(p) \end{equation} find \begin{equation} \mathcal{M}[ln(x)\cdot x^3\cdot \frac{d^2}{dx^2} F(2x^3)] \end{equation} atleast where can i start??...
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Mellin transform involving $\sinh({A_1}/2)$

So I need to figure out how to take the Mellin transform of $$ f(x)=\int_2^x \sin(A_1/2)+\sinh(A_1/2)dt,$$ where $A_1=1/\ln(t).$ I'd also like to know how well the Mellin transform of $f(x)$ ...
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Existence of classical Dirichlet series from Lebesgue integrable inverse Mellin transform

Let $f(s)$ be meromorphic in $\mathbb{C}$. Let the following inverse Mellin transform be Lebesgue integrable for all real positive $x$ at some complex point $s$ with some real $c$: $\frac{1}{2\pi i} \...
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Mellin inverse of $\sum_{n=0}^{\infty}\frac{\Gamma(n+s)\zeta(n+s)\zeta(n+1+s)}{\zeta(s)n!}\left(-\omega\right)^{n}$

I am trying to compute the inverse Mellin transform of : $$\sum_{n=0}^{\infty}\frac{\Gamma(n+s)\zeta(n+s)\zeta(n+1+s)}{\zeta(s)n!}\left(-\omega\right)^{n}$$ w.r.t. the complex number $s$. $\omega$ ...
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Question related to derived formula for zeta-zero counting function

This question assumes the definitions of the Mellin and Fourier transforms illusrated in (1) and (2) below and the corresponding relationship illustrated in (3) below. (1) $\quad\mathcal{M}_x[f(x)](s)...
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Proving $\mathcal{M}\left(\sin(x)\right)(s) = \Gamma(s)\sin\left(\frac{\pi}{2}s \right)$ using Real Analysis

recently I've been investigating Mellin Transforms and this morning solved for case of $\sin(x)$ using Ramunajan's Master Theorem. I was curious if there were any Real based methods to evaluate this ...
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Is there a preference if one of the functions in convolution of Mellin transform is divergent?

The convolution of Mellin transform is $$ \sigma \left( x \right) = \int _x^1 f \left( \epsilon \right) h \left( \frac{x}{\epsilon} \right) \frac{1}{\epsilon} \mathrm{d} \epsilon , $$ if both $f \...
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Questions related to the Dirichlet series for $\frac{\zeta'(s)}{\zeta(s)^2}$

This question is related to the following two functions evaluated with the coefficient function $a(n)=\mu(n)\log(n)$. (1) $\quad f(x)=\sum\limits_{n=1}^x a(n)$ (2) $\quad\frac{\zeta'(s)}{\zeta(s)^2}=...
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What is the relationship between simple prime-power counting function and $\log\zeta(s)$?

This question assumes the following definitions of prime-counting functions where $p$ denotes a prime number, $k$ denotes a positive integer, and $\theta(y)$ is the Heaviside step function which takes ...
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2answers
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Methods on finding closed form of $\int_0^\infty x^{s-1}e^{-a(x+\frac{1}{x})}dx$

I was playing around with Mellin Transforms and encountered this interesting integral$$I(s,a)=\int_0^\infty x^{s-1}e^{-a(x+\frac{1}{x})}dx$$ Since $f(x)=e^{-a(x+\frac{1}{x})}=f(1/x)$, I found that ...
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What is the Explicit Formula for $\log(\zeta(s))$?

The explicit formula for the logarithmic derivative $\frac{\zeta'(s)}{\zeta(s)}$ is illustrated in (1) below. (1) $\quad\frac{\zeta'(s)}{\zeta(s)}=\frac{s}{1-s}+\log(2\,\pi)-\frac{1}{2}H_{\frac{s}{2}}...
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Fourier and Mellin transforms of Hilbert Transform

I am reading Hilbert transform recently and meet two questions. The book I am reading is Debnath and Bhatta "Integral Transforms and Their Applications". If we define the Hilbert transform on the ...
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Question on Probability Distributions Related to the Riemann Xi Function $\xi(s)$

This question is related to the $g_i(x)$ functions which are defined below both in expanded form and in terms of the $f_i(x)$ functions defined in my previous question at ref(1) where all $f_i(x)$ ...
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1answer
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Polynomial growth of L-function

Let $f$ be a newform, $L(f,s)$ the related L-function with Ramanujan-Petersson conjunction $|\lambda(n)|\leq \sigma_0(n)$ (divisor counting function). How can I see that it grows only polynomially in ...
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Uniqueness of $L$-series of cusp forms

For a cusp form $f$, one gets an $L$-series by taking the Mellin transform as we have $$ \tilde{f}(s) = (2\pi)^{-s} \Gamma(s) L(s,f). $$ My question is: is this operation injective? It seems to me ...
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Questions related to the Riemann Xi function $\xi(s)$ and Jacobi theta functions $\vartheta_3(0,q)$

This question assumes the following definitions. (1) $\quad\psi(x)=\sum\limits_{n=1}^\infty e^{-\pi\,n^2\,x}=\frac{1}{2} \left(\vartheta_3\left(0,e^{-\pi\,x}\right)-1\right)$ (2) $\quad f(x)=\sum\...
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multiple harmonic sum with arbitrary starting index

I am considering the following multiple harmonic sum: $T(N,p,\ell)=\underbrace{\sum_{n_1=\ell}^N\sum_{n_2=n_1}^N \cdots \sum_{n_p=n_{p-1}}^N}_{p-\text{sums}} \frac{1}{n_1 n_2 \cdots n_p}$ Using the ...
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Questions related to $f(x)$ where the Riemann Xi function $\xi(s)=s\int\limits_0^\infty f(x)\,x^{-s-1}\,dx$

I realize this question is a bit long and contains quite a few formulas, but I believe a considerable amount of background and context are needed to fully understand my questions below. Also, I ...
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What is the convergence of the explicit formula for $\frac{\zeta'(s)}{\zeta(s)}$?

The explicit formula for $\frac{\zeta'(s)}{\zeta(s)}$ illustrated in (3) below was derived from the relationship illustrated in (1) below using the explicit formula for $\psi(x)$ defined in (2) below. ...
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1answer
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isometry relating $L^2(\mathbb{R}, dx)$ and $L^2([0, \infty), \frac{dx}{x})$

What is the basis for functions on the Hilbert space $L^2([0,\infty), \frac{dx}{x})$. I am studying the Mellin transform and I'm trying to understand the role of the functions $n^{it} = e^{it \, \log ...
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“Bilateral Mellin convolution”

The Mellin convolution of two functions, when it exists, is of the form $$ (f \ast_M g)(t) = \int_0^\infty f\left( \frac{t}{\tau} \right) g(\tau) \frac{\mathrm{d}\tau}{\tau} $$ and ...
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Mellin transform - change of variables in double integral

Together with a friend, I am trying to derive the Lamperti distribution from a ratio of stable random variables, and got stuck in a part of the proof which involves a substitution in a double integral ...
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What are the easiest inverse Mellin transforms?

I'm looking to familiarize myself with the inverse Mellin transform \begin{align*} \mathcal{M}^{-1}[\varphi](t) = \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty} t^{-s}\varphi(s)\, \mathrm{d}s \end{...
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Is this expression of the Riemann zeta function as a Mellin transform true for $\Im\{s\}\neq 0$?

For sure, we can write \begin{align} \zeta(s)&=\int_0^\infty \sum_{n\geq 1}\delta\big(\log \frac{x}{n}\big)x^{-s-1}dx\\ &=\int_0^\infty \sum_{n\geq 1}\delta(x-n)x^{-s}dx \nonumber \\ &=\...
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2answers
193 views

Mellin transform of a Gaussian Hypergeometric Function with negative $x$-argument

I am quite fascinated by the formula for the Mellin transform of the Gaussian Hypergeometric Function, which is given by: $$\mathcal M [_2F_1(\alpha,\beta;\gamma;-x)] = \frac {B(s,\alpha-s)B(s,\...
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Generalization of hypergeometric type differential equation

I am aware that hypergeometric type differential equations of the type: can be solved e.g. by means of Mellin transforms when σ(s) is at most a 2nd-degree polynomial and τ(s) is at most 1st-degree, ...
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Proof that Integral over the arc vanishes as $R\rightarrow\infty$ in Inverse Mellin Transform of $\Gamma(s)$

It´s a very well know result that $$e^{-x}=\frac{1}{2\pi i}\int_{\gamma-i\infty}^{\gamma+i\infty} \Gamma(s)x^{-s}ds$$ In order to solve this integral we have to close the contour to the left and show ...
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Gamma Identities from Inverse Transform

I have been working on a transform (the notes are a little rough and the hosting website has some fraction formatting issues recently) but the idea is there: Basically, the transform $\mathcal{I}_x[f(...
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Mellin Transform

If $$ H(x)=\sum_{n \leq x} \frac{1}{n} $$ what is its Mellin transform? I was able to find the Mellin transform of $\log(x+1)$ and of $\frac{1}{x+1}$, but I'm quite a bit inexperienced so I haven't ...
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A name for $W_{m,n}=\binom{n}{m} \sum_{k = 0}^{\max (m, n) + 1} \zeta (k) \operatorname{Res} (\hat{y}_{n, k} (s), s = - m)$?

Does this type of series have a name or any references? $$\begin{array}{ll} W_{m, n} & = \binom{n}{m} \sum_{k = 1}^{\infty} \frac{1}{k^{m + 1} (k + 1)^{n + 1}}\\ & = \binom{n}{m} \...
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Mellin transform of polynomials over the unit interval. How to invert?

Let's take the mellin transform of $1 - 2 x$ over the interval $x = 0 \ldots 1$ \begin{equation} \int_0^1 (1 - 2 x) x^{s - 1} d x = - \frac{s - 1}{s (s + 1)} \end{equation} How can we use the mellin ...
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Using the Laplace transform to calculate the distribution function of hitting time.

Let $B$ be a standard Brownian motion, and consider the drift process $X_t=B_t+ct$ for $c\in \mathbb R$. For $x>0$, set $H_x=\inf\{t>0: X_t=x\}$. I have been able to show that this is a stopping ...
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Inverse Mellin Transform of gamma function products

I have been trying to solve for f(x) by doing the Inverse Mellin Transform of F(s) by summing up residues, but I don't think I'm on the right track. $$f(x)=\frac{1}{2i\pi}\int_{c-i\infty}^{c+i\infty}...
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Non-vanishing of K-Bessel function

I don't know much about the spectral theory of $\text{SL}_2(\mathbb{Z}) \backslash \mathbb{H}$ nor do I know much about Bessel functions, hence the following question. Suppose $f$ is Maass form of ...
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Integral with solution in terms of hypergeometric functions

I am trying to solve the following integral; $\int_0^\infty \frac{1-\cos(x)}{x \left(\left(\frac{x}{\Lambda}\right)^2 +1\right)^{\frac{1}{2}}}\,dx = I(\Lambda), \Lambda > 0$ My approach thus far ...
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212 views

Questions on Prime Counting Functions, Explicit Formulas, and Related Zeta Functions

This question is related to the prime-counting functions defined in (1) to (4) below. (1) $\quad A(x)=\sum\limits_{n=2}^\infty\frac{\Lambda(n)}{\log(n)^2}\,\theta(x-n)$ (2) $\quad\Pi(x)=\sum\limits_{...
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Questions on Riemann's Prime-Power Counting Function $\Pi(x)$ and a Related Staircase Function

This question is related to the prime-counting function $\Pi(x)$ and staircase function $Q(x)$ defined in (1) and (2) below respectively. (1) $\quad\Pi(x)=\sum\limits_{n=2}^\infty\frac{\Lambda(n)}{\...
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Can a Mellin / Laplace Transform-like method be done with functions beside $x^{s-1}, e^{-st}$?

Was wondering if using other kernel functions beside these would result in illucidating other types of formulas than what the above two well-known transform methods typically handle. I don't ...
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64 views

Closed form of Integral of ellipticK and log using Mellin transform? $\int_{0}^4 K(1-u^2) \log[1+u z] \frac{du}{u}$

I am trying to evaluate the integral: $\mathcal{I}(z)=\int_{0}^a K(1-u^2) \log[1+u z] \frac{du}{u}$, $\qquad $ ($a$ fixed, $a>0$ and $K$ is the complete elliptic integral of the first kind) in ...
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3answers
213 views

Riemann zeta for real argument between 0 and 1 using Mellin, with short asymptotic expansion

The following would appear to be true. For real $0 < \sigma < 1,$ we seem to have a very satisfying sum minus integral limit, $$ \zeta(\sigma) \; \; = \; \; \lim_{n \rightarrow \infty} \; \;...
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1answer
62 views

Integral inside Mellin Transform?

Hello I have this formula for Mellin Transform : $$\mathcal{M}_x[f(x)](s)=-\frac{\mathcal{M}_x\left[\frac{\partial f(x)}{\partial x}\right](s+1)}{s}$$ but I need formula like so (integral inside ...
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1answer
121 views

Questions on Mellin convolutions involving $g(x)=\log(x)$

The questions below assume the following definitions of the Mellin transform and associated convolutions. (1) $\quad F(s)=\mathcal{M}_x[f(x)](s)=\int\limits_0^\infty f(x)\,x^{s-1}dx$ (2) $\quad f(x)*...
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2answers
305 views

Questions on Mellin Transform of $x^j$ and Interpretation of Distributions with Complex Arguments

The Mellin transform/inverse transform pair are defined as follows: (1) $\quad F(s)=\mathcal{M}_x[f(x)](s)=\int\limits_0^\infty f(x)\,x^{s-1}\,dx$ (2) $\quad f(x)=\mathcal{M}_s^{-1}[F(s)](x)=\frac{1}...
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86 views

Floor function as an inverse Mellin transform of Riemann zeta function

We have $$\lfloor x \rfloor=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\zeta(s)\frac{x^{s}}{s}ds\;\;\;(c>1)$$ If I apply residue theorem I get: $$\text{Res}\left(\frac{x^s \zeta (s)}{s},0\right)=-\...