Questions tagged [mellin-transform]

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

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What is the relationship between the Mellin transforms of $h(x)$, $f(x)$ and $g(x)$ where $h(y)=\int\limits_0^\infty f(x)\ g(y-(x-1))\,dx$

The Mellin transform is defined in (1) below and the standard and alternate Mellin convolutions are defined in (2) and (3) below respectively. (1) $\quad\mathcal{M}[f(x)](s)=\int\limits_0^\infty f(x)\...
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40 views

Roots of trancendental equations and their relation to Dirichlet series and Mellin transforms

Consider a function of the form $F(x)=x^{\alpha}f(\ln(x))$, with $0<\alpha<1$ and $c_1<f(\ln(x))<c_2$ for some positive constants $c_1,c_2$, such that $F(x)$ is strictly increasing. ...
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14 views

What is the Schwartz-type space for Mellin transform?

It is well known that for $f\in S(\mathbb R)$, the Schwartz space, one can assert that $f^{(a)}$, $Ff^{(a)}$ (the Fourier transform of $f^{(a)}$) are also in $S(\mathbb R)$ for any $a=0,1,2,\ldots$. ...
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Is there a transform based on Mellin convolution analogous to the Hilbert transform which is based on Fourier convolution?

Fourier convolution is defined as follows. (1) $\quad\left(f(x)\ *_\mathcal{F}\ g(x)\right)(y)=\int\limits_{-\infty}^\infty f(x)\ g(y-x)\ dx$ The Hilbert transform of $f(x)$ defined in (2) below is ...
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53 views

Is this formula related to the Dirac delta function $\delta(s)$ nearly globally convergent?

Formula (2) below for $f(s)$ was derived from formula (1) below for $s$ which is proven in this answer to this question on Math Overflow. Formula (2) was derived from formula (1) by evaluating the ...
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1answer
31 views

Mellin transform and Bessel function

In this article, the following identities are stated without proof: $$\int_0^\infty x^{s - 1}K_0(4\pi x^{1/2}) dx = \frac{1}{2}(2\pi)^{-s}\Gamma(s)^2$$$$\int_0^\infty x^{s - 1}Y_0(4\pi x^{1/2}) dx = -\...
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How can we prove that $\int_0^\infty e^{-ix}x^{s-1} \ \mathrm{d}x = i^{-s}\Gamma(s)?$ [duplicate]

I have seen $$\int_0^\infty e^{-ix}x^{s-1} \ \mathrm{d}x = i^{-s}\Gamma(s)$$ in a few posts regarding Mellin tranforms or a few difficult integrals. How can we prove this equality? I assume it can ...
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154 views

Help with the Euler-type integral $\int_{0}^{m}\frac{1-e^{2\pi i x}}{x-j}\frac{x^{s-1}}{(1+x)^{z}}dx$

Consider the integral : $$I=\int_{0}^{m}\frac{1-e^{2\pi i x}}{x-j}\frac{x^{s-1}}{(1+x)^{z}}dx\;\;\;\;s,z\in\mathbb{C}\;\;\;\;j,m \in \mathbb{N}\;\;0\leq j\leq m$$ i have tried using the Mellin ...
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Is there a broader definition of the convolution operation?

The convolution operator is defined as $${\displaystyle (f*g)(t)\triangleq \ \int _{-\infty }^{\infty }f(\tau )g(t-\tau )\,d\tau .} $$ where it shares a relationship with the Laplace transform such ...
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Dirac delta as a “mixture” of half-normal distributions

Let's say I have a function $f$ which I want to represent as a "mixture" of half-normal distributions. It's not the mixture exactly because we allow negative "mixture density". To put it another way, ...
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1answer
67 views

Reference request: why integral transforms?

I understand there have been previous questions on why/how exactly integral transforms arise, but I am here asking specifically for reference requests to sources other than full-length books. My ...
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Question related to nested Fourier series representation of $h(s)=\frac{i s}{s^2-1}$

In this question I use the term "nested Fourier series representation" to refer to an infinite series of Fourier series versus a single Fourier series. Whereas a single Fourier series is periodic, an ...
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29 views

How does one invert a p-adic mellin transform, and what does it say about function asymptotics?

In ordinary analysis, given a sufficiently nice $f:\left[0,\infty\right)\rightarrow\mathbb{C}$, if we can compute the Mellin transform: $$\mathscr{M}\left\{ f\right\} \left(s\right)=\int_{0}^{\infty}x^...
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Mellin/Bromwich Integral (Inverse Laplace Transform) problem

I have a solution in laplace images: \begin{align} &p_f(x,s) = - \frac{1}{s}\frac{b}{a} \frac{1}{\sqrt{\frac{s}{a}+ \frac{b}{a} \frac{\sqrt{ s}}{1+c\sqrt{s}}}} e^{-x \sqrt{\frac{s}{a}+ \frac{b}{...
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Asymptotic expansion of the Volterra integral equation with series ansatz

Problem I have a problem which can be boiled down to the Volterra integral equation $$ \begin{aligned} w(\eta) &\sim \eta \int _0^\infty K(s)F(\eta s)ds \end{aligned} $$ for $\alpha \in (\frac ...
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Is there a closed form expression of the parallel DGLAP evolution equation?

The parallel DGLAP evolution equation is given by $$t\frac{\partial f(x,t)}{\partial t} = \frac{\alpha_s(t)}{2\pi} \int_x^1 \frac{d\hat{x}}{\hat{x}} P_{qq}(\hat{x})f\left(\frac{x}{\hat{x}},t\right)$$ ...
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1answer
62 views

How to get Mellin transform of the given function?

Could anyone help me how to find Mellin transform of the following function $$f(x)=\frac{1}{(x^a-A)^3},$$ i.e. the integral $$\int_0^{+\infty} \frac{x^{s-1}\,{\rm d}x}{(x^a-A)^3}$$ where $a,A>0\,.$
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Is there a valid explicit formula for $f(x)=\sum\limits_{n=1}^x \frac{1}{n}\sum\limits_{d|n} \mu(d)\,d$?

This question is related to the function $f(x)$ defined in (1) below where A023900(n) is the Dirichlet inverse of Euler totient function $\phi(n)$. I believe the related Dirichlet series illustrated ...
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An inverse Mellin transform with Gauss hypergeometric function

buddies, I meet an inverse Mellin transform problem with Gauss hypergeometric function in term of $z$ as follows: $ \mathcal M^{-1}\left\{ {\beta ^{ - z}}{\Gamma ^a}\left( z \right){\left[ {{}_2{F_1}...
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Mellin Transform of $h(x-a)$

The question is from Integral Transform For Engineers By Andrews and Shivamoggi. Evaluate the Mellin transform of the given function. When possible, use known integral results from previous ...
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112 views

Use of Mellin transform for evaluation of a series

Show that $$\sum_{n=1}^\infty \frac{\sin an}{n}=\frac{\pi-a}{2} \ , \ 0<a<2\pi$$ I was asked to use Mellin transform to prove this result. So I used a formula related to the general series as ...
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Questions on formulas for the reciprocal gamma function $\frac{1}{\Gamma(s)}$

This question is related to the following formulas for the reciprocal gamma function $\frac{1}{\Gamma(s)}$ where formula (2) represents the analytic continuation of the sum over $k$ in formula (1). ...
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105 views

Questions on convergence of formula for $\zeta(s)$

This question assumes definition (1) below and relationship (2) below. With respect to the integral in (2) below, I selected $\frac{1}{2}$ as the lower integration bound because this is the ideal ...
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integral equation and series

let be the integral equation $$ f(s)= \int_{0}^{\infty}\frac{g(t)}{(s+t)}$$ here 's' is a non negative real number i know that if $ f(s)= \sum_{n=0}^{\infty}\frac{m(n)}{s^{n+1}} $ then we have ...
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What is $s\int_1^\infty\sin(2\,\pi\,n\,x)\,x^{-s-1}\,dx$?

This response to my question Are these formulas for the Riemann zeta function $\zeta(s)$ globally convergent? didn't answer my question, but rather proposed an alternate approach which was intended ...
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functions growing faster than Gamma function

is there a Mellin transform of a certain function $ g(t) $ so $ M(n)= \int_{0}^{\infty}t^{n} g(t)dt $ so for integers a,b,d,n and real positive constant C we have that $$ (\Gamma (na+b))^{d} \le C ...
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Computing Mellin transform/linear combination of incomplete gamma values

Consider the function $f(t) = e^{\pi t} (8 \gamma(it+2,-2)+12 \gamma(it +3,-2) + 6 \gamma(it+4,-2) + \gamma(it+5,-2))$. Not surprisingly, the way it's written, it's an invitation to numerical ...
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Higher-order asymptotic of Mellin transform

Consider a function $F(t)$ such that for some constant $C$ we have $$ F(t) \sim C t^b \qquad t\rightarrow \infty $$ The asymptotic behavior of it's Mellin transform can be deduced directly by ...
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1answer
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What is the definition of $Li(x)$ in these MathOverflow questions on $I_s =\int_{1}^{\infty} (\pi(x)-Li(x))\,x^{-s-1} dx$?

Question: What is the definition of $Li(x)$ in the MathOverflow questions linked below and their related answers? $Li(x)=\int\limits_0^x\frac{1}{\log(t)}\,dt$, $Li(x)=\int\limits_1^x\frac{1}{\log(t)}\,...
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1answer
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Is there a Mellin transform or an analogue on $L^2([0,2\pi])$ or $\ell^2(\mathbb{Z})$?

From Wikipedia, the Mellin transform is an isometry $M : L^2(\mathbb{R}^+) \mapsto L^2(\mathbb{R})$, $$\{M f\} (s) := \frac{1}{\sqrt{2\pi}} \int_{\mathbb{R}^+} x^{-1/2 + \mathrm{i} s} f(x) dx.$$ ...
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1answer
116 views

Find the inverse of the integral transform $ S_n\{f\}=\int_{0}^1f(x)x^{n}\,dx$

Let $S$ denote the integral transform that maps every function $f$ integrable on (0,1) to a sequence $\left(S_n\{f\}\right)_{n\in\mathbb{N}}$: $$S:f\mapsto \left(S_n\{f\}\right)_{n\in\mathbb{N}}\text{ ...
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1answer
66 views

Is there “multiplicative” analogue of Fourier transform?

Let $f(x)$ switch between $-1$ and $1$ as together the number of digits after the decimal point is odd or even. Let's see: $$ f(x) = \left\{ \begin{array}{r|rcccr} 1 & 4^m &< &x&...
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1answer
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How Does Ramanujan’s Master Theorem Work?

I am trying to understand Ramanujan' Master theorem, but I really can't seem to wrap my head around it. So the theorem states that if you have a function $F$ such that you can write it as $$F(x)=\...
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Questions about the Dirichlet series $K(s)=\sum\limits_{p^k} p^{\,-k\,s}$

This question is related to Riemann's prime-power counting function $J(x)$, the fundamental prime-counting function $\pi(x)$, and the simple prime-power counting function $k(x)$ defined below where $p\...
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1answer
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Residue of function with $\frac{1}{\ln(z^{-2})}$ and $z^{-2}$ dependence

In my current research, I have come across an integral that is refusing to submit to any method I can think of for evaluation. It is an integral that comes from an operator product expansion of gluon ...
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Analytic Continuation of Mellin Transforms and the Prime Zeta Function

Lemma 1.1.1. on page 4 of these notes sates that for a $\mathcal{C^{\infty}}$-smooth function $f:\mathbb{R}_{+}\rightarrow\mathbb{C}$ such that $\lim_{x\rightarrow\infty}x^{n}f\left(x\right)=0$ for ...
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Does the notion of Mellin transform make sense for complex functions as well?

In most of the sources I looked regarding the Mellin transform, they say let $f$ be a function defined on the positive real axis $0 < t < \infty$. And then say the Mellin transform ...
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1answer
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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 [![enter image description here][...
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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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1answer
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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 $M[\log(x)f(x)] = \frac{d}{ds}F(s)$

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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51 views

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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1answer
31 views

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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1answer
164 views

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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2answers
127 views

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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1answer
141 views

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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1answer
57 views

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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159 views

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}=...