# Questions tagged [mellin-transform]

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

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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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### 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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### 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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### 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][...
### 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}=...