# Questions tagged [mellin-transform]

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

125 questions
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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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### 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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### 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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### 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 ...
### 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 ...