Questions tagged [mellin-transform]
The Mellin transform is an integral transform similar to Laplace and Fourier transforms.
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Writing $\ln(x+1)e^{-ax}$ in terms of Meijer-G function
Is there any way to write $f(x)=\ln(x+1)e^{-ax}$ in terms of Meijer-G function? I tried calculating Mellin transform of $f(x)$ to no avail. Frustrated, I used Mathematica to get the following answer
$$...
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Deriving the value of the series $\sum _{n=1}^{\infty }\left[\frac{2n}{e^{2\pi n}+1}+\frac{2n-1}{e^{\left(2n-1\right)\pi }-1}\right]$
I have been working on trying to show that
$\displaystyle \sum _{n=1}^{\infty }\left[\frac{2n}{e^{2\pi n}+1}+\frac{2n-1}{e^{\left(2n-1\right)\pi }-1}\right]=\frac{\varpi ^2}{4\pi ^2}-\frac{1}{8}$
...
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Mellin transform of exponential and logarithm
I am trying to calculate the Mellin transform of the function $f(x) = e^{-ax}\ln\left(1+x\right)$. The mathematica gives me the answer
$$\int_{0}^{\infty}x^{s-1}f(x)dx = \frac{G_{3,5}^{5,2}\left(\frac{...
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Question regarding terms with negative coefficients in multivariate Fox H function
I was studying a paper where an integral expression in terms of Fox H function of multiple variable were used. The definition of multivariate Fox H (extracted from appendix A-1 of Mathai-Saxena) is as ...
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Mellin transform of confluent Lauricella hypergeometric function
The $F_D^{(n)} $ Lauricella's hypergeometric function can be defined as follow
$$F_D\left(a,b_1,\cdots,b_n;c;x_1,\cdots,x_n\right) = \sum_{m_1=0,\cdots,m_n=0}^{\infty}\frac{\left(a\right)_{m_1+\cdots+...
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2 dimensional Fourier transform of $\frac{x}{(x^2+y^2+c^2)^{3/2}}$
I'm trying to calculate the 2D Fourier transform of this function:
$$\frac{x}{(x^2+y^2+c^2)^{3/2}}$$,
where $x$ and $y$ are independent variables and $c$ is a positive constant,
I know the answer ...
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What is the inverse Mellin transform of $\Gamma(1+i s)$?
Mathematica and WolframAlpha both indicate
$$
\mathcal{M}_{s}^{-1}\left[\Gamma\left(1 + {\rm i}s\right)\right]\left(x\right) =
-\,{\rm i}
\operatorname{G}_{\,0,1}^{\,1,0}\,\left(x,{\rm i}\,
\left\vert\...
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Mellin-Barnes representation of the ratio of first kind Bessel functions
I have a question regarding the Mellin-Barnes representation of Bessel functions of the first kind. I know that the product of these functions admits the following integral representation
$J_{\mu}(x)...
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Condition for series expansion of a parametric integral
Suppose you have a function $h:(0,1)\times (0,1)\to \mathbb{R}$ and a parametric integral of the form:
$$
I(x)=\int_x^1 h(x,y)\,\mathrm{d}y
$$
Question: What would be the conditions on $h$ so that the ...
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Inverse Mellin transform of $\Delta(t):=\sum_{s=1}^\infty d(s)\sqrt{\frac{t}{s}}M_1(4\pi \sqrt{ts})$
Define
$$ \Delta(t):=\sum_{s=1}^\infty d(s)\sqrt{\frac{t}{s}}M_1(4\pi \sqrt{ts}) $$
where $M_1(z)=-Y_1{(z)}-\frac{2}{\pi}K_1(z)$ and $d(s)$ is the divisor function.
What is the inverse Mellin ...
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Problem with Mellin transform of convolution production formula
I have came across a case where the standard formula for the Mellin transform of a convolution of two functions is not the product of the Mellin transforms and I don't understand exactly where the ...
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Sum of Meijer G-functions with index in the arguments
I have a hunch that the following sum can be expressed as a single $G$-function:
$$
S(t,y) = \sum_{k=0}^\infty \frac{t^{2k}}{(2k)!} G_{p+2,q+2}^{m+2,n}\left(y\left|\begin{smallmatrix}\vec{a}-k,-k,-k-\...
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Mistake computing $\sum_{n=1}^{+\infty} \frac{n}{e^{2\pi n}-1} = \frac{1}{24}-\frac{1}{8\pi}$
I recently gave a try to show that
$$\sum_{n=1}^{+\infty} \frac{n}{e^{2\pi n}-1}=\frac{1}{24}-\frac{1}{8\pi} $$
without using the Theta function or Mellin transform, but I ended up with twice the ...
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Practical integration along vertical lines
I want to ask a practical integration and I hope the answer could be a line by line form to for me to understand.
Consider complex integration. Assume the integration is consist of a vertical line $K: ...
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Inverse Mellin transform of the Mellin transform of the binomial PMF
The probability mass function of the Binomial distribution is given by
$$\begin{equation}
f(x)=\binom{n}{x} p^x (1-p)^{n-x},
\end{equation}$$
where $p \in [0,1]$ and $x=\{0,1,\dots,n\}$ (finite ...
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