The Mellin transform is an integral transform similar to Laplace and Fourier transforms.
The Mellin transform of a function $f(x)$ is defined as:
$$ { [\mathcal{M}f]} (s)=\int_0^{\infty}x^{s-1}f(x)dx $$
Questions tagged [mellin-transform]
260 questions
9
votes
1 answer
Mellin convolution and Mellin transform
How can I prove that the Mellin transform of the function defined by
$$ \int_{0}^{\infty}K(xy)f(y)dy $$
is equal to the product $ K(s)F(1-s)$
and that the Mellin transform of $$ \int_{0}^{\infty}K(x/y)f(y)dy/y $$
is just the product of $ K(s)F(s)…
Jose Garcia
- 8,506
2
votes
1 answer
Inverse Mellin transform of $\Gamma(s)$ $\zeta(s/2)$
How would you find an approximation of the Inverse Mellin transform of $\Gamma(s)$ $\zeta(s/2)$ near $x=0$?
sigma
- 191
1
vote
1 answer
Specific instance of Mellin Inversion
In my number theory notes, I have the following integral formula
$$
\frac{1}{2\pi i}\int_{(2)}\frac{t^s}{s(s+1)...(s+r)}ds=\begin{cases}
\frac{1}{r!}(1-\frac{1}{t})^r & t\geq 1\\
0 & 0
Steven Creech
- 2,243
1
vote
1 answer
problem with a Mellin tranform
i have a problem with 2 different functions that give the similar mellin transform, for example let be the floor function $ [x] $
then $ \int_{0}^{\infty}dt t^{s-1}[1/x]= \frac{\zeta(s)}{s} $
but also $ \int_{0}^{\infty}dt t^{s-1}frac(1/x)=…
Jose Garcia
- 8,506
0
votes
1 answer
Mellin Transform problem
The question is : if \begin{equation}
\mathcal{M}[F(x)]=f(p)
\end{equation}
find \begin{equation}
\mathcal{M}[ln(x)\cdot x^3\cdot \frac{d^2}{dx^2} F(2x^3)]
\end{equation}
atleast where can i start??
m7waath
- 1
0
votes
2 answers
mellin transform of a function related to the derivative of Riemann zeta function
if we know that $$ \frac{\zeta(s)}{s}=\int_{0}^{\infty}dx \frac{[x]}{x^{s+1}}$$
then how could we get the function so
$$ \frac{d\zeta(s)}{ds}=s\int_{0}^{\infty}dx \frac{f(x)}{x^{s+1}}$$
the function whose mellin transform is the derivative of the…
Jose Garcia
- 8,506