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The Mellin and Inverse Mellin transforms mellin and invmellin are part of the inttrans package. The Mellin transform is closely related to the Laplace and Fourier transforms and has applications in many areas, including:

digital data structures

probabilistic algorithms

asymptotics of Gamma-related functions

coefficients of Dirichlet series

asymptotic estimation of integral forms

asymptotic analysis of algorithms

communication theory

The Mellin transform, as a function $M\left(s\right)$ of $s$, of a function $m\left(x\right)$ of $x$, is defined by the integral

$M\left(s\right)\={\int }_{-0}^{\infty }m\left(x\right)x^{s-1}\ⅆx$

The Inverse Mellin transform is defined by the contour integral

$F\left(s\right)\=\frac{1\mathrm{\pi }I\left({\int }_{-Ic\+\infty }^{Ic\+\infty }f\left(t\right)t^{-s}\ⅆt\right)}{2}$

for a function $f\left(t\right)$ of $t$.

Here are a few examples of invmellin, the inverse Mellin transform, in action.

Try an assumption on a:

Try changing the range:

In the above, we see that the correct assumptions on parameters and the correct range must be specified for the inverse Mellin transform.

Continuing with another example:

Check to see that the Mellin transform of this is our original expression:

The following is an example of a Mellin transform which does not simplify:

We try taking the inverse Mellin transform of this, with the valid range, and check to see that we get the original function:

The mellin and invmellin functions can also handle the Whittaker functions:

Try some general formulae: