The function mellin computes the Mellin transform (M(s)) of expr (m(x)) with respect to x, using the definition

Some expressions involving exponentials, polynomials, algebraic functions, trigonometrics (sin, cos, $\sinh$, cosh) or various special functions can be transformed.  The procedure will be able to obtain the Mellin transforms of all the functions of the type $K{\ln \left(x\right)}^{n}f\left(a{x}^b\right)x^c$ as long as the Mellin transform of $f\left(x\right)$ is known.

The mellin function attempts to reduce the expression according to a set of simplification rules and then tries to match the reduced expression against an internal table of basic Mellin transforms.

Users can add their own functions to mellin's internal lookup table by using the addtable function.

If the option opt is set to 'NO_INT', then the program will not resort to integration of the original problem if all other methods fail.  This will increase the speed at which the transform will run.

The command with(inttrans,mellin) allows the use of the abbreviated form of this command.

$\mathrm{with}\left(\mathrm{inttrans}\right)\:$

$\mathrm{assume}\left(0<s\right)\&colon;$

$\mathrm{mellin}\left(a{x}^b\exp \left(-x^{\frac{1}{4}}\right)\&comma;x\&comma;s\right)$

$4a\mathrm{\Gamma }\left(4\mathrm{s~}+4b\right)$

$\mathrm{mellin}\left(\frac{x}{x^2+1}\&comma;x\&comma;s\right)$

$\frac{\mathrm{\pi }}{2\sin \left(\frac{\mathrm{\pi }\left(\mathrm{s~}+1\right)}{2}\right)}$

$\mathrm{mellin}\left(\frac{\ln \left(x\right)x}{x^2+1}\&comma;x\&comma;s-2\right)$

$-\frac{{\mathrm{\pi }}^2\cos \left(\frac{\mathrm{\pi }\left(\mathrm{s~}-1\right)}{2}\right)}{4{\sin \left(\frac{\mathrm{\pi }\left(\mathrm{s~}-1\right)}{2}\right)}^2}$

$\mathrm{mellin}\left(\frac{1}{x^3-x+1}\&comma;x\&comma;s\right)$

$\frac{\left(\sum _{\mathrm{\_\&alpha;}=\mathrm{RootOf}\left({\mathrm{\_Z}}^3-\mathrm{\_Z}+1\right)}\frac{\left(6{\mathrm{\_\&alpha;}}^2+9\mathrm{\_\&alpha;}-4\right){\left(-\mathrm{\_\&alpha;}\right)}^{\mathrm{s~}}}{23\mathrm{\_\&alpha;}}\right)\mathrm{\pi }}{\sin \left(\mathrm{\pi }\mathrm{s~}\right)}$

$\mathrm{mellin}\left(\frac{\exp \left(-3x^2\right)}{\exp \left(x^2\right)-1}\&comma;x\&comma;s\right)$

$\frac{\mathrm{\Gamma }\left(\frac{\mathrm{s~}}{2}\right)\left(\mathrm{\zeta }\left(\frac{\mathrm{s~}}{2}\right)2^{\frac{\mathrm{s~}}{2}}{3}^{\frac{\mathrm{s~}}{2}}-2^{\frac{\mathrm{s~}}{2}}{3}^{\frac{\mathrm{s~}}{2}}-2^{\frac{\mathrm{s~}}{2}}-3^{\frac{\mathrm{s~}}{2}}\right)}{22^{\frac{\mathrm{s~}}{2}}{3}^{\frac{\mathrm{s~}}{2}}}$

$\mathrm{mellin}\left(\frac{\ln \left(x\right)\exp \left(-3x^2\right)}{\exp \left(x^2\right)-1}\&comma;x\&comma;s\right)$

$\frac{\mathrm{\Gamma }\left(\frac{\mathrm{s~}}{2}\right)\left(\mathrm{\Psi }\left(\frac{\mathrm{s~}}{2}\right)\mathrm{\zeta }\left(\frac{\mathrm{s~}}{2}\right)2^{\frac{\mathrm{s~}}{2}}{3}^{\frac{\mathrm{s~}}{2}}-\mathrm{\Psi }\left(\frac{\mathrm{s~}}{2}\right)2^{\frac{\mathrm{s~}}{2}}{3}^{\frac{\mathrm{s~}}{2}}+{\mathrm{\zeta }}^{\left(1\right)}\left(\frac{\mathrm{s~}}{2}\right)2^{\frac{\mathrm{s~}}{2}}{3}^{\frac{\mathrm{s~}}{2}}-\mathrm{\Psi }\left(\frac{\mathrm{s~}}{2}\right)2^{\frac{\mathrm{s~}}{2}}-\mathrm{\Psi }\left(\frac{\mathrm{s~}}{2}\right)3^{\frac{\mathrm{s~}}{2}}+\ln \left(2\right)3^{\frac{\mathrm{s~}}{2}}+\ln \left(3\right)2^{\frac{\mathrm{s~}}{2}}\right)}{42^{\frac{\mathrm{s~}}{2}}{3}^{\frac{\mathrm{s~}}{2}}}$

$\mathrm{addtable}\left(\mathrm{mellin}\&comma;f\left(t\right)\&comma;F\left(s\right)\&comma;t\&comma;s\right)\&colon;$

$\mathrm{mellin}\left(f\left(x\right)\&comma;x\&comma;s\right)$

$F\left(\mathrm{s~}\right)$