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inttrans

 mellin
 Mellin transform

 Calling Sequence mellin(expr, x, s)

Parameters

 expr - expression to be transformed x - variable expr is transformed with respect to x s - parameter of transform opt - option to run this under (optional)

Description

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

$M\left(s\right)={\int }_{0}^{\mathrm{\infty }}m\left(x\right){x}^{s-1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆx$

 • Some expressions involving exponentials, polynomials, algebraic functions, trigonometrics (sin, cos, $\mathrm{sinh}$, 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 $K{\mathrm{ln}\left(x\right)}^{n}f\left(a{x}^{b}\right){x}^{c}$ 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.

Examples

 > $\mathrm{with}\left(\mathrm{inttrans}\right):$
 > $\mathrm{assume}\left(0
 > $\mathrm{mellin}\left(a{x}^{b}{ⅇ}^{-{x}^{\frac{1}{4}}},x,s\right)$
 ${4}{}{a}{}{\mathrm{\Gamma }}{}\left({4}{}{\mathrm{s~}}{+}{4}{}{b}\right)$ (1)
 > $\mathrm{mellin}\left(\frac{x}{{x}^{2}+1},x,s\right)$
 $\frac{{\mathrm{\pi }}}{{2}{}{\mathrm{sin}}{}\left(\frac{{\mathrm{\pi }}{}\left({\mathrm{s~}}{+}{1}\right)}{{2}}\right)}$ (2)
 > $\mathrm{mellin}\left(\frac{\mathrm{ln}\left(x\right)x}{{x}^{2}+1},x,s-2\right)$
 ${-}\frac{{{\mathrm{\pi }}}^{{2}}{}{\mathrm{cos}}{}\left(\frac{{\mathrm{\pi }}{}\left({\mathrm{s~}}{-}{1}\right)}{{2}}\right)}{{4}{}{{\mathrm{sin}}{}\left(\frac{{\mathrm{\pi }}{}\left({\mathrm{s~}}{-}{1}\right)}{{2}}\right)}^{{2}}}$ (3)
 > $\mathrm{mellin}\left(\frac{1}{{x}^{3}-x+1},x,s\right)$
 $\frac{\left({\sum }_{{\mathrm{_α}}{=}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{3}}{-}{\mathrm{_Z}}{+}{1}\right)}{}\frac{\left({6}{}{{\mathrm{_α}}}^{{2}}{+}{9}{}{\mathrm{_α}}{-}{4}\right){}{\left({-}{\mathrm{_α}}\right)}^{{\mathrm{s~}}}}{{23}{}{\mathrm{_α}}}\right){}{\mathrm{\pi }}}{{\mathrm{sin}}{}\left({\mathrm{\pi }}{}{\mathrm{s~}}\right)}$ (4)
 > $\mathrm{mellin}\left(\frac{{ⅇ}^{-3{x}^{2}}}{{ⅇ}^{{x}^{2}}-1},x,s\right)$
 $\frac{{\mathrm{\Gamma }}{}\left(\frac{{\mathrm{s~}}}{{2}}\right){}\left({\mathrm{\zeta }}{}\left(\frac{{\mathrm{s~}}}{{2}}\right){}{{2}}^{\frac{{\mathrm{s~}}}{{2}}}{}{{3}}^{\frac{{\mathrm{s~}}}{{2}}}{-}{{2}}^{\frac{{\mathrm{s~}}}{{2}}}{}{{3}}^{\frac{{\mathrm{s~}}}{{2}}}{-}{{2}}^{\frac{{\mathrm{s~}}}{{2}}}{-}{{3}}^{\frac{{\mathrm{s~}}}{{2}}}\right)}{{2}{}{{2}}^{\frac{{\mathrm{s~}}}{{2}}}{}{{3}}^{\frac{{\mathrm{s~}}}{{2}}}}$ (5)
 > $\mathrm{mellin}\left(\frac{\mathrm{ln}\left(x\right){ⅇ}^{-3{x}^{2}}}{{ⅇ}^{{x}^{2}}-1},x,s\right)$
 $\frac{{\mathrm{\Gamma }}{}\left(\frac{{\mathrm{s~}}}{{2}}\right){}\left({\mathrm{\Psi }}{}\left(\frac{{\mathrm{s~}}}{{2}}\right){}{\mathrm{\zeta }}{}\left(\frac{{\mathrm{s~}}}{{2}}\right){}{{2}}^{\frac{{\mathrm{s~}}}{{2}}}{}{{3}}^{\frac{{\mathrm{s~}}}{{2}}}{-}{\mathrm{\Psi }}{}\left(\frac{{\mathrm{s~}}}{{2}}\right){}{{2}}^{\frac{{\mathrm{s~}}}{{2}}}{}{{3}}^{\frac{{\mathrm{s~}}}{{2}}}{+}{\mathrm{\zeta }}{}\left({1}{,}\frac{{\mathrm{s~}}}{{2}}\right){}{{2}}^{\frac{{\mathrm{s~}}}{{2}}}{}{{3}}^{\frac{{\mathrm{s~}}}{{2}}}{-}{\mathrm{\Psi }}{}\left(\frac{{\mathrm{s~}}}{{2}}\right){}{{2}}^{\frac{{\mathrm{s~}}}{{2}}}{-}{\mathrm{\Psi }}{}\left(\frac{{\mathrm{s~}}}{{2}}\right){}{{3}}^{\frac{{\mathrm{s~}}}{{2}}}{+}{\mathrm{ln}}{}\left({2}\right){}{{3}}^{\frac{{\mathrm{s~}}}{{2}}}{+}{\mathrm{ln}}{}\left({3}\right){}{{2}}^{\frac{{\mathrm{s~}}}{{2}}}\right)}{{4}{}{{2}}^{\frac{{\mathrm{s~}}}{{2}}}{}{{3}}^{\frac{{\mathrm{s~}}}{{2}}}}$ (6)
 > $\mathrm{addtable}\left(\mathrm{mellin},f\left(t\right),F\left(s\right),t,s\right):$
 > $\mathrm{mellin}\left(f\left(x\right),x,s\right)$
 ${F}{}\left({\mathrm{s~}}\right)$ (7)