inttrans - Maple Help

Home : Support : Online Help : Mathematics : Calculus : Transforms : inttrans/mellin

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
 > $\mathrm{mellin}\left(a{x}^{b}{ⅇ}^{-{x}^{\frac{1}{4}}},x,s\right)$
 ${4}{}{a}{}{\mathrm{Γ}}{}\left({4}{}{\mathrm{s~}}{+}{4}{}{b}\right)$ (1)
 > $\mathrm{mellin}\left(\frac{x}{{x}^{2}+1},x,s\right)$
 $\frac{{1}}{{2}}{}\frac{{\mathrm{π}}}{{\mathrm{sin}}{}\left(\frac{{1}}{{2}}{}{\mathrm{π}}{}\left({\mathrm{s~}}{+}{1}\right)\right)}$ (2)
 > $\mathrm{mellin}\left(\frac{\mathrm{ln}\left(x\right)x}{{x}^{2}+1},x,s-2\right)$
 ${-}\frac{{1}}{{4}}{}\frac{{{\mathrm{π}}}^{{2}}{}{\mathrm{cos}}{}\left(\frac{{1}}{{2}}{}{\mathrm{π}}{}\left({\mathrm{s~}}{-}{1}\right)\right)}{{{\mathrm{sin}}{}\left(\frac{{1}}{{2}}{}{\mathrm{π}}{}\left({\mathrm{s~}}{-}{1}\right)\right)}^{{2}}}$ (3)
 > $\mathrm{mellin}\left(\frac{1}{{x}^{3}-x+1},x,s\right)$
 $\frac{\left({\sum }_{{\mathrm{_α}}{=}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{3}}{-}{\mathrm{_Z}}{+}{1}\right)}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{1}}{{23}}{}\frac{\left({6}{}{{\mathrm{_α}}}^{{2}}{+}{9}{}{\mathrm{_α}}{-}{4}\right){}{\left({-}{\mathrm{_α}}\right)}^{{\mathrm{s~}}}}{{\mathrm{_α}}}\right){}{\mathrm{π}}}{{\mathrm{sin}}{}\left({\mathrm{π}}{}{\mathrm{s~}}\right)}$ (4)
 > $\mathrm{mellin}\left(\frac{{ⅇ}^{-3{x}^{2}}}{{ⅇ}^{{x}^{2}}-1},x,s\right)$
 $\frac{{1}}{{2}}{}\frac{{\mathrm{Γ}}{}\left(\frac{{1}}{{2}}{}{\mathrm{s~}}\right){}\left({\mathrm{ζ}}{}\left(\frac{{1}}{{2}}{}{\mathrm{s~}}\right){}{{2}}^{\frac{{1}}{{2}}{}{\mathrm{s~}}}{}{{3}}^{\frac{{1}}{{2}}{}{\mathrm{s~}}}{-}{{2}}^{\frac{{1}}{{2}}{}{\mathrm{s~}}}{}{{3}}^{\frac{{1}}{{2}}{}{\mathrm{s~}}}{-}{{2}}^{\frac{{1}}{{2}}{}{\mathrm{s~}}}{-}{{3}}^{\frac{{1}}{{2}}{}{\mathrm{s~}}}\right)}{{{2}}^{\frac{{1}}{{2}}{}{\mathrm{s~}}}{}{{3}}^{\frac{{1}}{{2}}{}{\mathrm{s~}}}}$ (5)
 > $\mathrm{mellin}\left(\frac{\mathrm{ln}\left(x\right){ⅇ}^{-3{x}^{2}}}{{ⅇ}^{{x}^{2}}-1},x,s\right)$
 $\frac{{1}}{{4}}{}\frac{{\mathrm{Γ}}{}\left(\frac{{1}}{{2}}{}{\mathrm{s~}}\right){}\left({\mathrm{Ψ}}{}\left(\frac{{1}}{{2}}{}{\mathrm{s~}}\right){}{\mathrm{ζ}}{}\left(\frac{{1}}{{2}}{}{\mathrm{s~}}\right){}{{2}}^{\frac{{1}}{{2}}{}{\mathrm{s~}}}{}{{3}}^{\frac{{1}}{{2}}{}{\mathrm{s~}}}{-}{\mathrm{Ψ}}{}\left(\frac{{1}}{{2}}{}{\mathrm{s~}}\right){}{{2}}^{\frac{{1}}{{2}}{}{\mathrm{s~}}}{}{{3}}^{\frac{{1}}{{2}}{}{\mathrm{s~}}}{+}{\mathrm{ζ}}{}\left({1}{,}\frac{{1}}{{2}}{}{\mathrm{s~}}\right){}{{2}}^{\frac{{1}}{{2}}{}{\mathrm{s~}}}{}{{3}}^{\frac{{1}}{{2}}{}{\mathrm{s~}}}{-}{\mathrm{Ψ}}{}\left(\frac{{1}}{{2}}{}{\mathrm{s~}}\right){}{{2}}^{\frac{{1}}{{2}}{}{\mathrm{s~}}}{-}{\mathrm{Ψ}}{}\left(\frac{{1}}{{2}}{}{\mathrm{s~}}\right){}{{3}}^{\frac{{1}}{{2}}{}{\mathrm{s~}}}{+}{\mathrm{ln}}{}\left({2}\right){}{{3}}^{\frac{{1}}{{2}}{}{\mathrm{s~}}}{+}{\mathrm{ln}}{}\left({3}\right){}{{2}}^{\frac{{1}}{{2}}{}{\mathrm{s~}}}\right)}{{{2}}^{\frac{{1}}{{2}}{}{\mathrm{s~}}}{}{{3}}^{\frac{{1}}{{2}}{}{\mathrm{s~}}}}$ (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)