MellinTransform

The Mellin transform $F(s) = \int_0^\infty f(x)\,x^{s-1}\,dx$, as a rule table over nine standard pairs plus the scaling and power-shift theorems.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageMellinTransform✓↗

MellinTransform(f, x, s)$F(s) = \int_0^\infty f(x)\,x^{s-1}\,dx$, for f a function of x.

Typed signatures

Type	Package	Applies
(expression, expression, expression) -> expression	analytic	always

Domain: Transforms

Details

• Covered pairs (each strip of convergence confirmed against wolframscript's MellinTransform[..., GenerateConditions -> True]): $e^{-x}\to\Gamma(s)$ ($\mathrm{Re}(s)>0$), $e^{-x^2}\to\Gamma(s/2)/2$ ($\mathrm{Re}(s)>0$), $1/(1+x)\to\pi\csc(\pi s)$ ($0<\mathrm{Re}(s)<1$), $1/(1+x)^A\to\Gamma(A-s)\Gamma(s)/\Gamma(A)$ ($0<\mathrm{Re}(s)<\mathrm{Re}(A)$), $\sin(x)\to\Gamma(s)\sin(\pi s/2)$ ($-1<\mathrm{Re}(s)<1$), $\cos(x)\to\Gamma(s)\cos(\pi s/2)$ ($0<\mathrm{Re}(s)<1$), $\log(1+x)\to\pi\csc(\pi s)/s$ ($-1<\mathrm{Re}(s)<0$).
• Two theorems generalize every pair above: scaling ($f(ax)\to a^{-s}F(s)$, any provably positive a) and the power shift ($x^c f(x)\to F(s+c)$) — together they cover e.g. $e^{-ax}$, $\sin(ax)$, and $x^c e^{-x}\to\Gamma(c+s)$.
• Declined: an unknown-sign scale or shift parameter, a quadratic or higher argument to Sin/Cos/Ln, an opaque function, and anything outside these nine shapes.

Examples

Basic #

$\mathcal{M}\{e^{-x}\}(s) = \Gamma(s)$, $\mathrm{Re}(s)>0$

live ↗

In$\mathrm{MellinTransform}(\exp(-x), x, s)$Out$Γ(s)$

$\mathcal{M}\{1/(1+x)\}(s) = \pi\csc(\pi s)$, $0<\mathrm{Re}(s)<1$

live ↗

In$\mathrm{MellinTransform}(\frac{1}{1+x}, x, s)$Out$π\csc(πs)$

$\mathcal{M}\{e^{-x^2}\}(s) = \Gamma(s/2)/2$

live ↗

In$\mathrm{MellinTransform}(\exp(-x^2), x, s)$Out$\frac{1}{2}(Γ(\frac{s}{2}))$

$\mathcal{M}\{\log(1+x)\}(s) = \pi\csc(\pi s)/s$, $-1<\mathrm{Re}(s)<0$

live ↗

In$\mathrm{MellinTransform}(\ln(1+x), x, s)$Out$\frac{1}{s}(π\csc(πs))$

Scope #

power shift — $\mathcal{M}\{x^a e^{-x}\}(s) = \Gamma(a+s)$

live ↗

In$\mathrm{MellinTransform}(x^{a}\exp(-x), x, s)$Out$Γ(a+s)$

The Mellin transform of $\frac1{1+x}$ at $s=\frac12$ is $\frac{\pi}{\sin(\pi s)}=\pi$

live ↗

In$\mathrm{MellinTransform}(\frac{1}{x+1}, x, \frac{1}{2})$Out$π$

About

Examples	6
Checked against	Wolfram Language 6 agree
Crosswalk	Wolfram Language

Implementation

primitive · kernela kernel — an algorithm over mutable state that would not be readable as a tree

nativeenginetypescriptpackages/symbols/analysis/analytic/src/mellin-transform.ts

See also: InverseMellinTransform, LaplaceTransform, FourierTransform