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 #

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

\mathrm{MellinTransform}(\exp(-x), x, s)

Out

Γ(s)

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

\mathrm{MellinTransform}(\frac{1}{1+x}, x, s)

Out

π\csc(πs)

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

\mathrm{MellinTransform}(\exp(-x^2), x, s)

Out

\frac{1}{2}(Γ(\frac{s}{2}))

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

\mathrm{MellinTransform}(\ln(1+x), x, s)

Out

\frac{1}{s}(π\csc(πs))

Scope #

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

\mathrm{MellinTransform}(x^{a}\exp(-x), x, s)

Out

Γ(a+s)

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

\mathrm{MellinTransform}(\frac{1}{x+1}, x, \frac{1}{2})

Out

π

About

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

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