MeijerGReduce

Rewrites a handful of elementary and special functions into MeijerG form, using identities that hold for the whole operand (not just a linear argument), so each is unconditional.

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

MeijerGReduce(expr, x)expr (a function of x) rewritten as a MeijerG call, for the heads below.

Typed signatures

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

Domain: Special functions

Details

• Covered: $e^g \to G^{1,0}_{0,1}(-g \mid {}; 0)$, $\sin(g) \to \sqrt{\pi}\,G^{1,0}_{0,2}(g^2/4 \mid {}; 1/2, 0)$, $\cos(g) \to \sqrt{\pi}\,G^{1,0}_{0,2}(g^2/4 \mid {}; 0, 1/2)$, $\ln(1+g) \to G^{1,2}_{2,2}(g \mid 1,1; 1,0)$, and $J_n(g) \to G^{1,0}_{0,2}(g^2/4 \mid {}; n/2, -n/2)$ — for any g (and, for the Bessel case, any n), since each identity holds for the operand as a whole rather than requiring a linear argument.
• Emits the plain 4-argument MeijerG form. Wolfram's own MeijerGReduce often prefers a generalized 5-argument MeijerG[…, z, r] with $r \neq 1$ to keep the argument free of an explicit square root; both are correct, and what's verified here is that ours round-trips numerically through this package's own MeijerG evaluator, not that it matches Wolfram's textual form.
• A shifted or scaled argument (sin(2x), e^{x+1}, log(1+x^2), BesselJ(n, 3x), …) is still covered — each identity holds for the operand g as a whole, not just a linear one. Declined: a bare log(x) (no 1 + x shift — none of the identities above apply), any other head, and an expr that doesn't depend on x.

Examples

Basic #

$e^{x} = G^{1,0}_{0,1}(-x \mid {}; 0)$

live ↗

In$\mathrm{MeijerGReduce}(e^{x}, x)$Out$\mathrm{MeijerG}(\bigl\lbrack\bigl\lbrack\bigr\rbrack, \bigl\lbrack\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack0\bigr\rbrack, \bigl\lbrack\bigr\rbrack\bigr\rbrack, -x)$

$\sin(x) = \sqrt{\pi}\,G^{1,0}_{0,2}(x^2/4 \mid {}; 1/2, 0)$

live ↗

In$\mathrm{MeijerGReduce}(\sin(x), x)$Out$\mathrm{MeijerG}(\bigl\lbrack\bigl\lbrack\bigr\rbrack, \bigl\lbrack\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack\frac{1}{2}\bigr\rbrack, \bigl\lbrack0\bigr\rbrack\bigr\rbrack, \frac{x^2}{4})\sqrt{π}$

$\ln(1+x) = G^{1,2}_{2,2}(x \mid 1,1; 1,0)$

live ↗

In$\mathrm{MeijerGReduce}(\ln(1+x), x)$Out$\mathrm{MeijerG}(\bigl\lbrack\bigl\lbrack1, 1\bigr\rbrack, \bigl\lbrack\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack1\bigr\rbrack, \bigl\lbrack0\bigr\rbrack\bigr\rbrack, x)$

live ↗

In$\mathrm{MeijerGReduce}(\operatorname{J}_{1}(ax), x)$Out$\mathrm{MeijerG}(\bigl\lbrack\bigl\lbrack\bigr\rbrack, \bigl\lbrack\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack\frac{1}{2}\bigr\rbrack, \bigl\lbrack-\frac{1}{2}\bigr\rbrack\bigr\rbrack, \frac{1}{4}(a^2x^2))$

Scope #

$J_n(x) = G^{1,0}_{0,2}(x^2/4 \mid {}; n/2, -n/2)$, symbolic order $n$

live ↗

In$\mathrm{MeijerGReduce}(\operatorname{J}_{n}(x), x)$Out$\mathrm{MeijerG}(\bigl\lbrack\bigl\lbrack\bigr\rbrack, \bigl\lbrack\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack\frac{n}{2}\bigr\rbrack, \bigl\lbrack-\frac{n}{2}\bigr\rbrack\bigr\rbrack, \frac{x^2}{4})$

Cosine is a Meijer G function: $\cos(ax)=\sqrt\pi\,G(\tfrac{a^2x^2}{4})$

live ↗

In$\mathrm{MeijerGReduce}(\cos(ax), x)$Out$\mathrm{MeijerG}(\bigl\lbrack\bigl\lbrack\bigr\rbrack, \bigl\lbrack\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack0\bigr\rbrack, \bigl\lbrack\frac{1}{2}\bigr\rbrack\bigr\rbrack, \frac{1}{4}(a^2x^2))\sqrt{π}$

A function with no reduction is left unevaluated

live ↗

In$\mathrm{MeijerGReduce}(\frac{\arcsin(x)}{\sqrt{1-x^2}}, x)$Out$\mathrm{MeijerGReduce}(\frac{\arcsin(x)}{\sqrt{1-x^2}}, x)$

live ↗

In$\mathrm{MeijerGReduce}(\sin(ax), x)$Out$\mathrm{MeijerG}(\bigl\lbrack\bigl\lbrack\bigr\rbrack, \bigl\lbrack\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack\frac{1}{2}\bigr\rbrack, \bigl\lbrack0\bigr\rbrack\bigr\rbrack, \frac{1}{4}(a^2x^2))\sqrt{π}$

live ↗

In$\mathrm{MeijerGReduce}(\exp(cx), x)$Out$\mathrm{MeijerG}(\bigl\lbrack\bigl\lbrack\bigr\rbrack, \bigl\lbrack\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack0\bigr\rbrack, \bigl\lbrack\bigr\rbrack\bigr\rbrack, -(cx))$

Properties #

live ↗

In$\mathrm{MeijerGReduce}(\exp(-x), x)$Out$\mathrm{MeijerG}(\bigl\lbrack\bigl\lbrack\bigr\rbrack, \bigl\lbrack\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack0\bigr\rbrack, \bigl\lbrack\bigr\rbrack\bigr\rbrack, x)$

About

Examples	10
Checked against	Wolfram Language 4 agree3 differ2 unevaluated
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/meijer-g-reduce.ts

See also: MeijerG