FunctionExpand

Rewrite special functions in expr in terms of more elementary or better-known ones. Provided by @enumeratio/analytic.

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

FunctionExpand(expr)expr with a handful of named special-function identities applied.

Typed signatures

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

Domain: Transformations

Details

Not a general special-function identity engine: DirichletEta and DirichletBeta rewrite in terms of the (Hurwitz) zeta this package already declares, BarnesG(½) in Glaisher's constant, and two special angles (Sin(π/15), Cos(π/24)) past compute-engine's automatic table -- each is a named identity, not a derivation.
The two special angles are pinned lookups at exactly those arguments, not a general nested-radical solver for an arbitrary rational multiple of π.
Pochhammer(x, 3) and Binomial(n, 2) need no rule here: compute-engine's own evaluator already expands a concrete nonnegative integer length into the product before FunctionExpand runs.

Examples

1 more kept as data, checked against the oracles

Basic #

$\eta(s) = (1 - 2^{1-s})\zeta(s)$

In

\mathrm{FunctionExpand}(\mathrm{DirichletEta}(s))

Out

(1-2^{1-s})ζ(s)

$\beta(s) = 4^{-s}(\zeta(s, \tfrac14) - \zeta(s, \tfrac34))$

In

\mathrm{FunctionExpand}(\mathrm{DirichletBeta}(s))

Out

\frac{ζ(s, \frac{1}{4})-ζ(s, \frac{3}{4})}{4^{s}}

$G(\tfrac12)$ in Glaisher's constant

In

\mathrm{FunctionExpand}(\mathrm{BarnesG}(\frac{1}{2}))

Out

\frac{\sqrt[24]{2}\sqrt[8]{e}}{\sqrt{\mathrm{ConstGlaisher}}^{3}\sqrt[4]{π}}

Radicals past the automatic special-angle table

In

\mathrm{FunctionExpand}(\sin(\frac{π}{15}))

Out

\frac{1}{8}(-\sqrt{15}+\sqrt{3}+\sqrt{10+2\sqrt{5}})

Half-angle radicals: $\cos\frac{\pi}{24} = \tfrac12\sqrt{2 + \tfrac{\sqrt2 + \sqrt6}{2}}$

In

\mathrm{FunctionExpand}(\cos(\frac{π}{24}))

Out

\frac{1}{4}(\sqrt{2}\sqrt{4+\sqrt{2}+\sqrt{6}})

Scope #

A rising factorial with an integer length becomes a polynomial

In

\mathrm{FunctionExpand}(\mathrm{Pochhammer}(x, 3))

Out

x×(x+1)(x+2)

$\binom n2 = \frac{n(n-1)}{2}$

In

\mathrm{FunctionExpand}(\binom{n}{2})

Out

\frac{n×(n-1)}{2}

Nothing to expand, the square root of a square stays

In

\mathrm{FunctionExpand}(\frac{\sin(\sqrt{x^2})}{\sqrt{x^2}})

Out

\frac{\sin(\sqrt{x^2})}{\sqrt{x^2}}

The reflection factor is left as written, not expanded

In

\mathrm{FunctionExpand}(\frac{ζ(1-z)}{ζ(z)})

Out

\frac{ζ(1-z)}{ζ(z)}

In

\mathrm{FunctionExpand}(\bigl\lbrack\sqrt{-x^2}, \ln(\sqrt{1-x^2})\bigr\rbrack)

Out

\bigl\lbrack\sqrt{-x^2}, \frac{1}{2}(\ln(1-x^2))\bigr\rbrack

In

\mathrm{FunctionExpand}(\operatorname{Ai}(x×(-1)^{\frac{4}{3}}))

Out

\operatorname{Ai}(x)

Properties #

Only $\zeta(2)$ expands, to $\pi^2/6$ ; the reflection formula is not applied

In

\mathrm{FunctionExpand}(\operatorname{Li}_{2}(z)+\operatorname{Li}_{2}(1-z)-ζ(2))

Out

\operatorname{Li}_{2}(z)+\operatorname{Li}_{2}(1-z)-\frac{π^2}{6}

About

Examples	13
Checked against	Wolfram Language 11 agree2 differ
Crosswalk	Wolfram Language

See also: FullSimplify, DirichletEta, DirichletBeta, BarnesG