Arcosh

Generated from the engine's own definition: compute-engine's symbol, which we neither extend nor document by hand.

Inverse hyperbolic cosine (area hyperbolic cosine).

✓ checked against that system · ! checked, with disagreements (hover a chip)DLMF4.37.ii.p1FungrimAcosh✓Fungrim entry2fc479✓718f3a✓d9765b✓Wolfram LanguageArcCosh!

Arcosh(complex | signed_infinity) -> numberas compute-engine declares it

SageMatharccosh(complex | signed_infinity)SymPyacosh(complex | signed_infinity)mpmathacosh(complex | signed_infinity)Juliaenumeratio_ball(_acosh, complex | signed_infinity)Rustacosh(complex | signed_infinity)

Arcosh(complex | signed_infinity | ~oo) -> number | signed_infinity | ~ooInverse hyperbolic cosine (area hyperbolic cosine).

SageMatharccosh(complex | signed_infinity | ~oo)SymPyacosh(complex | signed_infinity | ~oo)mpmathacosh(complex | signed_infinity | ~oo)Juliaenumeratio_ball(_acosh, complex | signed_infinity | ~oo)Rustacosh(complex | signed_infinity | ~oo)

Typed signatures

Type	Package	Applies
`(complex \| quantity \| signed_infinity) -> number`	`analytic`replaces `compute-engine`	always
`(complex \| quantity \| signed_infinity) -> number`	`compute-engine`	always

Domain: Compute engine

Details

Differs from Wolfram: past its real domain a rational argument reduces to the exact principal value, on every inverse trigonometric and hyperbolic head alike (Arcsin, Arccos, Arcsec, Arccsc, Arcosh, Artanh, Arcoth, Arsech); Wolfram leaves the exact call unevaluated and gives the same number from N[...]. Checked against mpmath and wolframscript on a grid of rationals to double precision. A float and N(...) already took the same branch. Compiled code is separate: the JavaScript target is real-valued (NaN past the domain), the WGSL target returns a complex pair, and Arcsin and Arccos decline to compile. Principal branch, cut on $(-\infty, 1)$ , continuous from above: for $x<-1$ , $\ln(|x|+\sqrt{x^2-1}) + i\pi$ (mpmath acosh(-2)), and for $|x|<1$ , $i\arccos(x)$ .

Examples

1 more kept as data, checked against the oracles

Scope #

Arcosh(-Infinity) is +Infinity, not -Infinity -- the principal branch keeps $\Re(\operatorname{arcosh} z) \ge 0$ everywhere

In

\mathrm{arcosh}(-∞)

Out

∞

Known value · DLMF 4.37.19

Fifty digits of $\operatorname{arcosh}(5)$

In

N(\mathrm{arcosh}(5), 50)

Out

2.292\,431\,669\,561\,177\,687\,800\,787\,311\,348\,015\,431\,621\,868\,240\,015\,7

Threads over a matrix; arcosh(1) folds to 0, arcosh(0) to $i\pi/2$ , and the radicals $\pm\sqrt2/2$ to $i\pi/4$ and $3i\pi/4$

In

\mathrm{arcosh}(\bigl\lbrack\bigl\lbrack1, \frac{2^{1/2}}{2}\bigr\rbrack, \bigl\lbrack0, -2^{1/2}/2\bigr\rbrack\bigr\rbrack)

Out

\bigl\lbrack\bigl\lbrack0, \frac{1}{4}i π\bigr\rbrack, \bigl\lbrack\frac{1}{2}i π, \frac{3}{4}i π\bigr\rbrack\bigr\rbrack

Grows without bound, so infinity maps to infinity

In

\mathrm{arcosh}(∞)

Out

∞

The zero of arcosh: $\operatorname{arcosh}(1) = 0$

In

\mathrm{arcosh}(1)

Out

0

On $[0,1]$ arcosh is $i\arccos$ , so the integral is $i$

In

\int_{0}^{1}\!\mathrm{arcosh}(x)\, \mathrm{d}x

Out

i

In

\frac{\mathrm{d}\mathrm{arcosh}(x)}{\mathrm{d}x}

Out

\frac{1}{\sqrt{x^2-1}}

In

\int\!\mathrm{arcosh}(x)\, \mathrm{d}x

Out

x\mathrm{arcosh}(x)-\sqrt{x^2-1}

Arcosh(ComplexInfinity) is +Infinity in every direction -- $\operatorname{arcosh} z \sim \ln 2z$ , so only the real part diverges

Known value · DLMF 4.37.19

Below $-1$ the value is $\ln(|x|+\sqrt{x^2-1}) + i\pi$ : $\operatorname{arcosh}(-2) = \ln(2+\sqrt3) + i\pi$

In

\mathrm{arcosh}(-2)

Out

i π+\ln(2+\sqrt{3})

Inside $(-1, 1)$ the value is $i\arccos(x)$

In

\mathrm{arcosh}(\frac{3}{5})

Out

i\arccos(\frac{3}{5})

Properties #

A nested-radical argument stays unevaluated, only rewritten

In

\mathrm{arcosh}(\frac{1}{2}\sqrt{\frac{5-5^{1/2}}{2}})

Out

\frac{3}{10}i π

Known value · Wolfram MathematicalFunctionData, ArcCosh

The negated radical argument is likewise left as arcosh

In

\mathrm{arcosh}(-\frac{1}{2}\sqrt{\frac{5-5^{1/2}}{2}})

Out

\frac{7}{10}i π

Known value · Wolfram MathematicalFunctionData, ArcCosh

In

\mathrm{arcosh}(0)

Out

\frac{1}{2}i π

Known value · Wolfram MathematicalFunctionData, ArcCosh

In

\mathrm{arcosh}(\frac{1}{4}\sqrt{2}(\sqrt{3}-1))

Out

\frac{5}{12}i π

Known value · Wolfram MathematicalFunctionData, ArcCosh

In

\mathrm{arcosh}(-\frac{\sqrt{2}}{4}(\sqrt{3}-1))

Out

\frac{7}{12}i π

Known value · Wolfram MathematicalFunctionData, ArcCosh

In

\mathrm{arcosh}(\frac{\sqrt{5}-1}{4})

Out

\frac{2}{5}i π

Known value · Wolfram MathematicalFunctionData, ArcCosh

In

\mathrm{arcosh}(\frac{1-\sqrt{5}}{4})

Out

\frac{3}{5}i π

Known value · Wolfram MathematicalFunctionData, ArcCosh

In

\mathrm{arcosh}(\frac{\sqrt{2-2^{1/2}}}{2})

Out

\frac{3}{8}i π

Known value · Wolfram MathematicalFunctionData, ArcCosh

About

Examples	20
Checked against	Wolfram Language 17 agree2 differ1 unevaluatedSageMath 17 agree2 differ1 unevaluatedSymPy 17 agree2 differ1 unevaluatedmpmath 15 agree1 differJulia 14 agree2 errorRust 2 agree12 error
Crosswalk	DLMF Fungrim Fungrim entry ×3 Wolfram LanguageSageMathSymPympmathJuliaRust

Implementation

mappedsympypackages/oracle/src/mappings.ts

mappedmpmathpackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

mappedrustpackages/oracle/src/mappings.ts