Artanh

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

Inverse hyperbolic tangent (area hyperbolic tangent).

✓ checked against that system · ! checked, with disagreements (hover a chip)DLMF4.37.ii.p1↗FungrimAtanh!↗Fungrim entryWolfram LanguageArcTanh!↗

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

SageMatharctanh(complex | signed_infinity)SymPyatanh(complex | signed_infinity)mpmathatanh(complex | signed_infinity)Juliaenumeratio_ball(_atanh, complex | signed_infinity)Rustatanh(complex | signed_infinity)

Artanh(complex | signed_infinity | ~oo) -> Indeterminate | number | signed_infinityInverse hyperbolic tangent (area hyperbolic tangent).

SageMatharctanh(complex | signed_infinity | ~oo)SymPyatanh(complex | signed_infinity | ~oo)mpmathatanh(complex | signed_infinity | ~oo)Juliaenumeratio_ball(_atanh, complex | signed_infinity | ~oo)Rustatanh(complex | signed_infinity | ~oo)

Typed signatures

Type	Package	Applies
(complex | quantity | signed_infinity) -> number	analyticreplaces 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, cuts on $(-\infty,-1]$ and $[1,\infty)$: past 1 the value is the limit from below, $\tfrac12\ln\frac{x+1}{x-1} - \tfrac{i\pi}{2}$ (mpmath atanh(2); Wolfram's ArcTanh[2.] agrees), and oddness gives the other cut.

Examples

Scope #

Fifty digits of $\operatorname{artanh}(1/2)$

live ↗

In$N(\mathrm{artanh}(\frac{1}{2}), 50)$Out$0.549\,306\,144\,334\,054\,845\,697\,622\,618\,461\,262\,852\,323\,745\,278\,911\,37$

The limit at infinity is a purely imaginary constant, $-i\pi/2$

live ↗

In$\mathrm{artanh}(∞)$Out$(-\frac{1}{2}i)π$

Vanishes at the origin

live ↗

In$\mathrm{artanh}(0)$Out$0$

Derivative is $1/(1-x^2)$

live ↗

In$\frac{\mathrm{d}\mathrm{artanh}(x)}{\mathrm{d}x}$Out$\frac{1}{1-x^2}$

The addition formula is not recognised; the tanh term stays

live ↗

In$\mathrm{FullSimplify}(\tanh(\mathrm{artanh}(x)+\mathrm{artanh}(y))-\frac{x+y}{xy+1})$Out$\frac{-x-y}{xy+1}+\tanh(\mathrm{artanh}(x)+\mathrm{artanh}(y))$

No exponential rewrite is applied; artanh(z) is returned as is

live ↗

In$\mathrm{TrigToExp}(\mathrm{artanh}(z))$Out$\mathrm{artanh}(z)$

Antiderivative $x\operatorname{artanh}x + \tfrac12\ln(1-x^2)$

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In$\int\!\mathrm{artanh}(x)\, \mathrm{d}x$Out$x\mathrm{artanh}(x)+\frac{1}{2}\ln(1-x^2)$

Past $[-1, 1]$ the value is $\tfrac12\ln\frac{x+1}{x-1} - \tfrac{i\pi}{2}$: $\operatorname{artanh}(2) = \tfrac12\ln3 - \tfrac{i\pi}{2}$

live ↗

In$\mathrm{artanh}(2)$Out$\frac{\ln(3)}{2}+(-\frac{1}{2}i)π$

Odd: $\operatorname{artanh}(-2)$ is the negative of $\operatorname{artanh}(2)$

live ↗

In$\mathrm{artanh}(-2)$Out$\frac{1}{2}i π-\frac{\ln(3)}{2}$

Possible issues #

Artanh(ComplexInfinity) is Indeterminate -- the sign of the boundary value $\pm i\pi/2$ depends on which side of the branch cut z approaches from

live ↗

Known value · DLMF 4.37.24

About

Examples	10
Checked against	Wolfram Language 8 agree2 differSageMath 7 agree2 differ1 errorSymPy 8 agree1 differmpmath 5 agreeJulia 4 agree1 errorRust 2 agree2 error
Crosswalk	DLMFFungrimFungrim entry ×7Wolfram 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