Arcoth

Inverse hyperbolic cotangent, the inverse of Coth.

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

Arcoth(x)the value $y$ with $\coth(y) = x$.

SageMatharccoth(x)SymPyacoth(x)mpmathacoth(x)Juliaenumeratio_ball(_acoth, x)Rustatanh(n(1) / x)

Typed signatures

Type	Package	Applies
(complex | infinity | quantity) -> number	analyticreplaces compute-engine	always
(complex | infinity | quantity) -> number	compute-engine	always

Domain: Elementary functions

Details

• Stays symbolic at plain evaluation, same as Arsinh-family functions; wrap in N(...).
• 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. For $0<|x|<1$, $\operatorname{arcoth}(x) = \operatorname{sign}(x)\left(\tfrac12\ln\frac{1+|x|}{1-|x|} - \tfrac{i\pi}{2}\right)$, which is $\operatorname{artanh}(1/x)$ (mpmath acoth(0.5)).

Examples

Basic #

live ↗

In$N(\mathrm{arcoth}(2))$Out$0.549\,306\,144\,334\,054\,845\,7$

live ↗

In$\operatorname{Series}(\mathrm{arcoth}(x), x, ∞, 5)$Out$\frac{1}{x}+\frac{1}{3x^3}+\frac{1}{5x^5}+O\left(\frac{1}{x^7}\right)$

Scope #

A logarithmic singularity at 1

live ↗

In$\mathrm{arcoth}(1)$Out$∞$

live ↗

In$\mathrm{arcoth}(∞)$Out$0$

A float argument evaluates directly

live ↗

In$\mathrm{arcoth}(3.5)$Out$0.293\,893\,332\,451\,059\,504\,095$

As a logarithm, $\operatorname{arcoth} 2 = \tfrac12\ln\frac{2+1}{2-1} = \tfrac12\ln 3$

live ↗

In$\mathrm{TrigToExp}(\mathrm{arcoth}(2))$Out$\frac{\ln(3)}{2}$

Oddness pulls the sign out of a symbolic argument: $\operatorname{arcoth}(-x) = -\operatorname{arcoth} x$

live ↗

In$\mathrm{arcoth}(-x)$Out$-\mathrm{arcoth}(x)$

Derivative: $\frac{1}{1 - x^2}$

live ↗

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

Fifty digits of a real value beyond the unit interval

live ↗

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

At complex infinity the value is 0

live ↗

In$\mathrm{arcoth}(\tilde∞)$Out$0$

A logarithmic pole at $-1$ gives negative infinity

live ↗

In$\mathrm{arcoth}(-1)$Out$-∞$

live ↗

In$\mathrm{arcoth}(\bigl\lbrack\bigl\lbrack i, -i\bigr\rbrack, \bigl\lbrack0, -i\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack\mathrm{arcoth}(i), \mathrm{arcoth}(-i)\bigr\rbrack, \bigl\lbrack\frac{1}{2}i π, \mathrm{arcoth}(-i)\bigr\rbrack\bigr\rbrack$

live ↗

In$\mathrm{MatrixFunction}(\operatorname{\_1}↦\mathrm{arcoth}(\operatorname{\_1}), \bigl\lbrack\bigl\lbrack i, -i\bigr\rbrack, \bigl\lbrack0, -i\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack\mathrm{arcoth}(i), \frac{\mathrm{arcoth}(-i)}{2}-\frac{\mathrm{arcoth}(i)}{2}\bigr\rbrack, \bigl\lbrack0, \mathrm{arcoth}(-i)\bigr\rbrack\bigr\rbrack$

For $0<|x|<1$ the value is $\tfrac12\ln\frac{1+x}{1-x} - \tfrac{i\pi}{2}$: $\operatorname{arcoth}(1/2) = \tfrac12\ln3 - \tfrac{i\pi}{2}$

live ↗

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

Odd: $\operatorname{arcoth}(-1/2)$ is the negative of $\operatorname{arcoth}(1/2)$

live ↗

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

live ↗

In$\mathrm{arcoth}(0)$Out$\frac{1}{2}i π$

About

Examples	16
Checked against	Wolfram Language 14 agree2 differSageMath 12 agree2 differ1 unevaluatedSymPy 14 agree1 errormpmath 10 agreeJulia 7 agree3 errorRust 3 agree5 error
Crosswalk	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

atanh of the reciprocal, the principal value off 0.

See also: Artanh