Arsech

Inverse hyperbolic secant, the inverse of Sech.

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

Arsech(x)the value $y$ with $\operatorname{sech}(y) = x$.

SageMatharcsech(x)SymPyasech(x)mpmathasech(x)Juliaenumeratio_ball(_asech, x)Rustacosh(n(1) / x)

Typed signatures

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

Domain: Elementary functions

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. For $x<0$, $\operatorname{arsech}(x) = \operatorname{arcosh}(1/x)$ (mpmath asech(-2) is $\tfrac{2\pi i}{3}$).

Examples

Basic #

live ↗

In$N(\mathrm{arsech}(\frac{1}{2}))$Out$1.316\,957\,896\,924\,816\,708\,63$

Scope #

$\operatorname{arsech} 1 = 0$ exactly

live ↗

In$\mathrm{arsech}(1)$Out$0$

A logarithmic singularity at 0

live ↗

In$\mathrm{arsech}(0)$Out$∞$

A float argument evaluates directly

live ↗

In$\mathrm{arsech}(0.25)$Out$2.063\,437\,068\,895\,560\,546\,73$

Past 1 the value is imaginary: $\operatorname{arsech} 2 = \operatorname{arcosh}\tfrac12 = i\pi/3$

live ↗

In$\mathrm{arsech}(2)$Out$\frac{1}{3}i π$

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

live ↗

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

Fifty digits of an inverse hyperbolic secant

live ↗

In$N(\mathrm{arsech}(\frac{1}{3}), 50)$Out$1.762\,747\,174\,039\,086\,050\,465\,218\,649\,959\,584\,618\,056\,320\,656\,523\,3$

Threads over a matrix; zero gives infinity and 2 and -2 give imaginary angles

live ↗

In$\mathrm{arsech}(\bigl\lbrack\bigl\lbrack2, 0\bigr\rbrack, \bigl\lbrack1, -2\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack\frac{1}{3}i π, ∞\bigr\rbrack, \bigl\lbrack0, \frac{2}{3}i π\bigr\rbrack\bigr\rbrack$

At infinity the value tends to $i\pi/2$

live ↗

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

Stays unevaluated; no exponential rewrite is applied

live ↗

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

As a matrix function, the zero entry becomes 0

live ↗

In$\mathrm{MatrixFunction}(\operatorname{\_1}↦\mathrm{arsech}(\operatorname{\_1}), \bigl\lbrack\bigl\lbrack2, 0\bigr\rbrack, \bigl\lbrack1, -2\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack\frac{1}{3}i π, 0\bigr\rbrack, \bigl\lbrack(-\frac{1}{12}i)π, \frac{2}{3}i π\bigr\rbrack\bigr\rbrack$

live ↗

In$\int\!\mathrm{arsech}(x)\, \mathrm{d}x$Out$x\mathrm{arsech}(x)+\arcsin(x)$

For negative $x$ the value is $\operatorname{arcosh}(1/x)$: $\operatorname{arsech}(-2) = \frac{2\pi i}{3}$

live ↗

In$\mathrm{arsech}(-2)$Out$\frac{2}{3}i π$

And $\operatorname{arsech}(-1/2) = \operatorname{arcosh}(-2) = \ln(2+\sqrt3) + i\pi$

live ↗

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

Properties #

Rewritten with a single radical in the denominator

live ↗

In$\mathrm{arsech}(2\sqrt{\frac{1}{2+\sqrt{2}}})$Out$\mathrm{arsech}(\frac{2}{\sqrt{2+\sqrt{2}}})$

Known value · Wolfram MathematicalFunctionData, ArcSech

The negated argument is rewritten the same way

live ↗

In$\mathrm{arsech}(-2\sqrt{\frac{1}{2+\sqrt{2}}})$Out$\mathrm{arsech}(-\frac{2}{\sqrt{2+\sqrt{2}}})$

Known value · Wolfram MathematicalFunctionData, ArcSech

live ↗

In$\mathrm{arsech}(-1)$Out$i π$

Known value · Wolfram MathematicalFunctionData, ArcSech

Possible issues #

Arsech(ComplexInfinity) is Indeterminate, by $\operatorname{arsech}(z) = \operatorname{arcosh}(1/z)$ at the same branch-cut ambiguity as Artanh

live ↗

Known value · DLMF 4.37.8

About

Examples	18
Checked against	Wolfram Language 14 agree3 differ1 unevaluatedSageMath 14 agree1 differ1 unevaluated1 errorSymPy 15 agree1 differ1 unevaluatedmpmath 11 agree2 errorJulia 10 agree3 errorRust 4 agree8 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

acosh of the reciprocal, the principal value off 0.

See also: Arcosh