Arcsch

Inverse hyperbolic cosecant, the inverse of Csch.

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

Arcsch(x)the value

y

with

\operatorname{csch}(y) = x

SageMatharccsch(x)SymPyacsch(x)mpmathacsch(x)Juliaenumeratio_ball(_acsch, x)Rustasinh(n(1) / x)

Typed signatures

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

Domain: Elementary functions

Examples

Basic #

N(\mathrm{arcsch}(2))

Out

0.481\,211\,825\,059\,603\,447\,495

Scope #

A pole at 0

\mathrm{arcsch}(0)

Out

\tilde∞

\mathrm{arcsch}(∞)

Out

0

A float argument evaluates directly

\mathrm{arcsch}(1.5)

Out

0.625\,145\,117\,250\,416\,687\,636

As a logarithm, $\operatorname{arcsch} 1 = \ln(1 + \sqrt2)$

\mathrm{TrigToExp}(\mathrm{arcsch}(1))

Out

\ln(1+\sqrt{2})

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

\mathrm{arcsch}(-x)

Out

-\mathrm{arcsch}(x)

Derivative, for real $x \ne 0$

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

Out

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

Fifty digits of an inverse hyperbolic cosecant

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

Out

0.198\,690\,110\,349\,241\,406\,474\,636\,915\,950\,206\,968\,221\,308\,794\,224\,45

At complex infinity the value is 0

\mathrm{arcsch}(\tilde∞)

Out

0

\mathrm{arcsch}(\bigl\lbrack\bigl\lbrack2i, 0\bigr\rbrack, \bigl\lbrack i, -2i\bigr\rbrack\bigr\rbrack)

Out

\bigl\lbrack\bigl\lbrack\mathrm{arcsch}(2i), \tilde∞\bigr\rbrack, \bigl\lbrack\mathrm{arcsch}(i), \mathrm{arcsch}(-2i)\bigr\rbrack\bigr\rbrack

\mathrm{MatrixFunction}(\operatorname{\_1}↦\mathrm{arcsch}(\operatorname{\_1}), \bigl\lbrack\bigl\lbrack2i, 0\bigr\rbrack, \bigl\lbrack i, -2i\bigr\rbrack\bigr\rbrack)

Out

\bigl\lbrack\bigl\lbrack\mathrm{arcsch}(2i), 0\bigr\rbrack, \bigl\lbrack\frac{\mathrm{arcsch}(2i)}{4}-\frac{\mathrm{arcsch}(-2i)}{4}, \mathrm{arcsch}(-2i)\bigr\rbrack\bigr\rbrack

Properties #

Negation is carried through unchanged

\mathrm{arcsch}(-i\sqrt{2-\frac{2\sqrt{5}}{5}})

Out

\mathrm{arcsch}((-\frac{\sqrt{10}}{5}i)\sqrt{5-\sqrt{5}})