Arsinh

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

Inverse hyperbolic sine (area hyperbolic sine).

✓ checked against that system · ! checked, with disagreements (hover a chip)DLMF4.37.ii.p1↗FungrimAsinh✓↗Fungrim entry423b36✓↗Wolfram LanguageArcSinh!↗

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

SageMatharcsinh(complex | signed_infinity)SymPyasinh(complex | signed_infinity)mpmathasinh(complex | signed_infinity)Juliaenumeratio_ball(_asinh, complex | signed_infinity)Rustasinh(complex | signed_infinity)

Arsinh(complex | signed_infinity | ~oo) -> number | signed_infinity | ~ooInverse hyperbolic sine (area hyperbolic sine).

SageMatharcsinh(complex | signed_infinity | ~oo)SymPyasinh(complex | signed_infinity | ~oo)mpmathasinh(complex | signed_infinity | ~oo)Juliaenumeratio_ball(_asinh, complex | signed_infinity | ~oo)Rustasinh(complex | signed_infinity | ~oo)

Typed signatures

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

Domain: Compute engine

Examples

Scope #

A hundred digits of $\operatorname{arsinh}(1)$

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In$N(\mathrm{arsinh}(1), 100)$Out$0.881\,373\,587\,019\,543\,025\,232\,609\,324\,979\,792\,309\,028\,160\,328\,261\,635\,410\,753\,295\,608\,653\,377\,184\,222\,026\,087\,833\,706\,891\,910\,256\,042\,9$

Infinity maps to infinity

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In$\mathrm{arsinh}(∞)$Out$∞$

An odd function through the origin: $\operatorname{arsinh}(0)=0$

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

Derivative is $1/\sqrt{x^2+1}$

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

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In$\mathrm{arsinh}(\bigl\lbrack\bigl\lbrack i, (2^{1/2}/2)i\bigr\rbrack, \bigl\lbrack0, -(2^{1/2}/2)i\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack\mathrm{arsinh}(i), \mathrm{arsinh}((2^{1/2}/2)i)\bigr\rbrack, \bigl\lbrack0, \mathrm{arsinh}(-(2^{1/2}/2)i)\bigr\rbrack\bigr\rbrack$

Properties #

A nested radical on the imaginary axis stays symbolic

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In$\mathrm{arsinh}(\frac{1}{2}i\sqrt{\frac{5-5^{1/2}}{2}})$Out$\mathrm{arsinh}(\frac{\sqrt{2}}{4}i\sqrt{5-\sqrt{5}})$

Known value · Wolfram MathematicalFunctionData, ArcSinh

The negated radical is left unevaluated as well

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In$\mathrm{arsinh}((-\frac{1}{2}i)\sqrt{\frac{5-5^{1/2}}{2}})$Out$\mathrm{arsinh}((-\frac{\sqrt{2}}{4}i)\sqrt{5-\sqrt{5}})$

Known value · Wolfram MathematicalFunctionData, ArcSinh

Possible issues #

Arsinh(ComplexInfinity) is ComplexInfinity: $|\operatorname{arsinh} z| \sim \ln|2z|$ in every direction

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Known value · DLMF 4.37.16

About

Examples	8
Checked against	Wolfram Language 5 agree3 differSageMath 7 agree1 differSymPy 7 agree1 differmpmath 6 agreeJulia 5 agree1 errorRust 2 agree
Crosswalk	DLMFFungrimFungrim entryWolfram 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