ArcSinh
ArcSinh[z]
gives the inverse hyperbolic sine of the complex number
.
Details

- Mathematical function, suitable for both symbolic and numerical manipulation.
- For certain special arguments, ArcSinh automatically evaluates to exact values.
- ArcSinh can be evaluated to arbitrary numerical precision.
- ArcSinh automatically threads over lists.
- ArcSinh[z] has branch cut discontinuities in the complex
plane running from
to
and
to
.
Background & Context
- ArcSinh is the inverse hyperbolic sine function. For a real number
, ArcSinh[x] represents the hyperbolic angle measure
such that
.
- ArcSinh automatically threads over lists. For certain special arguments, ArcSinh automatically evaluates to exact values. When given exact numeric expressions as arguments, ArcSinh may be evaluated to arbitrary numeric precision. Operations useful for manipulation of symbolic expressions involving ArcSinh include FunctionExpand, TrigToExp, TrigExpand, Simplify, and FullSimplify.
- ArcSinh is defined for complex argument
by
. ArcSinh[z] has branch cut discontinuities in the complex
plane.
- Related mathematical functions include Sinh, ArcCosh, and ArcSin.
Examples
open allclose allBasic Examples (6)
Plot over a subset of the reals:
Plot over a subset of the complexes:
Asymptotic expansion at Infinity:
Scope (38)
Numerical Evaluation (6)
Specific Values (4)
Visualization (3)
Function Properties (5)
ArcSinh is defined for all real and complex values:
ArcSinh achieves all real values:
Function range for arguments from the complex domain:
ArcSinh is an odd function:
ArcSinh has the mirror property :
TraditionalForm formatting:
Integration (3)
Series Expansions (4)
Integral Transforms (2)
Function Identities and Simplifications (3)
Function Representations (5)
Represent using ArcCsch:
Representation through inverse Jacobi functions:
Represent using Hypergeometric2F1:
ArcSinh can be represented in terms of MeijerG:
ArcSinh can be represented as a DifferentialRoot:
Applications (2)
Properties & Relations (4)
Compose with the inverse function:
Use PowerExpand to disregard multivaluedness of the ArcSinh:
Alternatively, evaluate under additional assumptions:
Use TrigToExp to express ArcSinh using logarithm:
Use Reduce to solve an equation in terms of ArcSinh:
ArcSinh is a special case of some special functions:
Possible Issues (2)
Text
Wolfram Research (1988), ArcSinh, Wolfram Language function, https://reference.wolfram.com/language/ref/ArcSinh.html.
BibTeX
BibLaTeX
CMS
Wolfram Language. 1988. "ArcSinh." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/ArcSinh.html.
APA
Wolfram Language. (1988). ArcSinh. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ArcSinh.html