gives the inverse hyperbolic sine of the complex number .
- 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 .
- ArcSinh can be used with Interval and CenteredInterval objects. »
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.
Examplesopen allclose all
Basic Examples (6)
Plot over a subset of the reals:
Plot over a subset of the complexes:
Asymptotic expansion at Infinity:
Numerical Evaluation (6)
The precision of the output tracks the precision of the input:
Evaluate for complex arguments:
Evaluate ArcSinh efficiently at high precision:
ArcSinh threads elementwise over lists and matrices:
ArcSinh can be used with Interval and CenteredInterval objects:
Specific Values (4)
Plot the ArcSinh function:
Function Properties (12)
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 :
is an analytic function of over the reals:
It is neither analytic nor meromorphic over the complexes:
ArcSinh is non-decreasing:
ArcSinh is injective:
ArcSinh is surjective:
ArcSinh is neither non-negative nor non-positive:
ArcSinh has no singularities or discontinuities:
ArcSinh is neither convex nor concave:
Series Expansions (4)
Integral Transforms (2)
Compute the Fourier transform using FourierTransform:
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:
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)
Wolfram Research (1988), ArcSinh, Wolfram Language function, https://reference.wolfram.com/language/ref/ArcSinh.html (updated 2021).
Wolfram Language. 1988. "ArcSinh." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2021. https://reference.wolfram.com/language/ref/ArcSinh.html.
Wolfram Language. (1988). ArcSinh. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ArcSinh.html