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)
Asymptotic expansion at Infinity:
Numerical Evaluation (6)
Evaluate ArcSinh efficiently at high precision:
ArcSinh threads elementwise over lists and matrices:
Specific Values (4)
Plot the ArcSinh function:
Function Properties (12)
ArcSinh is defined for all real and complex values:
ArcSinh achieves all real values:
ArcSinh is an odd function:
ArcSinh has the mirror property :
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)
Function Identities and Simplifications (3)
Properties & Relations (4)
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