gives the inverse hyperbolic cosecant of the complex number .
- Mathematical function, suitable for both symbolic and numerical manipulation.
- For certain special arguments, ArcCsch automatically evaluates to exact values.
- ArcCsch can be evaluated to arbitrary numerical precision.
- ArcCsch automatically threads over lists.
- ArcCsch[z] has a branch cut discontinuity in the complex plane running from to .
- ArcCsch can be used with Interval and CenteredInterval objects. »
Background & Context
- ArcCsch is the inverse hyperbolic cosecant function. For a real number , ArcCsch[x] represents the hyperbolic angle measure such that .
- ArcCsch automatically threads over lists. For certain special arguments, ArcCsch automatically evaluates to exact values. When given exact numeric expressions as arguments, ArcCsch may be evaluated to arbitrary numeric precision. Operations useful for manipulation of symbolic expressions involving ArcCsch include FunctionExpand, TrigToExp, TrigExpand, Simplify, and FullSimplify.
- ArcCsch is defined for complex argument by . ArcCsch[z] has a branch cut discontinuity in the complex plane.
- Related mathematical functions include Csch, ArcSech, and ArcCsc.
Examplesopen allclose all
Basic Examples (6)
Plot over a subset of the reals:
Plot over a subset of the complexes:
Series expansion at the origin:
Asymptotic expansion at Infinity:
Numerical Evaluation (6)
The precision of the output tracks the precision of the input:
Evaluate for complex arguments:
Evaluate ArcCsch efficiently at high precision:
ArcCsch threads elementwise over lists and matrices:
ArcCsch can be used with Interval and CenteredInterval objects:
Specific Values (4)
Plot the ArcCsch function:
Function Properties (11)
ArcCsch is defined for all real values except 0:
ArcCsch achieves all real values except 0:
Function range for arguments from the complex domain:
ArcCsch is an odd function:
ArcCsch has the mirror property :
ArcCsch is not an analytic function:
ArcCsch is neither non-decreasing nor non-increasing:
ArcCsch is injective:
ArcCsch is neither non-negative nor non-positive:
It has both singularity and discontinuity at zero:
ArcCsch is neither convex nor concave:
Series Expansions (3)
Integral Transforms (2)
Compute the Mellin transform using MellinTransform:
Function Identities and Simplifications (3)
Function Representations (5)
Represent using ArcSinh:
Representation through inverse Jacobi functions:
Represent using Hypergeometric2F1:
ArcCsch can be represented in terms of MeijerG:
ArcCsch can be represented as a DifferentialRoot:
Branch cut structure of ArcCsch runs along the imaginary axis:
Solve a differential equation:
Solution of the sinh‐Gordon equation :
Properties & Relations (2)
Compose with the inverse function:
Use PowerExpand to disregard multivaluedness of the ArcCsch:
Alternatively, evaluate under additional assumptions:
Use TrigToExp to express in terms of logarithm:
Wolfram Research (1988), ArcCsch, Wolfram Language function, https://reference.wolfram.com/language/ref/ArcCsch.html (updated 2021).
Wolfram Language. 1988. "ArcCsch." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2021. https://reference.wolfram.com/language/ref/ArcCsch.html.
Wolfram Language. (1988). ArcCsch. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ArcCsch.html