gives the inverse hyperbolic cosine of the complex number .
- Mathematical function, suitable for both symbolic and numerical manipulation.
- For certain special arguments, ArcCosh automatically evaluates to exact values.
- ArcCosh can be evaluated to arbitrary numerical precision.
- ArcCosh automatically threads over lists.
- ArcCosh[z] has a branch cut discontinuity in the complex plane running from to .
- ArcCosh can be used with Interval and CenteredInterval objects. »
Background & Context
- ArcCosh is the inverse hyperbolic cosine function. For a real number , ArcCosh[x] represents the hyperbolic angle measure such that .
- ArcCosh automatically threads over lists. For certain special arguments, ArcCosh automatically evaluates to exact values. When given exact numeric expressions as arguments, ArcCosh may be evaluated to arbitrary numeric precision. Operations useful for manipulation of symbolic expressions involving ArcCosh include FunctionExpand, TrigToExp, TrigExpand, Simplify, and FullSimplify.
- ArcCosh is defined for complex argument by . ArcCosh[z] has a branch cut discontinuity in the complex plane.
- Related mathematical functions include Cosh, ArcSinh, and ArcCos.
Examplesopen allclose all
Basic Examples (5)
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 ArcCosh efficiently at high precision:
ArcCosh threads elementwise over lists:
ArcCosh can be used with Interval and CenteredInterval objects:
Specific Values (4)
Plot the ArcCosh function:
Function Properties (10)
ArcCosh is defined for all real values greater than or equal to 1:
Complex domain is the whole plane:
ArcCosh achieves all real values greater than or equal to 0:
Function range for arguments from the complex domain:
ArcCosh is not an analytic function:
ArcCosh is increasing over its real domain:
ArcCosh is injective:
ArcCosh is not surjective:
ArcCosh is non-negative over its real domain:
It has both singularity and discontinuity in (-∞,1]:
ArcCosh is concave over its real domain:
Series Expansions (4)
Function Identities and Simplifications (3)
Function Representations (5)
Represent using ArcSech:
Representation through inverse Jacobi functions:
Represent using Hypergeometric2F1:
ArcCosh can be represented in terms of MeijerG:
ArcCosh can be represented as a DifferentialRoot:
Find the rapidity of a boost that makes the energy of a body twice its rest energy:
Fraction of the speed of light required:
Width at the base of the inverted catenary arch of Gateway Arch in St. Louis, Missouri, in feet:
Plot the real and imaginary part of ArcCosh:
Properties & Relations (5)
Compositions with the inverse function might need PowerExpand to simplify to an identity:
Alternatively, use additional assumptions:
This shows the branch cuts of the ArcCosh function:
Solve an inverse trigonometric equation:
Solve the differential equation satisfied by ArcCosh:
Verify it is satisfied by ArcCosh:
Possible Issues (2)
Neat Examples (1)
Wolfram Research (1988), ArcCosh, Wolfram Language function, https://reference.wolfram.com/language/ref/ArcCosh.html (updated 2021).
Wolfram Language. 1988. "ArcCosh." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2021. https://reference.wolfram.com/language/ref/ArcCosh.html.
Wolfram Language. (1988). ArcCosh. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ArcCosh.html