# ArcCosh

ArcCosh[z]

gives the inverse hyperbolic cosine of the complex number .

# Details

• 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.

# Examples

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## Basic Examples(5)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Asymptotic expansion at Infinity:

Asymptotic expansion at a singular point:

## Scope(41)

### Numerical Evaluation(6)

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Evaluate for complex arguments:

Evaluate ArcCosh efficiently at high precision:

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

Or compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix ArcCosh function using MatrixFunction:

### Specific Values(4)

Values of ArcCosh at fixed points:

Values at infinity:

Zero of ArcCosh:

Find the value of satisfying equation :

Substitute in the value:

Visualize the result:

### Visualization(3)

Plot the ArcCosh function:

Plot the real part of :

Plot the imaginary part of :

Polar plot with :

### 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:

Nor is it meromorphic:

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:

### Differentiation(3)

First derivative:

Higher derivatives:

Formula for the derivative:

### Integration(3)

Indefinite integral of ArcCosh:

Definite integral of ArcCosh over an interval outside of the real domain is imaginary:

More integrals:

### Series Expansions(4)

Find the Taylor expansion using Series:

Plot the first three approximations for ArcCosh around :

General term in the series expansion of ArcCosh around :

Find series expansions at branch points and branch cuts:

ArcCosh can be applied to power series:

### Function Identities and Simplifications(3)

Simplify expressions involving ArcCosh:

Use TrigToExp to express through logarithms and square roots:

Convert back:

Expand assuming real variables and :

### 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:

## Applications(4)

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:

Solve a differential equation:

## Properties & Relations(5)

Compositions with the inverse function might need PowerExpand to simplify to an identity:

This shows the branch cuts of the ArcCosh function:

Solve an inverse trigonometric equation:

Solve for zeros:

Solve the differential equation satisfied by ArcCosh:

Verify it is satisfied by ArcCosh:

## Possible Issues(2)

Generically :

On branch cuts, machine-precision inputs can give numerically wrong answers:

## Neat Examples(1)

Wolfram Research (1988), ArcCosh, Wolfram Language function, https://reference.wolfram.com/language/ref/ArcCosh.html (updated 2021).

#### Text

Wolfram Research (1988), ArcCosh, Wolfram Language function, https://reference.wolfram.com/language/ref/ArcCosh.html (updated 2021).

#### CMS

Wolfram Language. 1988. "ArcCosh." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2021. https://reference.wolfram.com/language/ref/ArcCosh.html.

#### APA

Wolfram Language. (1988). ArcCosh. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ArcCosh.html

#### BibTeX

@misc{reference.wolfram_2024_arccosh, author="Wolfram Research", title="{ArcCosh}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/ArcCosh.html}", note=[Accessed: 12-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_arccosh, organization={Wolfram Research}, title={ArcCosh}, year={2021}, url={https://reference.wolfram.com/language/ref/ArcCosh.html}, note=[Accessed: 12-September-2024 ]}