# ArcCoth

ArcCoth[z]

gives the inverse hyperbolic cotangent of the complex number .

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• For certain special arguments, ArcCoth automatically evaluates to exact values.
• ArcCoth can be evaluated to arbitrary numerical precision.
• ArcCoth automatically threads over lists.
• ArcCoth[z] has a branch cut discontinuity in the complex plane running from to .

# Background & Context

• ArcCoth is the inverse hyperbolic cotangent function. For a real number , ArcCoth[x] represents the hyperbolic angle measure such that .
• ArcCoth automatically threads over lists. For certain special arguments, ArcCoth automatically evaluates to exact values. When given exact numeric expressions as arguments, ArcCoth may be evaluated to arbitrary numeric precision. Operations useful for manipulation of symbolic expressions involving ArcCoth include FunctionExpand, TrigToExp, TrigExpand, Simplify, and FullSimplify.
• ArcCoth is defined for complex argument by . ArcCoth[z] has a branch cut discontinuity in the complex plane.
• Related mathematical functions include ArcTanh, Coth, and ArcCot.

# Examples

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## Basic Examples(6)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Asymptotic expansion at Infinity:

Asymptotic expansion at a singular point:

## Scope(43)

### 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 ArcCoth efficiently at high precision:

ArcCoth can deal with realvalued intervals:

ArcCoth threads elementwise over lists and matrices:

### Specific Values(5)

Values of ArcCoth at fixed points:

Values at infinity:

Singular points of ArcCoth:

Find the value of satisfying equation :

Simple exact values are generated automatically:

### Visualization(3)

Plot the ArcCoth function:

Plot the real part of :

Plot the imaginary part of :

Polar plot with :

### Function Properties(11)

ArcCoth is defined for all real values except from the interval :

Complex domain:

ArcCoth achieves all real values except 0:

Function range for arguments from the complex domain:

ArcCoth is an odd function:

ArcCoth is not an analytic function:

Nor is it meromorphic:

ArcCoth is neither non-decreasing nor non-increasing:

ArcCoth is injective:

ArcCoth is not surjective:

ArcCoth is neither non-negative nor non-positive:

It has both singularity and discontinuity in [-1,1]:

ArcCoth is neither convex nor concave:

### Differentiation(3)

First derivative:

Higher derivatives:

Formula for the  derivative:

### Integration(3)

Indefinite integral of ArcCoth:

A definite integral involving ArcCoth:

More integrals:

### Series Expansions(4)

Find the Taylor expansion using Series:

Plot the first three approximations for ArcCoth around :

General term in the series expansion of ArcCoth:

Find series expansions at branch points and branch cuts:

ArcCoth can be applied to power series:

### Function Identities and Simplifications(3)

Simplify expressions involving ArcCoth:

Express ArcCoth using Log:

Convert back:

Expand assuming real variables and :

### Function Representations(5)

Represent using ArcCoth:

Representation through inverse Jacobi functions:

ArcCoth is a special case of Hypergeometric2F1:

ArcCoth can be represented in terms of MeijerG:

ArcCoth can be represented as a DifferentialRoot:

## Applications(2)

Branch cuts of ArcCoth:

Solve a differential equation:

## Properties & Relations(1)

Express ArcCoth using Log:

Convert back: