# ArcTanh

ArcTanh[z]

gives the inverse hyperbolic tangent of the complex number .

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

# Background & Context

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

# Examples

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## Basic Examples(6)

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(42)

### 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 ArcTanh efficiently at high precision:

ArcTanh can deal with realvalued intervals:

ArcTanh threads elementwise over lists and matrices:

### Specific Values(4)

Values of ArcTanh at fixed points:

Values at infinity:

Zero of ArcTanh:

Find the value of satisfying equation :

Substitute in the value:

Visualize the result:

### Visualization(3)

Plot the ArcTanh function:

Plot the real part of :

Plot the imaginary part of :

Polar plot with :

### Function Properties(11)

ArcTanh is defined for all real values from the interval :

Complex domain:

ArcTanh achieves all real values:

Function range for arguments from the complex domain:

ArcTanh is an odd function:

ArcTanh is not an analytic function:

Nor is it meromorphic:

ArcTanh is increasing over its real domain:

ArcTanh is injective:

ArcTanh is surjective:

ArcTanh is neither non-negative nor non-positive:

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

ArcTanh is neither convex nor concave:

### Differentiation(3)

First derivative:

Higher derivatives:

Formula for the  derivative:

### Integration(3)

Indefinite integral of ArcTanh:

Definite integral of ArcTanh over an interval centered at the origin is 0:

More integrals:

### Series Expansions(4)

Find the Taylor expansion using Series:

Plots of the first three approximations for ArcTanh around :

General term in the series expansion of ArcTanh:

Find series expansions at branch points and branch cuts:

ArcTanh can be applied to power series:

### Function Identities and Simplifications(3)

Simplify expressions involving ArcTanh:

Express ArcTanh using Log:

Convert back:

Expand assuming real variables and :

### Function Representations(5)

Represent using ArcCoth:

Representation through inverse Jacobi functions:

ArcTanh is a special case of the hypergeometric function Hypergeometric2F1:

ArcTanh can be represented in terms of MeijerG:

ArcTanh can be represented as a DifferentialRoot:

## Applications(3)

Find the rapidity corresponding to a speed of 0.999 times the speed of light:

Branch cuts of ArcTanh:

Solve a differential equation:

## Properties & Relations(2)

Express ArcTanh using Log:

Convert back:

ArcTanh is a special case of some special functions: