# ArcSinh

ArcSinh[z]

gives the inverse hyperbolic sine of the complex number .

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

# Background & Context

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

# 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 0:

Asymptotic expansion at Infinity:

Asymptotic expansion at a singular point:

## Scope(45)

### 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 ArcSinh efficiently at high precision:

ArcSinh can deal with real-valued intervals:

ArcSinh threads elementwise over lists and matrices:

### Specific Values(4)

Values of ArcSinh at fixed points:

Values at infinity:

Zero of ArcSinh:

Find the value of satisfying equation :

Substitute in the value:

Visualize the result:

### Visualization(3)

Plot the ArcSinh function:

Plot the real part of :

Plot the imaginary part of :

Polar plot with :

### Function Properties(12)

ArcSinh is defined for all real and complex values:

ArcSinh achieves all real values:

Function range for arguments from the complex domain:

ArcSinh is an odd function:

ArcSinh has the mirror property : is an analytic function of over the reals:

It is neither analytic nor meromorphic over the complexes:

ArcSinh is non-decreasing:

ArcSinh is injective:

ArcSinh is surjective:

ArcSinh is neither non-negative nor non-positive:

ArcSinh has no singularities or discontinuities:

ArcSinh is neither convex nor concave:

### Differentiation(3)

First derivative:

Higher derivatives:

Formula for the  derivative:

### Integration(3)

Indefinite integral of ArcSinh:

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

More integrals:

### Series Expansions(4)

Find the Taylor expansion using Series:

Plot the first three approximations for ArcSinh around :

General term in the series expansion of ArcSinh:

Find series expansions at branch points and branch cuts:

ArcSinh can be applied to a power series:

### Integral Transforms(2)

Compute the Fourier transform using FourierTransform:

### Function Identities and Simplifications(3)

Simplify expressions involving ArcSinh:

Use TrigToExp to express ArcSinh using logarithm:

Expand assuming real variables and :

### Function Representations(5)

Represent using ArcCsch:

Representation through inverse Jacobi functions:

Represent using Hypergeometric2F1:

ArcSinh can be represented in terms of MeijerG:

ArcSinh can be represented as a DifferentialRoot:

## Applications(2)

Compute the length of hyperbola from the base to given :

Solve a differential equation:

## Properties & Relations(4)

Compose with the inverse function:

Use PowerExpand to disregard multivaluedness of the ArcSinh:

Use TrigToExp to express ArcSinh using logarithm:

Use Reduce to solve an equation in terms of ArcSinh:

ArcSinh is a special case of some special functions:

## Possible Issues(2)

Generically :

When using input in traditional form, parentheses are needed around the argument:

## Neat Examples(1)

Compute 100000 digits of , and show the first and last 20 digits: