# Sinh

Sinh[z]

gives the hyperbolic sine of z.

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• For certain special arguments, Sinh automatically evaluates to exact values.
• Sinh can be evaluated to arbitrary numerical precision.
• Sinh automatically threads over lists.

# Background & Context

• Sinh is the hyperbolic sine function, which is the hyperbolic analogue of the Sin circular function used throughout trigonometry. It is defined for real numbers by letting be twice the area between the axis and a ray through the origin intersecting the unit hyperbola . Sinh[α] then gives the vertical coordinate of the intersection point. Sinh may also be defined as , where is the base of the natural logarithm Log.
• Sinh automatically evaluates to exact values when its argument is the (natural) logarithm of a rational number. When given exact numeric expressions as arguments, Sinh may be evaluated to arbitrary numeric precision. Other operations useful for manipulation of symbolic expressions involving Sinh include TrigToExp, TrigExpand, Simplify, and FullSimplify.
• Sinh threads element-wise over lists and matrices. In contrast, MatrixFunction can be used to give the hyperbolic sine of a square matrix (i.e. the power series for the hyperbolic sine function with ordinary powers replaced by matrix powers) as opposed to the hyperbolic sines of the individual matrix elements.
• Sinh[x] decreases exponentially as x approaches and increases exponentially as x approaches . Sinh satisfies an identity similar to the Pythagorean identity satisfied by Sin, namely . The definition of the hyperbolic sine function is extended to complex arguments by way of the identity . The hyperbolic sine function is entire, meaning it is complex differentiable at all finite points of the complex plane. Sinh[z] has series expansion about the origin.
• The inverse function of Sinh is ArcSinh. Related mathematical functions include Cosh and Csch.

# Examples

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## Basic Examples(4)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at 0:

## Scope(40)

### Numerical Evaluation(6)

Evaluate numerically:

Evaluate to high precision:

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

Sinh can take complex number inputs:

Evaluate Sinh efficiently at high precision:

Sinh can deal with realvalued intervals:

Sinh threads elementwise over lists and matrices:

### Specific Values(4)

Values of Sinh at fixed, purely imaginary points:

Values at infinity:

Zero of Sinh:

Find the zero of Sinh using Solve:

Substitute in the value:

Visualize the result:

Simple exact values are generated automatically:

More complicated cases require explicit use of FunctionExpand:

### Visualization(3)

Plot the Sinh function:

Plot the real part of :

Plot the imaginary part of :

Polar plot with :

### Function Properties(5)

Sinh is defined for all real and complex values:

Sinh achieves all real values:

The range for complex values is the whole plane:

Sinh is an odd function:

Sinh has the mirror property :

### Differentiation(3)

First derivative:

Higher derivatives:

Formula for the  derivative:

### Integration(3)

Indefinite integral of Sinh:

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

More integrals:

### Series Expansions(4)

Find the Taylor expansion using Series:

Plot the first three approximations for Sinh around :

General term in the series expansion of Sinh:

A few first terms of Fourier series:

Sinh can be applied to power series:

### Integral Transforms(2)

Compute the Laplace transform using LaplaceTransform:

### Function Identities and Simplifications(6)

Sinh of a double angle:

Sinh of a sum:

Convert multipleangle expressions:

Convert sums of hyperbolic functions to products:

Expand assuming real variables and :

Convert to exponentials:

### Function Representations(4)

Representation through Sin:

Representation through Bessel functions:

Representation in terms of MeijerG:

Sinh can be represented as a DifferentialRoot:

## Applications(5)

Draw a hyperbola:

Rotation matrix in hyperbolic space:

Build from infinitesimal transformations:

Relativistic boost matrix:

The matrix is orthogonal with respect to the Minkowski metric:

Construct a relativistic coordinate transformation for rapidity :

Prolate spheroidal coordinates:

Special solution of the sineGordon equation:

## Properties & Relations(11)

Basic parity and periodicity properties of the hyperbolic sine function get automatically applied:

Complicated expressions containing hyperbolic functions do not autosimplify:

Compose with inverse functions:

Solve a hyperbolic equation:

Numerically find a root of a transcendental equation:

Reduce a hyperbolic equation:

Integral transforms:

Sinh appears in special cases of many mathematical functions:

Sinh is a numeric function:

The generating function for Sinh:

The exponential generating function for Sinh:

## Possible Issues(5)

Machine-precision input is insufficient to give a correct answer:

With exact input, the answer is correct:

A larger setting for \$MaxExtraPrecision can be needed: Machine number inputs can give highprecision results:

No power series exists at infinity, where Sinh has an essential singularity:

In TraditionalForm, parentheses are needed around the argument:

## Neat Examples(2)

Nested hyperbolic cosine over the complex plane:

Introduced in 1988
(1.0)