# SinhIntegral

SinhIntegral[z]

gives the hyperbolic sine integral function .

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• .
• SinhIntegral[z] is an entire function of z with no branch cut discontinuities.
• For certain special arguments, SinhIntegral automatically evaluates to exact values.
• SinhIntegral can be evaluated to arbitrary numerical precision.
• SinhIntegral automatically threads over lists.

# Examples

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## Basic Examples(5)

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:

## Scope(32)

### Numerical Evaluation(4)

Evaluate numerically to high precision:

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

SinhIntegral can take complex number inputs:

Evaluate SinhIntegral efficiently at high precision:

### Specific Values(3)

Value at the origin:

Values at infinity:

Find the real root of the equation :

### Visualization(2)

Plot the SinhIntegral function:

Plot the real part of :

Plot the imaginary part of :

### Function Properties(3)

SinhIntegral is defined for all real and complex values:

SinhIntegral takes all the real values:

SinhIntegral is an odd function:

### Differentiation(3)

First derivative:

Higher derivatives:

Formula for the  derivative:

### Integration(3)

Indefinite integral of SinhIntegral:

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

More integrals:

### Series Expansions(4)

Taylor expansion for SinhIntegral:

Plot the first three approximations for SinhIntegral around :

General term in the series expansion of SinhIntegral:

Find series expansions at infinity:

Give the result for an arbitrary symbolic direction :

SinhIntegral can be applied to power series:

### Integral Transforms(2)

Compute the Laplace transform using LaplaceTransform:

### Function Identities and Simplifications(3)

Primary definition of SinhIntegral:

Argument simplifications:

Simplify expressions to SinhIntegral:

### Function Representations(5)

Representation in terms of SinIntegral:

Series representation of SinhIntegral:

SinhIntegral can be represented in terms of MeijerG:

SinhIntegral can be represented as a DifferentialRoot:

## Applications(1)

Plot the real part in the complex plane:

## Properties & Relations(5)

Use FullSimplify to simplify expressions containing hyperbolic sine integrals:

Find a numerical root:

Obtain SinhIntegral from integrals and sums:

Integrals:

Laplace transform:

## Possible Issues(3)

SinhIntegral can take large values for moderatesize arguments:

A larger setting for \$MaxExtraPrecision can be needed: In traditional form, parentheses are required:

## Neat Examples(1)

Nested integrals: