# SinhIntegral

SinhIntegral[z]

gives the hyperbolic sine integral function .

# Examples

open allclose all

## 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(41)

### Numerical Evaluation(6)

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:

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

Or compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix SinhIntegral function using MatrixFunction:

### 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(10)

SinhIntegral is defined for all real and complex values:

SinhIntegral takes all the real values:

SinhIntegral is an odd function:

SinhIntegral is an analytic function of x:

SinhIntegral is non-decreasing:

SinhIntegral is injective:

SinhIntegral is surjective:

SinhIntegral is neither non-negative nor non-positive:

SinhIntegral has no singularities or discontinuities:

SinhIntegral is neither convex nor concave:

### 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(3)

Plot the real part in the complex plane:

Solve a differential equation:

Find the antiderivative using DSolveValue:

Compare with the answer given by Integrate:

## Properties & Relations(6)

SinhIntegral is bijective on the reals:

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:

Wolfram Research (1996), SinhIntegral, Wolfram Language function, https://reference.wolfram.com/language/ref/SinhIntegral.html (updated 2022).

#### Text

Wolfram Research (1996), SinhIntegral, Wolfram Language function, https://reference.wolfram.com/language/ref/SinhIntegral.html (updated 2022).

#### CMS

Wolfram Language. 1996. "SinhIntegral." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/SinhIntegral.html.

#### APA

Wolfram Language. (1996). SinhIntegral. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SinhIntegral.html

#### BibTeX

@misc{reference.wolfram_2024_sinhintegral, author="Wolfram Research", title="{SinhIntegral}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/SinhIntegral.html}", note=[Accessed: 07-August-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_sinhintegral, organization={Wolfram Research}, title={SinhIntegral}, year={2022}, url={https://reference.wolfram.com/language/ref/SinhIntegral.html}, note=[Accessed: 07-August-2024 ]}