# CoshIntegral

CoshIntegral[z]

gives the hyperbolic cosine integral .

# Details

• Mathematical function, suitable for both symbolic and numerical manipulation.
• , where is Eulers constant.
• CoshIntegral[z] has a branch cut discontinuity in the complex z plane running from - to 0.
• For certain special arguments, CoshIntegral automatically evaluates to exact values.
• CoshIntegral can be evaluated to arbitrary numerical precision.
• CoshIntegral automatically threads over lists.
• CoshIntegral can be used with Interval and CenteredInterval objects. »

# Examples

open allclose all

## Basic Examples(6)

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:

Asymptotic expansion at a singular point:

## Scope(38)

### Numerical Evaluation(6)

Evaluate to high precision:

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

Evaluate for complex arguments:

Evaluate CoshIntegral 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 CoshIntegral function using MatrixFunction:

### Specific Values(3)

Value at the origin:

Values at infinity:

Find the zero of CoshIntegral:

### Visualization(2)

Plot the CoshIntegral function:

Plot the real part of :

Plot the imaginary part of :

### Function Properties(9)

CoshIntegral is defined for all real positive values:

Complex domain:

CoshIntegral takes all the real values:

CoshIntegral is not an analytic function:

Nor is it meromorphic:

CoshIntegral is increasing on its real domain:

CoshIntegral is injective:

CoshIntegral is surjective:

CoshIntegral is neither non-negative nor non-positive:

It has both singularity and discontinuity in (-,0]:

CoshIntegral is neither convex nor concave:

### Differentiation(3)

First derivative:

Higher derivatives:

Formula for the derivative:

### Integration(3)

Indefinite integral of CoshIntegral:

Definite integral:

More integrals:

### Series Expansions(3)

Series expansion for CoshIntegral:

Plot the first three approximations for CoshIntegral around :

Find asymptotic series expansion at infinity:

CoshIntegral can be applied to power series:

### Integral Transforms(2)

Compute the Laplace transform using LaplaceTransform:

### Function Identities and Simplifications(3)

Primary definition of CoshIntegral:

Argument simplifications:

Simplify expressions to CoshIntegral:

### Function Representations(4)

Representation in terms of CosIntegral and Log:

CoshIntegral can be represented in terms of MeijerG:

CoshIntegral can be represented as a DifferentialRoot:

## Applications(3)

Plot the imaginary part in the complex plane:

Solve a differential equation:

Find the antiderivative using DSolveValue:

Compare with the answer given by Integrate:

## Properties & Relations(3)

Use FullSimplify to simplify expressions containing the hyperbolic cosine integral:

Use FunctionExpand to express CoshIntegral through other functions:

Find a numerical root:

Obtain CoshIntegral from integrals and sums:

## Possible Issues(2)

CoshIntegral can take large values for moderatesize arguments:

A larger setting for \$MaxExtraPrecision can be needed:

## Neat Examples(2)

Nested integrals:

Plot the logarithm of the absolute value in the complex plane:

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

#### Text

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

#### CMS

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

#### APA

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

#### BibTeX

@misc{reference.wolfram_2024_coshintegral, author="Wolfram Research", title="{CoshIntegral}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/CoshIntegral.html}", note=[Accessed: 14-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_coshintegral, organization={Wolfram Research}, title={CoshIntegral}, year={2022}, url={https://reference.wolfram.com/language/ref/CoshIntegral.html}, note=[Accessed: 14-September-2024 ]}