# 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

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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 the origin:

Asymptotic expansion at Infinity:

Asymptotic expansion at a singular point:

## Scope(37)

### Numerical Evaluation(5)

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:

CoshIntegral can be used with Interval and CenteredInterval objects:

### 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(2)

Plot the imaginary part in the complex plane:

Solve a differential equation:

## 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: