CoshIntegral

CoshIntegral[z]

gives the hyperbolic cosine integral .

Details

Mathematical function, suitable for both symbolic and numerical manipulation.
$TemplateBox[{z}, CoshIntegral]=gamma+log(z)+int_0^z(cosh(t)-1)/tdt$ , where is Euler’s 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 (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:

CoshIntegral threads elementwise over lists:

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:

HankelTransform:

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:

TraditionalForm formatting:

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 moderate‐size 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:

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