SinhIntegral

SinhIntegral[z]

gives the hyperbolic sine integral function .

Details

Mathematical function, suitable for both symbolic and numerical manipulation.
$TemplateBox[{z}, SinhIntegral]=int_0^zsinh(t)/tdt$ .
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.
SinhIntegral can be used with Interval and CenteredInterval objects. »

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 (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:

SinhIntegral 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 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:

HankelTransform:

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:

TraditionalForm formatting:

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 moderate‐size arguments:

A larger setting for $MaxExtraPrecision can be needed:

In traditional form, parentheses are required:

Neat Examples (1)

Nested integrals:

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