# SinIntegral

SinIntegral[z]

gives the sine integral function .

# Details

• Mathematical function, suitable for both symbolic and numerical manipulation.
• .
• SinIntegral[z] is an entire function of with no branch cut discontinuities.
• For certain special arguments, SinIntegral automatically evaluates to exact values.
• SinIntegral can be evaluated to arbitrary numerical precision.
• SinIntegral automatically threads over lists.
• SinIntegral can be used with Interval and CenteredInterval objects. »

# Examples

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## Basic Examples(6)

Evaluate numerically:

Plot :

Plot over a subset of the complexes:

Differentiate :

Series expansion at the origin:

Asymptotic expansion at Infinity:

## Scope(37)

### Numerical Evaluation(5)

Evaluate numerically to high precision:

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

Evaluate for complex arguments:

Evaluate SinIntegral 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 SinIntegral function using MatrixFunction:

### Specific Values(3)

Value at a fixed point:

Values at infinity:

Find a local maximum as a root of :

### Visualization(2)

Plot the SinIntegral function:

Plot the real part of :

Plot the imaginary part of :

### Function Properties(10)

SinIntegral is defined for all real and complex values:

Approximate function range of SinIntegral:

SinIntegral is an odd function:

SinIntegral is an analytic function of x:

SinIntegral is neither non-decreasing nor non-increasing:

SinIntegral is not injective:

SinIntegral is not surjective:

SinIntegral is neither non-negative nor non-positive:

SinIntegral has no singularities or discontinuities:

SinIntegral is neither convex nor concave:

### Differentiation(3)

First derivative:

Higher derivatives:

Formula for the derivative:

### Integration(3)

Indefinite integral of SinIntegral:

Definite integral of an odd integrand over an interval centered at the origin is 0:

More integrals:

### Series Expansions(4)

Taylor expansion for SinIntegral:

Plot the first three approximations for SinIntegral around :

General term in the series expansion of SinIntegral:

Find series expansion at infinity:

Give the result for an arbitrary symbolic direction :

SinIntegral can be applied to power series:

### Function Identities and Simplifications(3)

Use FullSimplify to simplify expressions containing sine integrals:

Simplify expressions to SinIntegral:

Argument simplifications:

### Function Representations(4)

Series representation of SinIntegral:

SinIntegral can be represented in terms of MeijerG:

SinIntegral can be represented as a DifferentialRoot:

## Generalizations & Extensions(1)

Find series expansions at infinity:

Give the result for an arbitrary symbolic direction :

## Applications(6)

Plot the absolute value in the complex plane:

Real part of the EulerHeisenberg effective action:

Find a leading term in :

Gibbs phenomenon for a square wave:

Magnify the overshoot region:

Compute the asymptotic overshoot:

Solve a differential equation:

Integrate a composition of trigonometric functions:

Plot Nielsen's spiral:

The curvature is a simple function of the parameter:

## Properties & Relations(7)

Parity transformation is automatically applied:

Use FullSimplify to simplify expressions containing sine integrals:

Find a numerical root:

Obtain SinIntegral from integrals and sums:

Obtain SinIntegral from a differential equation:

Calculate the Wronskian:

Compare with Wronskian:

Integrals:

Laplace transform:

## Possible Issues(2)

SinIntegral can take large values for moderatesize arguments:

A larger setting for \$MaxExtraPrecision can be needed:

## Neat Examples(1)

Nested integrals:

Wolfram Research (1991), SinIntegral, Wolfram Language function, https://reference.wolfram.com/language/ref/SinIntegral.html (updated 2022).

#### Text

Wolfram Research (1991), SinIntegral, Wolfram Language function, https://reference.wolfram.com/language/ref/SinIntegral.html (updated 2022).

#### CMS

Wolfram Language. 1991. "SinIntegral." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/SinIntegral.html.

#### APA

Wolfram Language. (1991). SinIntegral. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SinIntegral.html

#### BibTeX

@misc{reference.wolfram_2024_sinintegral, author="Wolfram Research", title="{SinIntegral}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/SinIntegral.html}", note=[Accessed: 14-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_sinintegral, organization={Wolfram Research}, title={SinIntegral}, year={2022}, url={https://reference.wolfram.com/language/ref/SinIntegral.html}, note=[Accessed: 14-September-2024 ]}