SinIntegral

SinIntegral[z]

gives the sine integral function $TemplateBox[{z}, SinIntegral]=int_0^zsin(t)/t dt$ .

Details

Mathematical function, suitable for both symbolic and numerical manipulation.
$TemplateBox[{z}, SinIntegral]=int_0^zsin(t)/t dt$ .
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:

TraditionalForm formatting:

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 Euler–Heisenberg 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 moderate‐size arguments:

A larger setting for $MaxExtraPrecision can be needed:

Neat Examples (1)

Nested integrals:

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