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


  • 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. »


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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:

SinIntegral threads elementwise over lists and matrices:

SinIntegral can be used with Interval and CenteredInterval objects:

Specific Values  (3)

Value at a fixed point:

Values at infinity:

Find a local maximum as a root of (dTemplateBox[{x}, SinIntegral])/(dx)=0:

Visualization  (2)

Plot the SinIntegral function:

Plot the real part of TemplateBox[{z}, SinIntegral]:

Plot the imaginary part of TemplateBox[{z}, SinIntegral]:

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 ^(th) 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 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:


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, (updated 2022).


Wolfram Research (1991), SinIntegral, Wolfram Language function, (updated 2022).


Wolfram Language. 1991. "SinIntegral." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022.


Wolfram Language. (1991). SinIntegral. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2024_sinintegral, author="Wolfram Research", title="{SinIntegral}", year="2022", howpublished="\url{}", note=[Accessed: 20-July-2024 ]}


@online{reference.wolfram_2024_sinintegral, organization={Wolfram Research}, title={SinIntegral}, year={2022}, url={}, note=[Accessed: 20-July-2024 ]}