SinIntegral
✖
SinIntegral
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
open allclose allBasic Examples (6)Summary of the most common use cases

https://wolfram.com/xid/0enwmonu8-db5


https://wolfram.com/xid/0enwmonu8-blk

Plot over a subset of the complexes:

https://wolfram.com/xid/0enwmonu8-kiedlx


https://wolfram.com/xid/0enwmonu8-h5nha8

Series expansion at the origin:

https://wolfram.com/xid/0enwmonu8-njw

Asymptotic expansion at Infinity:

https://wolfram.com/xid/0enwmonu8-d7gawi

Scope (37)Survey of the scope of standard use cases
Numerical Evaluation (5)
Evaluate numerically to high precision:

https://wolfram.com/xid/0enwmonu8-pbs

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

https://wolfram.com/xid/0enwmonu8-mp6

Evaluate for complex arguments:

https://wolfram.com/xid/0enwmonu8-e294kw

Evaluate SinIntegral efficiently at high precision:

https://wolfram.com/xid/0enwmonu8-di5gcr


https://wolfram.com/xid/0enwmonu8-bq2c6r

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

https://wolfram.com/xid/0enwmonu8-i6j67m


https://wolfram.com/xid/0enwmonu8-lmyeh7

Or compute average-case statistical intervals using Around:

https://wolfram.com/xid/0enwmonu8-cw18bq

Compute the elementwise values of an array:

https://wolfram.com/xid/0enwmonu8-thgd2

Or compute the matrix SinIntegral function using MatrixFunction:

https://wolfram.com/xid/0enwmonu8-o5jpo

Specific Values (3)

https://wolfram.com/xid/0enwmonu8-kjo


https://wolfram.com/xid/0enwmonu8-bdij6w


https://wolfram.com/xid/0enwmonu8-drqkdo

Find a local maximum as a root of :

https://wolfram.com/xid/0enwmonu8-f2hrld


https://wolfram.com/xid/0enwmonu8-dr32e4

Visualization (2)
Plot the SinIntegral function:

https://wolfram.com/xid/0enwmonu8-ecj8m7


https://wolfram.com/xid/0enwmonu8-bo5grg


https://wolfram.com/xid/0enwmonu8-hsbqli

Function Properties (10)
SinIntegral is defined for all real and complex values:

https://wolfram.com/xid/0enwmonu8-cl7ele


https://wolfram.com/xid/0enwmonu8-de3irc

Approximate function range of SinIntegral:

https://wolfram.com/xid/0enwmonu8-evf2yr

SinIntegral is an odd function:

https://wolfram.com/xid/0enwmonu8-dnla5q

SinIntegral is an analytic function of x:

https://wolfram.com/xid/0enwmonu8-h5x4l2

SinIntegral is neither non-decreasing nor non-increasing:

https://wolfram.com/xid/0enwmonu8-g6kynf

SinIntegral is not injective:

https://wolfram.com/xid/0enwmonu8-gi38d7


https://wolfram.com/xid/0enwmonu8-ctca0g

SinIntegral is not surjective:

https://wolfram.com/xid/0enwmonu8-hkqec4


https://wolfram.com/xid/0enwmonu8-b1r9xi

SinIntegral is neither non-negative nor non-positive:

https://wolfram.com/xid/0enwmonu8-84dui

SinIntegral has no singularities or discontinuities:

https://wolfram.com/xid/0enwmonu8-mdtl3h


https://wolfram.com/xid/0enwmonu8-mn5jws

SinIntegral is neither convex nor concave:

https://wolfram.com/xid/0enwmonu8-kdss3

Differentiation (3)

https://wolfram.com/xid/0enwmonu8-mmas49


https://wolfram.com/xid/0enwmonu8-nfbe0l


https://wolfram.com/xid/0enwmonu8-fxwmfc


https://wolfram.com/xid/0enwmonu8-odmgl1

Integration (3)
Indefinite integral of SinIntegral:

https://wolfram.com/xid/0enwmonu8-bponid

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

https://wolfram.com/xid/0enwmonu8-b9jw7l


https://wolfram.com/xid/0enwmonu8-e6pu7j


https://wolfram.com/xid/0enwmonu8-ft0ejz


https://wolfram.com/xid/0enwmonu8-b3elfw

Series Expansions (4)
Taylor expansion for SinIntegral:

https://wolfram.com/xid/0enwmonu8-ewr1h8

Plot the first three approximations for SinIntegral around :

https://wolfram.com/xid/0enwmonu8-binhar

General term in the series expansion of SinIntegral:

https://wolfram.com/xid/0enwmonu8-dznx2j

Find series expansion at infinity:

https://wolfram.com/xid/0enwmonu8-mhk4z

Give the result for an arbitrary symbolic direction :

https://wolfram.com/xid/0enwmonu8-qzjf2

SinIntegral can be applied to power series:

https://wolfram.com/xid/0enwmonu8-kivro1

Function Identities and Simplifications (3)
Use FullSimplify to simplify expressions containing sine integrals:

https://wolfram.com/xid/0enwmonu8-k7qf4l

Simplify expressions to SinIntegral:

https://wolfram.com/xid/0enwmonu8-fjcqm6


https://wolfram.com/xid/0enwmonu8-bztxyr


https://wolfram.com/xid/0enwmonu8-jgwst4

Function Representations (4)
Series representation of SinIntegral:

https://wolfram.com/xid/0enwmonu8-cud5sw

SinIntegral can be represented in terms of MeijerG:

https://wolfram.com/xid/0enwmonu8-dvgie


https://wolfram.com/xid/0enwmonu8-cgwzx

SinIntegral can be represented as a DifferentialRoot:

https://wolfram.com/xid/0enwmonu8-bgjnbg

TraditionalForm formatting:

https://wolfram.com/xid/0enwmonu8-teny7

Generalizations & Extensions (1)Generalized and extended use cases
Applications (6)Sample problems that can be solved with this function
Plot the absolute value in the complex plane:

https://wolfram.com/xid/0enwmonu8-mi7

Real part of the Euler–Heisenberg effective action:

https://wolfram.com/xid/0enwmonu8-dasdsl

https://wolfram.com/xid/0enwmonu8-iq2ahy

Gibbs phenomenon for a square wave:

https://wolfram.com/xid/0enwmonu8-d2d


https://wolfram.com/xid/0enwmonu8-ob9

Compute the asymptotic overshoot:

https://wolfram.com/xid/0enwmonu8-hq2i4x

Solve a differential equation:

https://wolfram.com/xid/0enwmonu8-gf5dvl

Integrate a composition of trigonometric functions:

https://wolfram.com/xid/0enwmonu8-xdp


https://wolfram.com/xid/0enwmonu8-e2lii

https://wolfram.com/xid/0enwmonu8-lu6dsb

The curvature is a simple function of the parameter:

https://wolfram.com/xid/0enwmonu8-g4w4zj

Properties & Relations (7)Properties of the function, and connections to other functions
Parity transformation is automatically applied:

https://wolfram.com/xid/0enwmonu8-hrak69


https://wolfram.com/xid/0enwmonu8-12pt5

Use FullSimplify to simplify expressions containing sine integrals:

https://wolfram.com/xid/0enwmonu8-c294y1


https://wolfram.com/xid/0enwmonu8-trk

Obtain SinIntegral from integrals and sums:

https://wolfram.com/xid/0enwmonu8-e7z


https://wolfram.com/xid/0enwmonu8-fii

Obtain SinIntegral from a differential equation:

https://wolfram.com/xid/0enwmonu8-fwo


https://wolfram.com/xid/0enwmonu8-hjy

Compare with Wronskian:

https://wolfram.com/xid/0enwmonu8-oo5dj7


https://wolfram.com/xid/0enwmonu8-nsq


https://wolfram.com/xid/0enwmonu8-s46

Possible Issues (2)Common pitfalls and unexpected behavior
SinIntegral can take large values for moderate‐size arguments:

https://wolfram.com/xid/0enwmonu8-nre

A larger setting for $MaxExtraPrecision can be needed:

https://wolfram.com/xid/0enwmonu8-hei



https://wolfram.com/xid/0enwmonu8-u9

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).
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.
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
Wolfram Language. (1991). SinIntegral. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SinIntegral.html
BibTeX
@misc{reference.wolfram_2025_sinintegral, author="Wolfram Research", title="{SinIntegral}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/SinIntegral.html}", note=[Accessed: 09-July-2025
]}
BibLaTeX
@online{reference.wolfram_2025_sinintegral, organization={Wolfram Research}, title={SinIntegral}, year={2022}, url={https://reference.wolfram.com/language/ref/SinIntegral.html}, note=[Accessed: 09-July-2025
]}