sine integral

The functionMathworldPlanetmath sine integralDlmfDlmfDlmfMathworldPlanetmath (in Latin sinus integralis) from to is defined as

Si x:=0xsintt𝑑t=0xsinc(t)𝑑t, (1)

or alternatively as

Si x:=01sintxt𝑑t.

It isn’t an elementary functionMathworldPlanetmath.  The equation (1) implies the Taylor seriesMathworldPlanetmath

Si z=z-z333!+z555!-z777!+-,

which converges for all complex values z and thus defines an entire transcendental function.

Si x satisfies the linear third differential equationMathworldPlanetmath


Remark 1.limxSi x=π2

Remark 2.  There is also another “sine integral”

si x:=xsintt𝑑t=Si x-π2

and the corresponding cosine integral

ci x:=xcostt𝑑t=γ+lnx+0xcost-1t𝑑t

where γ is the Euler–Mascheroni constant (

Title sine integral
Canonical name SineIntegral
Date of creation 2015-02-04 12:58:26
Last modified on 2015-02-04 12:58:26
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 17
Author pahio (2872)
Entry type Definition
Classification msc 30A99
Synonym sinus integralis
Synonym Si
Related topic SincFunction
Related topic SineIntegralInInfinity
Related topic LogarithmicIntegral2
Related topic CurvatureOfNielsensSpiral
Related topic LaplaceTransformOfIntegralSine
Related topic FresnelIntegrals
Related topic HyperbolicSineIntegral
Defines sine integral
Defines sinus integralis
Defines cosine integral