Laplace transform of sine integral

0.1 Derivation of {Sit}

If one performs the change of integration variable


in the defining integral (


of the sine integralDlmfDlmfDlmfMathworldPlanetmath functionMathworldPlanetmath, one obtains


getting limits (  We know (see the entry Laplace transform of sine and cosine) that


This transformation formula can be integrated with respect to the parametre x:


Thus we have the transformation formula of the sinus integralis:

{Sit}=1sarctan1s. (1)

0.2 Laplace transform of sinc function

By the formula  {f}=s{f}-limx0+f(x)  of the parent ( entry, we obtain as consequence of (1), that

{ddtSit}=s1sarctan1s-Si 0,


{sintt}=arctan1s. (2)

The formula (2) may be determined also directly using the definition of Laplace transformDlmfMathworldPlanetmath.  Take an additional parametre a to the defining integral


by setting


Now we have the derivative  φ(a)=0e-stcosatdt,  where one can partially integrate twice, getting


Thus we solve


and since  φ(0)=0, we obtain  φ(a)=arctanas.  This yields


i.e. the formula (2).

Formula (2) is derived here ( in a third way.

Title Laplace transform of sine integral
Canonical name LaplaceTransformOfSineIntegral
Date of creation 2014-11-07 15:36:06
Last modified on 2014-11-07 15:36:06
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 15
Author pahio (2872)
Entry type Example
Classification msc 44A10
Synonym Laplace transform of sinc function
Related topic SubstitutionNotation
Related topic SincFunction
Related topic TableOfLaplaceTransforms
Related topic SineIntegral