Sine Integral

Author

Eric W. Weisstein
September 30, 2009

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/SineIntegral.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/SineIntegral.html.

©2009 Wolfram Research, Inc. except for portions noted otherwise

Definition

Si(x)

Integrate[((Sin[t])/t),{t,0,x}]

si(x)

FullSimplify[(1/(2I))(ExpIntegralEi[I x]-ExpIntegralEi[-I x]),x<0]//Expand

-Integrate[((Sin[t])/t),{t,x,∞},Assumptions->x>0]//Simplify

Derivative

V7

D[SinIntegral[z],z]

V8

D[SinIntegral[z],z]

Integral

Integrate[SinIntegral[z],z]

Series

(∑_k^∞)(((((-1))^(k-1)) (x^(2 k-1)))/((2 k-1) (2 k-1)!))==Si(x)

V6

Sum[(((-1))^(k-1))((x^(2k-1))/((2k-1)(2k-1)!)),{k,∞}]

V7

Sum[(((-1))^(k-1))((x^(2k-1))/((2k-1)(2k-1)!)),{k,∞}]

2 x (∑_(n=0)^∞)(((Subscript[j, n])(x))^2)==Si(2 x)

V6

Pi Sum[BesselJ[n+1/2,x]^2,{n,0,∞}]

V7

Pi Sum[BesselJ[n+1/2,x]^2,{n,0,∞}]

SphericalBesselJ[n,x]^2//FunctionExpand

Sum[SphericalBesselJ[n,x]^2,{n,0,∞}]

Generating Function

SeriesCoefficient[SinIntegral[x],{x,0,n}]

FullSimplify[%]

FullSimplify[%,Mod[n,2]==1&&n>0]

Sum[((Sin[((n π)/2)])/((n^2) Gamma[n]))(x^n),{n,∞}]

FullSimplify[%]

GeneratingFunction[((Sin[((n π)/2)])/((n^2) Gamma[n])),n,x]

FullSimplify[%]

GeneratingFunction[If[n>0,((Sin[((n π)/2)])/((n^2) Gamma[n])),0],n,x]

FullSimplify[%]

Plots

<<Utilities`Plot`

Real Axis

Plot[SinIntegral[x],{x,-10,10},
PlotStyle->Red,
AxesLabel->TraditionalForm/@{x,SinIntegral[x]}]

Complex Plane

ComplexPlot3D[SinIntegral[z],{z,-2Pi,2Pi}]//Timing

ComplexContourPlot[SinIntegral[z],{z,-2Pi,2Pi}]//Timing