Shi

Author

Eric W. Weisstein
December 26, 2008

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

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

©2008 Wolfram Research, Inc. except for portions noted otherwise

Plots

<<Utilities`Plot`

Real Axis

Plot[SinhIntegral[x],{x,-5,5},
PlotStyle->Red,
AxesLabel->TraditionalForm/@{x,SinhIntegral[x]}]

Complex Plane

ComplexPlot3D[SinhIntegral[z],{z,-5,5}]//Timing

ComplexContourPlot[SinhIntegral[z],{z,-5,5}]//Timing

Series

Sum[((x^n)/((n^2) (Gamma[n]))),{n,1,∞,2}]==(SinhIntegral[x])

v5.0.1

Series[SinhIntegral[z],{z,0,10}]

v5.1

Series[SinhIntegral[z],{z,0,10}]

V8

SeriesCoefficient[SinhIntegral[z],{z,0,n}]

FullSimplify[%,Mod[n,2]==1&&n≥1&&n∈Integers]

Sum[(1/((n^2) Gamma[n]))(x^n),{n,1,∞,2}]

Sum[Piecewise[{{Sqrt[Pi]/(2^n*(n*Gamma[1/2 + n/2]*Gamma[1 + n/2])), Mod[-1 + n, 2] == 0 && n >= 1}}, 
  0],{n,0,∞}]

GeneratingFunction[(1/((((2n+1))^2) Gamma[2n+1])),n,x]

(∑_(n=0)^∞)((x^(2 n+1))/((((2 n+1))^2) (Gamma[2 n+1])))==(SinhIntegral[x])

Sum[(1/((((2n+1))^2) Gamma[2n+1]))(x^(2n+1)),{n,0,4}]

FullSimplify[(1/((((2n+1))^2) Gamma[2n+1])),n∈Integers&&n>0]

Sum[(1/((((2n+1))^2) Gamma[2n+1]))(x^(2n+1)),{n,0,∞}]

Derivative Integral

Derivative

D[SinhIntegral[z],z]

Integral

Integrate[SinhIntegral[z],z]

FullSimplify[%]