Modified Bessel Function of the First Kind

Author

Eric W. Weisstein
November 16, 2008

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

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

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

Plots

<<Utilities`Plot`

Real Axis

Plot[Evaluate[Table[BesselI[n,x],{n,0,5}]],{x,0,5},
AxesLabel->TraditionalForm/@{x,BesselI[n,x]},PlotStyle->{Black,Red,Yellow,Green,Blue,Violet}]

Complex Plane

ComplexPlot3D[BesselI[0,z],{z,-5,5}]//Timing

ComplexContourPlot[BesselI[0,z],{z,-5,5}]//Timing

Sums

(∑_(k=0)^∞)((((((z^2)/4)))^k)/(((k!))^2))==(BesselI[0, z])

Sum[(z^2/4)^k/k!^2,{k,0,∞}]

((((z/2)))^ν) (∑_(k=0)^∞)((((((z^2)/4)))^k)/(k! (Gamma[k+ν+1])))==(BesselI[ν, z])

((((1/2)z))^ν)Sum[(((((1/4)(z^2)))^k)/(k! Gamma[ν+k+1])),{k,0,∞}]