Ber

Author

Eric W. Weisstein
October 2, 2008

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

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

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

Definition

(Subscript[ber, 0])(z)

Sum[(-1)^n(z/2)^(4n)/((2n)!)^2,{n,0,Infinity}]

Simplify[1+Sum[(-1)^n(z/2)^(4n)/((2n)!)^2,{n,Infinity}]]

ber[z_?NumericQ]:=(1/2) (BesselI[0,(((-1))^(1/4)) z]+BesselJ[0,(((-1))^(1/4)) z])

(Subscript[ber, n])(z)

((((1/2)x))^ν)Sum[((Cos[((3/4)ν+(1/2)k)π])/(k! Gamma[ν+k+1]))((((1/4)(x^2)))^k),{k,0,∞}]

FullSimplify[%4]

FullSimplify[%,x∈Reals]

FunctionExpand[%5]//FullSimplify

FullSimplify[%5/.ν->0]

FullSimplify[With[{ν=0},((((1/2)x))^ν)Sum[((Cos[((3/4)ν+(1/2)k)π])/(k! Gamma[ν+k+1]))((((1/4)(x^2)))^k),{k,0,∞}]],x∈Reals]

FullSimplify[Subtract@@{(1/2) (BesselI[0,(((-1))^(1/4)) x]+BesselI[0,(((-1))^(3/4)) x]),(1/2) (BesselI[0,(((-1))^(1/4)) x]+BesselJ[0,(((-1))^(1/4)) x])}]

Special Values

Ber[0,z]

Plot

<<Utilities`Plot`
<<Utilities`Typesetting`

Real Axis

V5

Plot[Ber[0,z],{z,-10,10},AxesLabel->TraditionalForm/@{x,Ber[0,x]},
PlotStyle->Red]

Plot[Evaluate[Table[Ber[n,x],{n,0,5}]],{x,-10,10},AxesLabel->TraditionalForm/@{x,Ber[n,x]},PlotStyle->{Black,Red,Yellow,Green,Blue,Violet},PlotRange->{Automatic,{-40,40}}]

V7

Plot[Through[{Re,Im}[KelvinBer[0,z]]],{z,-10,10},AxesLabel->{x,KelvinBer[0,x]},Evaluated->True,PlotRange->10]

Plot[Table[KelvinBer[n,x],{n,0,5}],{x,-10,10},AxesLabel->{x,KelvinBer[n,x]},PlotStyle->{Black,Red,Yellow,Green,Blue,Purple},Evaluated->True]

Complex Plane

V5

ComplexPlot3D[Ber[z],{z,-10,10},PlotPoints->200]//Timing

ComplexContourPlot[Ber[z],{z,-10,10}]//Timing

V7

ComplexPlot3D[KelvinBer[z],{z,-10,10}]

ComplexPlot3D[KelvinBer[z],{z,-10,10},"PolygonReduction"->.1]

ComplexContourPlot[KelvinBer[z],{z,-10,10}]

ComplexContourPlot[KelvinBer[z],{z,-10,10},Contours->8,"MeshRegions"->False,"PackGraphicsComplex"->True]