Weierstrass Zeta Function

Author

Eric W. Weisstein
December 12, 2003

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

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

©2005 Wolfram Research, Inc. except for portions noted otherwise

Notation

WeierstrassZeta[z,{(Subscript[g, 2]),(Subscript[g, 3])}]//TraditionalForm

Plot

<<Utilities`Plot`

ComplexPlot3D[WeierstrassZeta[z,{4,0}],{z,-4,4},PlotPoints->100]//Timing

ComplexContourPlot[WeierstrassZeta[z,{4,0}],{z,-4,4}]//Timing

Derivative

D[WeierstrassZeta[z,{g2,g3}],z]

Integral

Integrate[WeierstrassZeta[z,{g2,g3}],z]

Limit

should be 0

Limit[WeierstrassZeta[z,{g2,g3}]-1/z,z->0]

Limit[WeierstrassZeta[z,{g2,g3}]-1/z,z->0,Direction->-1]

Series

Series[WeierstrassZeta[z,{g2,g3}],{z,0,3}]

Analytic

c[2]:=(Subscript[c, 2])
c[3]:=(Subscript[c, 3])
c[k_]:=c[k]=(3/((2k+1)(k-3)))Plus@@Table[c[m]c[k-m],{m,2,k-2}]

Simplify[Table[c[k],{k,2,10}]]//TableForm//TraditionalForm

Identities

ζ[2]

With[{z=-(x+y)},
(ζ[x]+ζ[y]+ζ[z])^2+dζ[x]+dζ[y]+dζ[z]
]/.{dζ[x_]:>Derivative[1][WeierstrassZeta][x,{g2,g3}],ζ[x_]:>WeierstrassZeta[x,{g2,g3}]}/.Thread[{x,y,g2,g3}->Table[Random[Complex,{-1,1}(1+I)2,20],{4}]]

Numerical Check

<< NumericalMath`NLimit`

weierstrassZeta[z_,{gg2_,gg3_},n_]:=Module[{k=2},
1/z-Plus@@Table[c[k]z^(2k-1)/(2k-1),{k,2,n}]
]/.Thread[{g2,g3}->{gg2,gg3}]

Withoff 24633

ListPlot[Table[c[k]/(2k-1),{k,2,30}]/.{g2->13,g3->-300},PlotJoined->True,PlotStyle->Red]

Plot[Im[WeierstrassZeta [z,{13,-300}]],{z,0,1},PlotStyle->Hue[0]]

N[weierstrassZeta [1,{13,-300},30],20]

N[WeierstrassZeta [1,{13,-300}],20]

Do[
Print[{n,N[weierstrassZeta [1,{13,-300},n],20]//Timing}],
{n,5,40,5}]

Weber

<<Graphics`Colors`

ListPlot[Table[c[k]/(2k-1),{k,2,30}]/.{g2->-28,g3->-23},PlotJoined->True,PlotStyle->Red]

ListPlot[Table[c[k]/(2k-1),{k,15,30}]/.{g2->-28,g3->-23},PlotJoined->True,PlotStyle->Red]

Plot[Im[WeierstrassZeta [z,{-28,-23}]],{z,0,1},PlotStyle->Red]

N[WeierstrassZeta [3/10,{-28,-23}],30]

N[weierstrassZeta [3/10,{-28,-23},35],20]

Plot[Im[ND[WeierstrassZeta[z,{-28,-23}],z,x]+WeierstrassP[x,{-28,-23}]],{x,.1,1},PlotStyle->Red]

Plot[Re[ND[WeierstrassZeta[z,{-28,-23}],z,x]+WeierstrassP[x,{-28,-23}]],{x,.1,1},PlotStyle->Red]