Weierstrass Elliptic Function

Author

Eric W. Weisstein
November 29, 2002

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

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

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

Plots

<<Utilities`Plot`

Real Axis

c=2.622057554370711`;

Show[GraphicsArray[{
PlotDiscontinuous[WeierstrassP[x,4,0],{x,-4,-Pi,0,Pi,4},
VerticalAsymptotes->{-c,0,c},
HorizontalAsymptotes->{1},
DisplayFunction->Identity,
PlotRange->{-1,10},
AxesLabel->TraditionalForm/@{x,WeierstrassP[x,4,0]}],
PlotDiscontinuous[WeierstrassPPrime[x,4,0],{x,-4,-c,0,c,4},
VerticalAsymptotes->{-c,0,c},
PlotRange->{-20,20},
DisplayFunction->Identity,
AxesLabel->TraditionalForm/@{x,WeierstrassPPrime[x,4,0]}]
}],GraphicsSpacing->-.05]

Complex Plane

℘(z;4,0)

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

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

℘'(z;4,0)

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

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

Plots of Half-Periods as a Function of (Subscript[g, 2])and (Subscript[g, 3])

Show[GraphicsArray[DensityPlot[#[First@WeierstrassHalfPeriods[{g1,g2}]],{g1,-1,1},{g2,-1,1},Mesh->False,PlotPoints->100,ColorFunction->Hue,DisplayFunction->Identity]&/@{Re,Im}]]//Timing

Show[GraphicsArray[DensityPlot[#[Last@WeierstrassHalfPeriods[{g1,g2}]],{g1,-1,1},{g2,-1,1},Mesh->False,PlotPoints->100,ColorFunction->Hue,DisplayFunction->Identity]&/@{Re,Im}]]//Timing

Extrema

Minimum of fuction

FindMinimum[WeierstrassP[x,4,0],{x,3,4}]

Asymptote of derivative

FindMinimum[WeierstrassPPrime[x,4,0],{x,3,4}]

FindMinimum[1/WeierstrassP[x,4,0],{x,3,4}]

c=x/.Last[%]

Periods for Special Cases

<<EISFormat`

Equianharmonic Case (Omega-2 Constant)

N[WeierstrassHalfPeriods[{0,1}],16]

N[WeierstrassHalfPeriods[{0,1}],30]

ω2=First[%]//Chop

N[ω2/2+I Sqrt[3]ω2/2,30]

First@RealDigits[Chop[First@N[WeierstrassHalfPeriods[{0,1}],100]]]

Lemniscate Case

WeierstrassHalfPeriods[{1,0}]

First@N[WeierstrassHalfPeriods[{1,0}],20]

N[Gamma[1/4]^2/4/Sqrt[Pi],20]

L:=Gamma[1/4]^2/Sqrt[2 Pi]

First@RealDigits[N[Gamma[1/4]^2/Sqrt[2 Pi],100]]

FormatSequence[First@RealDigits[N[Gamma[1/4]^2/Sqrt[2 Pi],102]],64853,0,
Name->"Lemniscate constant",
Mma->"First@RealDigits[N[Gamma[1/4]^2/Sqrt[2 Pi],102]]",
Keyw->{"easy","cons"},
html["LemniscateConstant"]
]

N[L/2/Sqrt[2],20]

Pseudolemniscate Case

I don't understand the notation here.  What are (ω,ω') and ((Subscript[ω, 1]),(Subscript[ω, 2]))?  What is Mathematica returning?

WeierstrassHalfPeriods[{-1,0}]

g2^3-27g3^2 /.{g2->-1,g3->0}

N[WeierstrassHalfPeriods[{-1,0}],20]

N[L/4,20]