3

Thanks to amazing code from "Guess who it is" here: Eisenstein Series in Mathematica? I'm able to make some nice plots using Eisenstein Series. What I'd like is a color plot of the argument of certain combinations of Eisenstein Series. What I have below, using DensityPlot, is nice, but what I was hoping for was something that emphasized more where the lines of constant argument are. Perhaps instead of a continuous variable for the angle, there is some way to partition the angles into 50 or so bins? Maybe that would show a little clearer where the lines of constant argument are.

The code I have so far is:

SetAttributes[EisensteinE, Listable];

EisensteinE[2, t_, s_] := 
  With[{q3 = EllipticTheta[3, 0, Exp[I*Pi*(t + I*s)]]^2}, 
    6/Pi EllipticE[InverseEllipticNomeQ[Exp[I*Pi*(t + I*s)]]] q3 - 
     q3^2 - EllipticTheta[4, 0, Exp[I*Pi*(t + I*s)]]^4]

EisensteinE[4, t_, s_] := 
  (EllipticTheta[2, 0, Exp[I*Pi*(t + I*s)]]^8 + 
   EllipticTheta[3, 0, Exp[I*Pi*(t + I*s)]]^8 + 
   EllipticTheta[4, 0, Exp[I*Pi*(t + I*s)]]^8)/2

K[i_, j_] = ((EisensteinE[2, i, j])^2) - EisensteinE[4, i, j];

DensityPlot[Arg[K[i, j]/144], {i, -1/2, 1/2}, {j, 0, 3}, 
  ColorFunction -> Hue, PlotLegends -> Automatic, ImageSize -> 300]

If my idea above of discretizing the angles is lousy, is there a more convenient way to achieve this? Also, any thoughts on why portions of the plot are simply white, as opposed to one of the colors labeling an angle?

Community
  • 1
Benighted
  • 1,327
  • 1
  • 10
  • 14
  • 1
    "lines of constant argument" - then you probably want the setting MeshFunctions -> {#3 &} and an appropriate setting for Mesh. Now a little technical note: InverseEllipticNomeQ[Exp[I π τ]] == ModularLambda[τ]. – J. M.'s missing motivation Aug 26 '15 at 07:33
  • @J. M. Was precisely what I was looking for thanks! I think the numerics break down for imaginary part of tau outside of 1/2<6, but within that region it looks perfect. – Benighted Aug 26 '15 at 18:04

0 Answers0