Addendum Figures for Berry's Construction

Nicholas Wheeler
March 2018

Figure 12a

r[n_]:=(Sqrt[2π(n-(1/8))])
th[n_]:=(π/4)+((Log[r[n](Sqrt[2π])])/(2 ((r[n])^2)))
z[n_]:=r[n]Exp[I th[n]]

N[Table[{n,z[n]},{n,1,7}]]//TableForm

BerryZeros=ListPlot[{{1.37061,1.90242},{2.19553,2.6383},{2.80223,3.19557},{3.30429,3.66456},{3.74191,4.07782},{4.13479,4.45166},{4.49428,4.79566}},PlotRange->{{0,5},{0,5}},PlotStyle->{PointSize[0.013]}];

NullCurves=ContourPlot[{Re[Erf[x+I y]]==0,Im[Erf[x+I y]]==0},{x,0,5},{y,0,5},PlotPoints->50];

Show[NullCurves,BerryZeros]

Figure 12b

ExactZeros=ListPlot[{{1.45062,1.88094},{2.24466,2.61658},{2.83974,3.17563},{3.33546,3.64617},{3.76901,4.06070}},PlotRange->{{0,5},{0,5}},PlotStyle->{PointSize[0.013]}];
Show[NullCurves,ExactZeros]

Figure 13a

antiStokes=ContourPlot[{Arg[N[Erf[x+I y]]]==(π/4),Arg[N[Erf[x+I y]]]==-(π/4)},{x,0,5},{y,0,5}]

Figure 13b

Show[antiStokes, ExactZeros]

Figure 13c

NullAndStokes=ContourPlot[{Re[Erf[x+I y]]==0,Im[Erf[x+I y]]==0,Arg[N[Erf[x+I y]]]==(π/4),Arg[N[Erf[x+I y]]]==-(π/4)},{x,0,5},{y,0,5},PlotPoints->50]

Figure 14

Plot3D[Arg[N[Erf[x+I y]]],{x,0,5},{y,0,5},PlotRange->{{0,5},{0,5},{-π,π}},Ticks->{{0,5},{0,5},{-π,π}}]