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Discontinuity of theta functions


Using Mathematica (Irix , version 3.0?), I have just discovered that the
theta-functions are dicontinuos, contradicting the long proven
analyticity!

You can see it by executing

ParametricPlot[
  Evaluate[{Re[#], Im[#]}&@EllipticTheta[1,Pi (0.9+0.3I),-0.1+t I]],{
    t,-.5,.5}]

The discontinuity occurs when t=0. (the straight segment in the plot is
another long-standing feature). Actually, a slightly more careful
analysis reveals that in Mathematica's implementation of
EllipticTheta[1,z,q], the values with Im[q]<0 are messed up. One can
repair this as follows:

ParametricPlot[
  Evaluate[{Re[#], Im[#]}&@
      If[t>=0,EllipticTheta[1,Pi (0.9+0.3I),-0.1+t I],
        I EllipticTheta[1,Pi (0.9+0.3I),-0.1+t I]]],{t,-.5,.5}]

Enjoy!


Matthias Weber


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