Discontinuity of theta functions

*To*: mathgroup at smc.vnet.net*Subject*: [mg14586] Discontinuity of theta functions*From*: weber at math.uni-bonn.de (Matthias Weber)*Date*: Mon, 2 Nov 1998 01:50:58 -0500*Organization*: RHRZ - University of Bonn (Germany)*Sender*: owner-wri-mathgroup at wolfram.com

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