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Solve an Analytic Equation in a Bounded Complex Region
Mathematica 7 finds all solutions of a univariate analytic equation in a bounded region.
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Click for copyable input
f = 2 Sin[Exp[z]] - Cos[Pi z] - 1/3;

rts = Reduce[f == 0 && Abs[z] < 1, z]
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In[2]:=

Click for copyable input
ContourPlot[{Re[f /. z -> x + I y] == 0, 

  Im[f /. z -> x + I y] == 0}, {x, -1, 1}, {y, -1, 1}, 

 RegionFunction -> (Abs[#1 + I #2] < 1 &), 

 Epilog -> {Red, PointSize[0.025], 

   Point[{Re[z], Im[z]}] /. {ToRules[rts]}}]
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