I am trying to solve the following function for 'r' and 'theta' at various values of 'a' and 'zeta' in order to detemine the saddle points (r,theta). I have tried the following and it works fine! However, the process takes quite a long time. I was wondering if there is another method...say Newtons Method or otherwise to go about solving the two simultaneous equations (eq1,eq2) for (r,theta)?
f:= (a,zeta,r,theta) -> -exp(I*zeta/(1 - exp(-r*exp(I*theta)))) * r * exp(I*(a*r*exp(I*theta) + theta))/(2*Pi*(1 - exp(r*exp(I*theta))));