I wish to describe the curves $|f|$=constant and arg$f$=constant for the following functions:
1.$f(z)=exp(z^2)$
2.$f(z)=exp\left(\cfrac{z+1}{z-1}\right)$
My thoughts: I can write down what the compositions do to a complex number $z=x-iy$. But then I'm not sure what to do with the conditions $|f|=$constant and arg$f$=constant. Does the former restrict me to a circle and the latter, to a ray from the origin? Also I'm not sure what 'describe the curve' means.
I also know some random facts that for holomorphic function $f$ defined on a connected domain, each of the conditions $|f|=$constant and arg$f$=constant imply that $f$ is constant. I also know that the exp map takes the rectangular co-ordinates to polar co-ordinates.
Any help is appreciated. Thanks.