About the behavior of $e^z = e^x e^{iy}$

Question

About the behavior of $e^z = e^x e^{iy}$ as

(a) $x$ tends to $- \infty$;

(b) $y$ tends to $ \infty$.

For $y$ tends to $ \infty$ I got that $e^z$ just rotates around the circle with radius $e^x$ over and over again.

Answer 1

The exponential map send lines parallel to $x-$axis (so $y$ is constant) into half lines starting from the origin, in particular you can verify that when $x\rightarrow -\infty$ and $y$ remain constant, then $e^z$ approach to zero along that half line. Infact $e^z=e^{x+iy}=e^xe^{iy}$, if $y$ is constant, then the argument remain constant, and the modulus decrease if $x$ goes to $-\infty$.

Answer 2

for $e^z=re^{iy}$, When $x\to-\infty$ the value of $r=e^{x}\to0$ that is with every constant argument, modulus $r$ will be smaller and smaller.