Past exam of the algebra, analysis and differential equations 1 course of the University of Oxford / 2007 / 6 / a

Parametrize the vertical line as $z=a+it$, $t\in \mathbb{R}$. Then $e^z=e^a(\cos t+i\sin t)$, whose modulus is $e^a$ and whose argument takes every value. The image isIncreasing $t$ traverses the circle counterclockwise, repeating with period $2\pi$. There is no injectivity on the full line.

Parametrize the horizontal line as $z=t+ib$, $t\in \mathbb{R}$. Then $e^z=e^t{e}^{ib}$, and $e^t$ takes every positive real value exactly once. The image isThis is the ray with argument $b$ modulo $2\pi$, excluding its endpoint zero. As $t$ increases, the image moves outward from arbitrarily small to arbitrarily large radius.