Complex exponential and lines Notation
$$\operatorname{exp}:\mathbb{C}\to\mathbb{C}:\operatorname{exp}(z):=e^z.$$

a) Show that $\operatorname{exp}$ maps vertical lines to circles with the origin as center.
b) Show that $\operatorname{exp}$ maps horizontal lines to half rays from the origin to infinity.

Of course, the first logical step would be to say that vertical lines are given by $z=a+i\cdot 0$ and horizontal ones by $z=0+bi$. Now, I'm not sure how to show that $e^z=e^a$ represent circles and $e^z = e^{ib} = \cos b + i\sin b$ half rays.
How do I best approach this?
 A: A vertical line in $\Bbb C$ is parallel to the $y$-axis; thus it is a set of points of the form
$a + it, \; a \in \Bbb R \; \text{fixed}, \; t \in \Bbb R,  -\infty \to t \to \infty; \tag 1$
we have
$e^{a + it} = e^a e^{it} = e^a(\cos t + i \sin t), \tag 2$
which traces out a circle in $\Bbb C$ of radius $e^a$ as $-\infty \to t \to \infty$.
A horizontal line in $\Bbb C$ is parallel to the $x$-axis; thus it is a set of points of the form
$t + ib, \; b \in \Bbb R \; \text{fixed}, \; t \in \Bbb R, -\infty \to t \to \infty, \tag 3$
and thus,
$e^{t + ib} = e^t e^{ib} = e^t(\cos b + i\sin b); \tag 4$
this traces out the half-ray of points of the form (in polar corrdinates) $re^{i \theta}$, where $0 < r < \infty$ and $\theta \in [0, 2\pi)$ is that unique angle such that
$\cos \theta = \cos b, \; \sin \theta = \sin b. \tag 5$
We note in closing that $a + 0i$ is a point on the real axis $\Bbb R$, not a line, and that $0 + bi$ is a point on the imaginary axis $i\Bbb R$, and that neither represents a line in $\Bbb C$.
