TITLE: Complex Analysis Sketch the image under the function $w = e^z$ of each of the following subsets of the z plane:
QUESTION [1 upvotes]: (a) $\{z \in \Bbb C: Re (z) = -2\}$
(b) $\{z \in \Bbb C: Im (z) = 5\pi\}$
(c) $\{z = x + iy : 0 \le x \le 1, 0 \le y \le \pi\}$
(d) $\{z = x + iy : -2 \le x \le -1, -\pi \le y \le 4\pi\}$
(e) $\{z : Im z \ge 0\}$
I know that (a) is mapped onto a circle of radius $\frac{1}{e^2}$, and b is mapped onto a ray from the origin with argument of $\pi$ (or 5$\pi$)
For the rest of the questions, I am unsure of what to do. Any help appreciated, and thanks in advance!

REPLY [0 votes]: a)$z = -2 + iy\\
w = e^z = e^{-2} e^{iy} = e^{-2}(\cos y + i\sin y)$
$(\cos y + i\sin y)$ desribes a circle
b) $z = x + 5\pi i\\
w = (e^x)e^{5\pi i} = -e^x$
The image of $w$  is the negative real numbers
c)
$x = e^x(\cos y + i\sin y)$ 
and I will let you guess what that maps to as the limits of $x,y$ change
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