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I came across this topic and I don't understand it at all. The point is, I'm supposed to draw a set of function values of a complex numbers.

I don't even need the solution itself, I need the explanation more, because I really want to understand this topic.

Examples:

  1. Depict the set $M = \{f (z) : z \in \Omega\}$, when

    $$\Omega = \{z \in C : |Im z| < \frac{\pi}{2} \}$$ $$f (z) := e^{iz}$$

  2. Depict the set $$\Omega$$ and $$f (\Omega) = \{f (z) : z \in \Omega\}$$ when

$$\Omega = U (1, 2)$$ $$f (z) := \frac{2z-1}{z+3}$$

My attempts:

I've attempted to draw the $\Omega$ sets:

Example 1:

Example 1

Example 2:

Example 2

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  • $\begingroup$ was the answer helpful? $\endgroup$ – Fede Poncio Nov 25 '15 at 23:51
  • $\begingroup$ Yes, thank you :-) $\endgroup$ – Eenoku Nov 26 '15 at 0:05
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Well, both the $\Omega$s are okay! About how to depict the first one, I thought it like this: Thinking $z=x+iy$, then $|Im(z)|<\frac{\pi}{2}$ is saying $-\frac{\pi}{2}<y<\frac{\pi}{2}$. Now, $$f(z)=e^{(iz)}=e^{i(x+iy)}=e^{ix-y}=e^{-y}e^{ix}=e^{-y}\left(Cos(x)+iSin(x)\right)$$ And if you think about the last two expressions, they are just a circle $(e^{ix})$ with radius $e^{-y}$. Of course, $y$ is not a constant, and therefore it is a changing radius, but it does have boundaries; since: $$-\frac{\pi}{2}<y<\frac{\pi}{2}$$ $$\frac{\pi}{2}>-y>-\frac{\pi}{2}$$ $$0.2079\approx e^{-\frac{\pi}{2}}<e^{-y}<e^{\frac{\pi}{2}}\approx 4.81$$ So in the end you will have an area trapped between a circle of radius $\approx0.2079$ and a bigger one of radius $\approx4.81$
Like this:
here's a Wolfram Alpha plot

As for the second one, I would parametrize $U(1,2)$ as $\left(x(t), y(t)\right)=(r(Cos(t)+1),rSin(t))$ where $0\le r<2$ , $0\le t \le2\pi$ and replace that into $f(z)$ for the $x$ and $y$s. First separate $f(z)$ into its complex and imaginary parts: $$f(z)=\frac{2z-1}{z+3}\frac{z+3}{z+3}=\frac{2x^2-2y^2-6x+3}{x^2-y^2-9}-6yi$$ That will give you a parametrization for the $(u,v)$ plane. I tried simplifying it but didn't succeed. An easy way is plotting the case where $r=2$ and testing where one point from the $U(1,2)$ goes, and then you can see which area corresponds to the domain $\Omega$ you are taking.
I really hope this helps!

EDIT: This is the parametrized plot for $r=2$ enter image description here

I just realized this second function is a Möbius transformation of the circle. That is another, simpler way of seeing how it is transformed.

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