I am new to complex analysis and am having hard time visualizing the image and preimage of some complex functions that i encountered while self preparing for this course. I have a map $f:\mathbb{C}\longrightarrow\mathbb{C}$ given by $f$$_n$($z$) = $z$$^n$ for various $n$.
now my first thought was to write $z$ in its polar form but that didnt get me any further because i never used polar expressions, second that doesn't help me to describe the mapping geometrically
Any hints or explanations are appreciated, even references to websites where where I can learn more about complex mappings are welcomed.
Find the image and preimage of complex mappings
complex-analysiscomplex-geometry
Best Answer
Here are a few examples:
Let $Q$ be the quarter circle of radius 2 and its interior in the upper right half plane:
$$Q=re^{i\theta}, \quad 0\le r \le 2, 0\le \theta \le \frac{\pi}{2 }$$
The image of $Q$ under the mapping $f_2(z) = z^2$ is the semicircle of radius $4$ in the upper half plane and its interior.
$$f_2(Q)=re^{i\theta}, \quad 0\le r \le 4,\,\, 0\le \theta \le \pi$$
The image of $Q$ under the mapping $f_3(z)=z^3$ is the union of three quarter circles, of radius $8$, and its "interior" spanning the first three quadrants.
$$f_3(Q)=re^{i\theta}, \quad 0\le r \le 8,\,\, 0\le \theta \le \frac{3\pi}{2 }$$
The image of $Q$ under the mapping $f_{-1}(z)=z^{-1}$ is the quarter circle, of radius $\frac{1}{2}$, and its "exterior" in the fourth quadrant.
$$f_{-1}(Q)=re^{i\theta}, \quad r \ge \frac{1}{2}, \,\, 0\ge \theta \ge -\frac{\pi}{2 }$$