Complex Analysis/Complex Functions/Complex Functions

A complex function is one that takes complex values and maps them onto complex numbers, which we write as $f:ℂ\to ℂ$ . Unless explicitly stated, whenever the term function appears, we will mean a complex function. A function can also be multi-valued – for example, $\sqrt{z}$ has two roots for every number. This notion will be explained in more detail in later chapters.

A plot of $|z^2|$ as $z$ ranges over the complex plane

A complex function $f(z):ℂ\to ℂ$ will sometimes be written in the form $f(z)=f(x+yi)=u(x,y)+v(x,y)i$ , where $u,v$ are real-valued functions of two real variables. We can convert between this form and one expressed strictly in terms of $z$ through the use of the following identities:

$x=\frac{z+\overline{z}}{2},y=\frac{1}{i}\frac{z-\overline{z}}{2}$

While real functions can be graphed on the x-y plane, complex functions map from a two-dimensional to a two-dimensional space, so visualizing it would require four dimensions. Since this is impossible we will often use the three-dimensional plots of $\mathrm{\Re }(z),\mathrm{\Im }(z)$ , and $|f(z)|$ to gain an understanding of what the function "looks" like.

For an example of this, take the function $f(z)=z^2=(x^2-y^2)+(2xy)i$ . The plot of the surface $|z^2|=x^2+y^2$ is shown to the right.

Another common way to visualize a complex function is to graph input-output regions. For instance, consider the same function $f(z)=z^2$ and the input region being the "quarter disc" $Q\cap 𝔻$ obtained by taking the region

$Q=\{x+yi:x,y\ge 0\}$ (i.e. $Q$ is the first quadrant)

and intersecting this with the disc $𝔻$ of radius 1:

$𝔻=\{z:|z|\le 1\}$

If we imagine inputting every point of $Q\cap 𝔻$ into $f$ , marking the output point, and then graphing the set $f(Q\cap 𝔻)$ of output points, the output region would be $UHP\cap 𝔻$ where

$UHP=\{x+yi:y\ge 0\}$ ($UHP$ is called the upper half plane).

So, the squaring function "rotationally stretches" the input region to produce the output region. This can be seen using the polar-coordinate representation of $ℂ$ , $z=r\text{cis}(\theta )$ . For example, if we consider points on the unit circle $S^1=\{z:|z|=1\}$ (i.e. the set "$r=1$") with $\theta \le \frac{\pi }{2}$ then the squaring function acts as follows:

$f(z)=1\text{cis}(\theta {)}^2=\text{cis}(2\theta )$

(here we have used $\text{cis}(\theta )\text{cis}(\varphi )=\text{cis}(\theta +\varphi )$). We see that a point having angle $\theta$ is mapped to the point having angle $2\theta$ . If $\theta$ is small, meaning that the point is close to $z=1$ , then this means the point doesn't move very far. As $\theta$ becomes larger, the difference between $\theta$ and $2\theta$ becomes larger, meaning that the squaring function moves the point further. If $\theta =\frac{\pi }{2}$ (i.e. $z=i$) then $2\theta =\pi$ (i.e. $z^2=-1$).

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