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 $f:\mathbb {C} \to \mathbb {C}$ . 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 complex function $f(z):\mathbb {C} \to \mathbb {C}$ 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+{\bar {z}}}{2}},y={\frac {1}{i}}{\frac {z-{\bar {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 $\Re (z),\Im (z)$ , 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" $Q\cap \mathbb {D}$ obtained by taking the region

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

and intersecting this with the disc $\mathbb {D}$ of radius 1:

\mathbb {D} =\{z:|z|\leq 1\}

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

$UHP=\{x+yi:y\geq 0\}$ ( $UHP$ 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 $\mathbb {C}$ , $z=r{\text{cis}}(\theta )$ . For example, if we consider points on the unit circle $S^{1}=\{z:|z|=1\}$ (i.e. the set " $r=1$ ") with $\theta \leq {\tfrac {\pi }{2}}$ then the squaring function acts as follows:

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

(here we have used ${\text{cis}}(\theta ){\text{cis}}(\phi )={\text{cis}}(\theta +\phi )$ ). 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 $\theta ={\tfrac {\pi }{2}}$ (i.e. $z=i$ ) then $2\theta =\pi$ (i.e. $z^{2}=-1$ ).