# 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 ${\displaystyle f:\mathbb {C} \to \mathbb {C} }$ . Unless explicitly stated, whenever the term function appears, we will mean a complex function. A function can also be multi-valued – for example, ${\displaystyle {\sqrt {z}}}$ has two roots for every number. This notion will be explained in more detail in later chapters.

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

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

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 ${\displaystyle \Re (z),\Im (z)}$ , and ${\displaystyle |f(z)|}$ to gain an understanding of what the function "looks" like.

For an example of this, take the function ${\displaystyle f(z)=z^{2}=(x^{2}-y^{2})+(2xy)i}$ . The plot of the surface ${\displaystyle |z^{2}|=x^{2}+y^{2}}$ 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 ${\displaystyle f(z)=z^{2}}$ and the input region being the "quarter disc" ${\displaystyle Q\cap \mathbb {D} }$ obtained by taking the region

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

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

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

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

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

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

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