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, z^{1 / 2} has two roots for every number. This notion will be explained in more detail in later chapters.

A plot of Abs(z²) as z ranges over the complex plane.

A complex function $f(z):\mathbb{C}\to\mathbb{C}$ will sometimes be written in the form f(z) = f(x + iy) = u(x,y) + iv(x,y), where u and 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 |z| to gain an understanding of what the function "looks" like.

For an example of this, take the function f(z) = z² = (x² − y²) + i(2xy). The plot of the surface | z² | = x² + y² 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) = z² and the input region being the "quarter disc" $Q \cap \Bbb{D}$ obtained by taking the region

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

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

$\Bbb{D} = \{z : |z| \le 1\}$ .

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

$UHP = \{x + iy : y \ge 0 \}$ (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 $\Bbb{C}$, z = rcis(θ). For example, if we consider points on the unit circle S¹ = {z: | z | = 1} (i.e. the set "r = 1") with $\theta \le \pi/2$ then the squaring function acts as follows:

f(z) = [1cis(θ)]² = cis(2θ)

(here we have used cis(θ)cis(ϕ) = cis(θ + ϕ) ). We see that a point having angle θ is mapped to the point having angle 2θ. If θ is small, meaning that the point is close to z = 1, then this means the point doesn't move very far. As θ becomes larger, the difference between θ and 2θ becomes larger, meaning that the squaring function moves the point further. If θ = π / 2 (i.e. z = i) then 2θ = π (i.e. z² = − 1).

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