Complex Analysis/Complex Functions/Complex Functions

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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)=z^2=(x^2-y^2)+i(2xy). 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 \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 = r 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 \pi/2 then the squaring function acts as follows:

 f(z) = [ 1 cis(\theta) ]^2 = cis( 2 \theta )

(here we have used  cis( \theta ) cis( \phi ) = 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 = \pi/2 (i.e. z = i) then  2 \theta = \pi (i.e.  z^2 = -1 ).