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 $$f:\Complex\to\Complex$$. 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 $$f(z):\Complex\to\Complex$$ 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:
 * $$x=\frac{z+\bar z}{2},y=\frac1i\frac{z-\bar z}{2}$$

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

$$Q=\{x+yi:x,y\ge0\}$$ (i.e. $$Q$$ is the first quadrant)

and intersecting this with the disc $$\mathbb D$$ of radius 1:
 * $$\mathbb D=\{z:|z|\le1\}$$

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

$$UHP=\{x+yi:y\ge0\}$$ ($$UHP$$ 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 $$\theta\le\tfrac{\pi}{2}$$ 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. $$z^2=-1$$).

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