Complex Analysis/Complex Functions/Complex Functions

We can extend the notion of a function to complex numbers. A complex function is one that takes complex values and maps them onto complex numbers, which we write as $f:ℂ\to ℂ$. 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 $f(z):ℂ\to ℂ$ 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:

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

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 $\mathrm{\Re }z$, $\mathrm{\Im }z$, 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.

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