Complex Functions

A complex function w = f(z) takes a complex number z and returns a complex number w. Since each complex number is a point in the plane, a complex function is a map from the plane to the plane — it picks up every point and moves it somewhere.

You cannot draw it as a single graph the way you draw y = f(x): input takes two real dimensions, output takes two more, so an honest graph would need four real dimensions. Instead we visualise a complex function by watching where points and regions go — the input plane on one side, the output plane on the other.

Three functions to keep in mind

Most of complex analysis is built from a handful of basic maps. Three are worth fixing immediately, because each one moves the plane in a clean geometric way:

Translation — f(z) = z + c. Every point slides by the fixed complex number c; the whole plane shifts rigidly, like a sheet of paper pushed across a desk.

Squaring — f(z) = z^2. Writing z = re^{i\theta}, the de Moivre rule gives

z^2 = \left(re^{i\theta}\right)^2 = r^2 e^{i\,2\theta}.

So squaring squares the modulus and doubles the argument — a point at radius r and angle \theta lands at radius r^2 and angle 2\theta.

Inversion — f(z) = 1/z. Since 1/z = \frac{1}{r}e^{-i\theta}, it inverts the modulus and flips the angle — points near the origin fly far out, distant points crowd in toward zero.

Splitting into real and imaginary parts

Because the output is itself a complex number, every complex function can be written as a pair of ordinary real-valued functions of the two real inputs x and y (where z = x + iy):

f(z) = u(x, y) + i\,v(x, y).

Here u is the real part of the output and v is the imaginary part. Let us extract u and v for the squaring map — this little computation comes back again and again.

Step 1 — substitute z = x + iy. Just expand the square:

z^2 = (x + iy)^2 = x^2 + 2ixy + (iy)^2.

Step 2 — use i^2 = -1. The last term becomes -y^2:

z^2 = x^2 - y^2 + 2ixy.

Step 3 — read off the two parts. Gather the term without i and the term carrying it:

u(x, y) = x^2 - y^2, \qquad v(x, y) = 2xy.

So f(z) = z^2 is the same as the pair of real surfaces u = x^2 - y^2 and v = 2xy. This (u, v) split is how every later page will analyse a complex function.

A complex function w = f(z):

maps \mathbb{C} to \mathbb{C} — a map from the plane to the plane, visualised by where points and regions go;
splits into two real functions of two real variables, f(z) = u(x, y) + i\,v(x, y);
for the squaring map z^2 = r^2 e^{i\,2\theta}, squaring the modulus and doubling the argument, with u = x^2 - y^2 and v = 2xy.

On the real line you can see a function: its graph is a curve in a plane. A complex function refuses that picture, because its graph lives in four-dimensional space. This is not a defect — it is a prompt to think differently. Instead of a frozen curve, a complex function is a transformation: feed it a grid, a circle, a region, and study the warped image that comes out. The famous pictures of complex analysis — conformal grids bending into flowers and fans — are exactly this idea, the input plane painted with its destination.

Watch a point get squared

Drag z = x + iy with the sliders. The first arrow is z; the second is its image f(z) = z^2. Notice how the image angle is always twice the input angle, and its length is the input length squared. The readout shows u = x^2 - y^2 and v = 2xy recomputed live.