Domain coloring

A complex function $w = f(z)$ takes a point of one plane to a point of another, so its graph would need four dimensions. Domain coloring keeps only the plane of $z$ and paints every point there with a colour that stands for the value $f(z)$. The hue is the argument of the value: in the default colour wheel, positive real numbers are red, and the colours run once round the wheel as the angle goes once round. The brightness is the modulus: values near 0 are dark, values with modulus near 1 have the wheel’s own colours, and large values are light. The inset is that key: the $w$-plane squeezed into a disc, each point painted its own colour, with 0 in the middle and ∞ at the rim.

A zero is where all the colours meet in a dark point, a pole where they meet in a light one. Near a zero $a$ of order $m$, the function behaves like $c\,(z-a)^m$, so its argument is about $\arg c + m\arg(z-a)$. Going once anticlockwise round a small circle that encloses only $a$, the colours run through the wheel $m$ times, in the wheel’s own order. Near a pole of order $n$, the function is like $c\,(z-b)^{-n}$, and the colours run round $n$ times the other way.

The red key walks the loop once and counts how many times the colour turns, forwards minus backwards. For a rational function

Arguments add when numbers multiply. Along the loop, each $\arg(z-a)$ gains one full turn if $a$ is inside and comes back to where it started if $a$ is outside. So the colour turns as many times as there are zeros inside, minus the poles inside, each counted with its order. This is the argument principle; it holds for any function that is analytic inside and on the loop apart from poles inside it, with no zero or pole on the loop itself. A zero and a pole inside together give no turn at all.