Drawing by Numbers

An easy attempt to make a model of color space is Philipp Otto Runge’s color sphere from 1810. The equator is colored by hue, brightness ranges from black at the north pole to white at the south pole.

Having such a color sphere suggests yet another way to visualize complex valued functions: Color a point z in the domain of the function by Runge’s color of the function value of that point, interpreted as a point on the Riemann sphere.

Square

For instance, up above is the coloring of the function f(z)=z^2 in the unit square. The white region at the center is caused by the zero, and that every color appears twice is of course a consequence of f being of degree 2. Let’s make it slowly more interesting. Here is the Möbius transformation f(z)=(z-1)/(z+1). Zero and poles are clearly visible.

Mobius

Locally, holomorphic functions are just as good or as bad as polynomials, so we shouldn’t expect anything more complicated to happen.

For me, the real excitement of complex analysis starts with essential singularities. There are the mind boggling theorems of Casorati-Weierstrass (images of neighborhoods of essential singularities are dense) and, much stronger and much harder to prove, Picard (images of neighborhoods of essential singularities miss at most two points).

Exp

Above is the coloring of f(z)= e^(i/z) in a thin rectangle centered at 0. We do see every color occurring infinitely often (more or less), but the image is still rather tame. After all, the exponential function is the simplest transcendental function. Things get truly wild if we look at the boundary behavior of gap series like

Gap2series

This series converges in the unit disk but cannot be homomorphically extended beyond.

Gap2

Unfortunately I don’t know enough about gap series to explain everything we can see here. Most puzzling are the circular arcs of increased brightness.

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