# Some holomorphic functions

Some time ago, I saw this beautiful video shared by Prof. Terrence Tao on Google plus. The video is from the youtube channel 3Blue1Brown, where you can find some very good mathematical videos.

A much appreciable component of the particular video was the visualization done to illustrate holomorphic functions. Here’s a clip from the video:

Looking at the said video, I was inspired to make my own holomorphic function visualizer. Here are some illustrations made from my program. In the images below are visualizations of the complex plane, the blue grid is the image of the gray grid. The vertical mark on the x-axis is (1+0i).

$f(z) = e^z$

$f(z) = z^2$

$f(z) = iz$

$f(z) = \frac{1}{z}$

$f(z) = cos(z)$

$f(z) = cos(z)$                     (Again!)

An interesting thing to note here is that the perpendicularity in the lines of the grid is maintained in the image. This is a trivial consequence of the Cauchy-Riemann equations for a holomorphic function. (This fact was also mentioned in the video).

I even tried to incorporate the logarithm defined for arguments in the region $-\pi <\theta <\pi$. Here’s what it looks like:

Of course, it behaves very badly when I try to go outside my domain.

The logarithm inspires us to define our coordinates on the associated Reimann surface (or, the covering space of complex plane with the zero removed. Picture is below), instead of the 2D coordinates used in the program. The determination of which “floor” of the surface the vertex is on can be done by recording the mouse movement history. This idea could perhaps occupy me for some time in the future.

A plot of the multi-valued imaginary part of the complex logarithm function (from Wikipedia)

Another interesting discussion is that of the animation of the zeta-function in this video. In the video, the animator makes a continuous change of the grid lines to the function values. To be technical, this sort of continuous change is a homotopy of the identity function to the given function. The animator uses a simple linear homotopy, but it doesn’t look very natural.

Here’s a discussion on StackExchange about how to do the above homotopy in such a manner that the intermediate functions are holomorphic. My initial goal with the program was to try to animate that (you can spot the reminiscent code on GitHub). It is a future goal, for now.

If you have any interesting comments, or if would like to see your favorite holomorphic function animated, or if you want a better explanation of anything said above, please comment below.