# Analytic Landscapes

## A brief history

A traditional concept for visualizing complex functions is the so called *analytic landscape*.
Probably introduced by Edmond Maillet in
1903, it
depicts the
graph of the absolute value of a function.

In the first half of the preceding century analytic landscapes became rather popular. The Figure 1
reproduces a
historical illustration from the book *Funktionentafeln mit Formeln und Kurven* by Jahnke and
Emde of 1909.
It shows the
analytic landscape of the complex Gamma
function and
reached an almost
iconic status. Today it is hard to believe that this detailed hand-drawn picture could be created
without the help
of computers!

In the era of black-and-white illustrations this shortcoming was often compensated by endowing the
analytic
landscape with lines of constant argument as in the previous figure, where the argument is
explicitly
indicated by its numerical value. Modern computer technology allows us to achieve the same effect
much better
using colors, which yields the *colored analytic landscape* shown in Figure 2.

## Dynamic exploration

Complex functions $ f:\mathbb{C}\to\mathbb{C} $ can be visualized by plotting the function $g: \mathbb{R^2} \to \mathbb{R}$ with $$g(x,y) = |f\left(x + i y\right)|.$$ The color of each point $\left(x, \,y, \,g(x,y)\right)$ indicates the phase (or argument) of the complex number $f(x + i y).$

In the applet below you can explore colored analytic landscapes. Use the blue slider on the right
side for
zooming out/in. The black slider defines a real scalar **a** $ \in [ -0.14, 1.14].$

In practice, it is often difficult to generate analytic landscapes which allow us to read off
properties of
the complex function easily and precisely. An alternative approach is not only simpler but even more
general: Instead of drawing a graph in $\mathbb R^3,$ we can depict a function directly on its
domain by
color coding its values completely. This method is called *domain coloring*.

**Note:** The last applet was written by Aaron Montag using CindyJS. The source code can be found at GitHub.