Notes

Chapter 10: Processes of Perception and Analysis

Section 11: Traditional Mathematics and Mathematical Formulas


Nested continuous functions

Most standard continuous mathematical functions never show any kind of nested behavior. Elliptic theta and elliptic modular functions are exceptions. Each of these functions has definite finite values only in a limited region of the complex plane, and on the boundary of this region they exhibit singularities at every single rational point. The picture below shows Im[ModularLambda[x+I y]]. Like other elliptic modular functions, ModularLambda satisfies f[z]==f[(a + b z)/(c + d z)] with a, b, c, d integers such that a c - b d == 1. The function can be obtained as the solution to a second-order nonlinear ordinary differential equation. Nested behavior is also found for example in EllipticTheta[3, 0, z], which is given essentially by Sum[z^n^2, {n, ∞}] .


From Stephen Wolfram: A New Kind of Science [citation]