In 1872, Weierstrass introduced a class of real-valued functions that are continuous but nowhere differentiable. This would now be identified as a fractal curve. His functions were of the form

,

where and are real parameters satisfying and . This Demonstration shows the graphs of these functions over the interval , along with the graphs of the companion functions obtained by replacing by or .

Given two parameters and satisfying and , the generalized complex Weierstrass function is defined by

.

The Weierstrass cosine and sine functions are defined, respectively, as the real and imaginary parts of :

,

.

All three types of Weierstrass functions are known to be continuous but nowhere differentiable functions, provided the parameters and satisfy . The graphs of the Weierstrass cosine and sine functions, regarded as subsets of , are fractal objects with box-counting dimension given by [1, Theorem 2.4]:

.

The dimension lies strictly between 1 and 2. Letting , the condition becomes . As approaches 1, the various graphs become smoother, and the dimension approaches 1.

The complex Weierstrass function can be represented by a path in the complex plane. In the case when is an integer, this function is periodic with period 1, so the corresponding path is a closed path.

Reference

[1] K. Barański, "Dimension of the Graphs of the Weierstrass-type Functions," Fractal Geometry and Stochastics V: Progress in Probability (C. Bandt, K. Falconer and M. Zähle, eds.), Cham: Birkhäuser, 2015 pp. 77–91. doi:10.1007/978-3-319-18660-3_5.