The Non-differentiable Functions of Weierstrass
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[more]
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 .[less]
Contributed by: Saurav Chittal, Malachi Robinson, Manisha Garg and A. J. Hildebrand.
(Based on an undergraduate research project at the Illinois Geometry Lab in Fall 2022) (June 13)
Open content licensed under CC BY-NC-SA
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.
 K. Bara\:0144ski, "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.