The Non-differentiable Functions of Weierstrass

Requires a Wolfram Notebook System
Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.
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]
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
Details
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. 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.
Snapshots
Permanent Citation