The Non-differentiable Functions of Weierstrass

​
function
WeierstrassComplex
WeierstrassSin
WeierstrassCos
number of sample points
100
300
1000
3000
10000
parameters
a
0.95
c
1.01
b = 2
show axes
show endpoints
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
∞
∑
n=0
n
a
sin(2π
n
b
t)
,
where
a
and
b
are real parameters satisfying
0<a<1
and
c=ab>1
. This Demonstration shows the graphs of these functions over the interval
[0,1]
, along with the graphs of the companion functions obtained by replacing
sinx
by
cosx
or
ix
e
.

Details

Given two parameters
a
and
b
satisfying
0<a<1
and
b>1/a
, the generalized complex Weierstrass function is defined by
W
a,b
(t)=
∞
∑
n=o
n
a
2πi
n
b
t
e
.
The Weierstrass cosine and sine functions are defined, respectively, as the real and imaginary parts of
W
a,b
(t)
:
Re
W
a,b
(t)=
∞
∑
n=o
n
a
cos(2πi
n
b
t)
,
Im
W
a,b
(t)=
∞
∑
n=o
n
a
sin(2π
n
b
t)
.
All three types of Weierstrass functions are known to be continuous but nowhere differentiable functions, provided the parameters
a
and
b
satisfy
ab>1
. The graphs of the Weierstrass cosine and sine functions, regarded as subsets of
2
R
, are fractal objects with box-counting dimension given by[1, Theorem 2.4]:
D=2+
loga
logb
.
The dimension
D
lies strictly between 1 and 2. Letting
c=ab
, the condition
b>1/a
becomes
c>1
. As
c
approaches 1, the various graphs become smoother, and the dimension
D
approaches 1.
The complex Weierstrass function
W
a,b
(t)
can be represented by a path in the complex plane. In the case when
b
is an integer, this function is periodic with period 1, so the corresponding path is a closed path.

References

[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.

External Links

Nowhere Differentiable Function (Wolfram MathWorld)
Riemann's Example of a Continuous but Nowhere Differentiable Function
The Generalized Weierstrass-Riemann Functions

Permanent Citation

Saurav Chittal, Malachi Robinson, Manisha Garg, A. J. Hildebrand
​
​"The Non-differentiable Functions of Weierstrass"​
​http://demonstrations.wolfram.com/TheNonDifferentiableFunctionsOfWeierstrass/​
​Wolfram Demonstrations Project​
​Published: March 1, 2023