Differentiability of a complex function.

In summary, the conversation discussed the search for a complex function that is nowhere differentiable except at the origin and on the circle of radius 1, centered at the origin. Suggestions were given to look into the Weierstrass function and to use a fractal-based approach.
  • #1
jeffreydk
135
0
(PROBLEM SOLVED)

I am trying to think of a complex function that is nowhere differentiable except at the origin and on the circle of radius 1, centered at the origin. I have tried using the Cauchy-Riemann equations (where f(x+iy)=u(x,y)+iv(x,y))

[tex]
\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y} \qquad \quad \frac{\partial v}{\partial x}=-\frac{\partial u}{\partial y}
[/tex]

to reduce it to a set of differential equations, but I haven't had any luck. Any suggestions on how to go about this? Thanks in advance for any input.
 
Last edited:
Physics news on Phys.org
  • #2


Hello,

I understand your frustration with trying to find a complex function that meets your criteria. I would suggest looking into the Weierstrass function, also known as the continuous nowhere differentiable function. This function is defined as:

f(x) = \sum_{n=0}^{\infty} a^n \cos (b^n \pi x)

where a and b are constants that can be chosen to create a function that is nowhere differentiable except at the origin and on the circle of radius 1, centered at the origin.

Another approach you could take is to use the concept of fractals to create a function that meets your criteria. Fractals are complex, self-similar patterns that are found in nature and can also be represented mathematically. By using a fractal-based approach, you may be able to create a function that is nowhere differentiable except at specific points.

I hope these suggestions help you in your search for a suitable complex function. If you have any further questions or need clarification, please do not hesitate to ask. Best of luck in your research!
 

1. What is the definition of differentiability for a complex function?

Differentiability for a complex function is defined as the ability for the function to have a derivative at a given point. This means that the function must be continuous at that point and have a unique tangent line.

2. How is differentiability related to continuity for a complex function?

In order for a complex function to be differentiable at a point, it must also be continuous at that point. However, the converse is not necessarily true. A function can be continuous at a point but not differentiable.

3. How do you determine if a complex function is differentiable?

To determine if a complex function is differentiable at a point, you can use the Cauchy-Riemann equations. These equations relate the partial derivatives of the real and imaginary parts of the function at that point.

4. Can a complex function be differentiable at some points but not others?

Yes, a complex function can be differentiable at some points and not others. This is because differentiability depends on the behavior of the function at a specific point, not the entire domain.

5. What is the geometric interpretation of differentiability for a complex function?

The geometric interpretation of differentiability for a complex function is that it represents the existence of a well-defined tangent line at a given point. This tangent line is perpendicular to the level curves of the function and represents the direction of greatest change.

Similar threads

Replies
3
Views
1K
Replies
1
Views
813
Replies
5
Views
271
  • Calculus
Replies
4
Views
876
Replies
6
Views
849
  • Calculus
Replies
3
Views
2K
  • Calculus
Replies
2
Views
2K
Replies
5
Views
1K
Replies
6
Views
1K
  • Calculus
Replies
2
Views
1K

Back
Top