Differentiability of a complex function.

[jeffreydk]

(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))

$$\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y} \qquad \quad \frac{\partial v}{\partial x}=-\frac{\partial u}{\partial y}$$

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.

 

[jvicens]

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!