# Limit / Differentiability of complex functions

• Sep 19th 2009, 11:35 PM
aman_cc
Limit / Differentiability of complex functions
I am new to delta-epsilon arguments to prove the existence of limits.

Q - Find all the points on the complex plain where f(z) = |z|^2 is differentiable? i.e. I seek all z0 where lim z->z0 [f(z)-f(z0)]/[z-z0] exisits

I did it without using any delta-epsilon argument. Answer I got is only z0 = 0 to be the only point.

Would really appreciate is someone can help me with a rigorous argument based on delta-epsilon definition of existence of limit.
• Sep 20th 2009, 02:58 AM
chisigma
If we write $\displaystyle z= x + i y$ and $\displaystyle f(z)= u(x,y) + iv(x,y)$ , then $\displaystyle f(z)$ is differentiable [analitic] where are satisfied the Cauchy-Riemann conditions...

$\displaystyle \frac{du}{dx} = \frac{dv}{dy}$

$\displaystyle \frac{du}{dy} = -\frac{dv}{dx}$ (1)

In this case is...

$\displaystyle u(x,y) = x^{2} + y^{2}$

$\displaystyle v(x,y)=0$ (2)

... and the (1) are satisfied only in $\displaystyle z=0$...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$
• Sep 20th 2009, 03:23 AM
aman_cc
Thanks. I do not want to use CR equations just want to use an epsilon-delta argument. I want to do this just to understand/practice how to use the most basic defintions of continity ans differentiability to prove this without any other tool at hand.