# Caucy-Riemann equations and differentiability question

bitrex
I'm doing a little self study on complex analysis, and am having some trouble with a concept.

From Wikipedia:

"In mathematics, the Cauchy–Riemann differential equations in complex analysis, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations that provides a necessary and sufficient condition for a differentiable function to be holomorphic in an open set."

But I have a function here, (z*)^2/z, which appears to satisfy the Cauchy-Riemann equations at z = 0, but the function is not differentiable there. Isn't this a contradiction that the above condition is both necessary and sufficient?

Homework Helper
Gold Member
But I have a function here, (z*)^2/z, which appears to satisfy the Cauchy-Riemann equations at z = 0, but the function is not differentiable there. Isn't this a contradiction that the above condition is both necessary and sufficient?

As you say, the derivatives in the CR equation don't exist at z=0, so I'm not sure why you believe that the CR equations are satisfied. A prerequisite to CR is that the real and imaginary parts of the function are differentiable with respect to the real variables.

bitrex
Thanks for your reply, I'm not sure exactly why I thought they were satisfied either! I see now that they are not.

A further question - some exercises in the problem set I'm working on ask me to determine where a complex function is differentiable, and some to determine where the function is analytic. If a complex function is differentiable at point $$Z_o$$, is that also a sufficient condition to assume that the function is analytic in the neighborhood of $$Z_o$$? The text I'm using is not clear on this, so I'm not sure if in the problem set they are essentially asking me the same thing with different terminology.

$$\lim_{h\rightarrow 0, h\in \mathbb{C}} \frac{f(z_0+h)-f(z_0)}{h} = f'(z_0) ~~~(*)$$