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?

 

[fzero]

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.

 

[fzero]

We need to differentiate between real differentiability and complex differentiability. Complex differentiability means that

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

exists. If the CR equations are satisfied, the function is both analytic and complex differentiable. If our function were not analytic, the derivative (*) would not make sense.