I'm doing some self-teaching in complex analysis, and I'm having a little difficulty with some of the terminology. A function is said to be "analytic" at a point \(Z_0\) if it is differentiable at every point within a neighborhood of\(Z_0\). If the function satisfies the Cauchy-Riemann equations, this implies that the function has a derivative at that point. Does this also imply that the function is analytic in the neighborhood of \(Z_0\)?

 

A function satisfying the C-R equations at a point is a necessary condition for a function to have a derivative at that point, but it's not sufficient. Churchill, "Complex Variables and Applications", 3rd ed., p. 45 gives the example function f(z) = (z*)^2/z if z is not 0 and f(z) = 0 if z = 0 (z* is the complex conjugate); this function satisfies the C-R equations at z = 0, but is not differentiable there.

 

Thanks very much for your reply, that is a good point. I'm still curious as to how to determine if a function is analytic at a point. The problem set I'm working about alternately asks me to determine where a function is differentiable, and where it is analytic, and I'm not sure what the distinction between the two conditions is. Edit: I did a little Googling and it seems that the Cauchy-Riemann equations are a sufficient condition to show that a function is analytic. So can a function be analytic in a neighborhood of \(Z_o\) but actually not be differentiable at \(Z_o\)?

 

I think that was the gist of the example from Churchill p.45. The other method was to look at the the coefficient of z^-1 in the Laurent series expansion to determine if a function was analytical.

http://en.wikipedia.org/wiki/Laurent_series

 

I did a little Googling and it seems that the Cauchy-Riemann equations are a sufficient condition to show that a function is analytic.

You need to make sure you understand the difference between "necessary" and "sufficient" in this context.

Section 17, "Sufficient Conditions", page 42, in the Churchill book is in the chapter on Analytic Functions. Here's the statement in a nutshell:

Satisfaction of the C-R equations at a point is not sufficient for the existence of a derivative at that point. He then proves a theorem that states that if the first partial derivatives of the function (in the usual Cartesian coordinates) exist in a neighborhood of the point and are continuous at the point, and the C-R equations are satisfied at the point, then the derivative at the point exists.

The proof is relatively elementary and takes less than a page. It's instructive to understand -- and it yields a practical formula for calculating the complex derivative.

You can then use it e.g. to show that \(f(z) = |z|^2\) has a zero derivative at z = 0.