# Complex Analysis question

Discussion in 'Math' started by bitrex, Feb 6, 2011.

1. ### bitrex Thread Starter Active Member

Dec 13, 2009
79
4
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 $Z_0$?

Jul 7, 2009
1,585
141
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.

3. ### bitrex Thread Starter Active Member

Dec 13, 2009
79
4
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 $Z_o$?

4. ### Papabravo Expert

Feb 24, 2006
11,083
2,159
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

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