Simple/trivial question on cauchy-riemann equations

Main Question or Discussion Point

This is a simple question, but I don't have a complex analysis book handy to verify, and I'm by no means very familar with complex analysis at all. Are the statements:

1. f(z) is analytic at a point z = z0 iff the cauchy-riemann equations hold in a neighborhood of z0
2. f(z) is analytic at a point z = z0 iff the cauchy-riemann equations hold at z0, and f(z) has continuous partials at z0

valid and equivalent? My gut says yes, but I have the feeling I'm missing something and that perhaps they're not equivalent. Wikipedia seems somewhat vague. Or are they not valid and I'm completely wrong? :)

BTW, this is not a homework question, I am not taking a complex analysis course, this is entirely for myself. Thanks!

Last edited:

HallsofIvy
Homework Helper
Yes, they are equivalent. It can be shown that if a function f(z) satisfies the Cauchy-Riemann equations hold in some neighborhood of z0 then f is in fact infinitely differentiable in that neighborhood. In fact, more: its Taylor series converges to the value of f(z) at every point in that neighborhood (which is the most basic definition of "analytic" on a neighborhood).

saltydog