# Can a function be analytic if it does not satisfy the Cauchy-Riemann conditions

I'm a student of electrical engineering, preparing for the theoretical exam covering complex analysis. I'm confused about the analyticity of a complex function, so I'm asking for clarification because our professor inadequately explained this concept.

What I understand from my lecture notes: suppose there's a complex-valued function $f(z)$. Such function is analytic if the following conditions are satisfied:

1. $f$ is differentiable at $z_0$, therefore $f'(z_0)$ exists
2. $f$ is differential at every point of some $\epsilon$-spherical neighborhood of $z_0$
3. $f$ can be expanded as a Taylor series in the vicinity of $z_0$

At that point, Cauchy-Riemann equations are nowhere mentioned. However, I've found online that C-R equations are a necessary condition for a complex-valued function being holomorphic (or analytic, although terms are used interchangeably). And here's where I'm starting to get lost.

If C-R equations are a necessary condition for holomorphicity, but they are not sufficient conditions for complex differentiability, then how can they ensure given function $f$ being holomorphic, if according to what I've read, such function must be differentiable at given point $z_0$ in order to be considered holomorphic? Can, therefore, a function be holomorphic (or analytic) if C-R conditions are not satisfied?

• C-R equations are necessary conditions for a function to be analytic. – Kavi Rama Murthy Jun 18 '18 at 10:03
• Alright, that's a start. So why is a complex differentiability mentioned? If a function must be smooth to be analytic (correct me if I'm wrong) and if C-R equations must be satisfied, what happens in a case when only first or second order derivatives exist? Is a function not analytic then despite satisfying C-R equations? – Pero Alex Jun 18 '18 at 10:09
• One of many interesting facts in Complex Analysis is if $f$ has continuous first partial derivatives satisfying C-R equations then $f$ is automatically infinitely differentiable. – Kavi Rama Murthy Jun 18 '18 at 10:18
• Well, that just made the whole thing clear for me. I hope I'm not asking too much, but can you provide a link to the proof for this claim? I'd be extremely helpful. Anyways, thank you for helping me understand this problem! Will mark the question solved. – Pero Alex Jun 18 '18 at 10:22
• Standard texts on CA are Conway and Rudin. Both these books have proofs of all the statements I have made. – Kavi Rama Murthy Jun 18 '18 at 10:28