Is it true that if function $f$ is analytic ($\sim$ holomorphic) in $\Omega \subset \Bbb C$, then it satisfies the Cauchy-Riemann (C-R) equations? And is it t true that if $f$ satisfies C-R equations and the functions $u(x,y)$ and $v(x,y)$ (the real and imaginary parts of $f(x,y)$ respectively) have first partial derivatives which are continuous, then $f$ is analytic ($\sim$ holomorphic)?
Is it true that if function $f$ is analytic( ~ holomorphic) in $\Omega \subset\Bbb C$, then it satisfies C-R equations?
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0Yes, but I am afraid I did not make myself clear what answer meant to me, so I decided to rewrite the most essential part in my previous question removing any unclear stuff. But next I would like to know how to use definition of complex differentiability( $f'(z_0)= \lim_{z \leftarrow z_0} \frac{f(z)-f(z_0)}{z-z_0}$) for example when $f(z)=\frac{1}{z}$? – 2012-08-24
1 Answers
Analytic and holomorphic are a priori different. A complex-analytic function is one that can be written as series with terms of the form $a_nz^n$. A holomorphic function is one that satisfies the Cauchy-Riemann equations. Your question is if they are equivalent, and the answer is yes.
Analytic functions are holomorphic, because you can differentiate the series term by term, and each term satisfies the Cauchy-Riemann equations. It is important here to distinguish complex-analyticity from real-analyticity, because for real-analytic functions this result does not hold. A real-analytic function in 2D is one that can be written as series with terms of the form $a_{n,m}x^ny^m$.
The result that says that holomorphic functions are analytic is called the Cauchy-Taylor theorem. One way to explain why it is true is that functions satisfying Cauchy-Riemann equations are very smooth, and their derivatives does not grow fast as you differentiate many times.
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1@alvoutila: Note that the main message here is not about how to name things, but rather about a relationship between two different concepts. References that follow this naming convention are Lang's *Complex analysis* and Remmert's *Theory of complex functions*. The latter book also has some historical material on how these names came to be. – 2012-08-28