Is it true that if function $f$ is analytic( ~ holomorphic) in $\Omega \subset\Bbb C$, then it satisfies C-R equations?

Question

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)?

Answer 1

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.