Margie Marx

Answered

2022-01-14

Analytic iff Holomorphic on open domainss of $\mathbb{C}$

Answer & Explanation

scoollato7o

Expert

2022-01-15Added 26 answers

Step 1
The definition that an infinitely differentiable function f is holomorphic on an open subset $U\subset \mathbb{C}$ if
$\frac{\partial f}{\partial \stackrel{―}{z}}=0$
a perfectly good definition and is essentially just a restatement of the Cauchy Riemann equations. It seems in Griffiths and Harris they prove a generalized version of the Cauchy integral formula which holds for any ${C}^{\mathrm{\infty }}$
function (which reduces to the standard version when $\frac{\partial f}{\partial \overline{z}}=0$) and use this to prove that such functions are analytic.
If you want to learn about the equivalent definitions of holomorphic/complex analytic functions and theorems about their equivalence I'd suggest a book specifically on complex analysis.

MoxboasteBots5h

Expert

2022-01-16Added 35 answers

Step 1
I do not have a copy of that text book at hande, but the standard definition of holomorphic function is that it is a differentiable function, that is, that, for each ${z}_{0}$ in its domain, the limit
$\underset{z\to {z}_{0}}{lim}\frac{f\left(z\right)-f\left({z}_{0}\right)}{z-{z}_{0}}$
exists.Quite often, but not always, it is added to the definition the condition that the domain of f is an open subset of $\mathbb{C}$

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