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Suppose I have a real-valued function $f$ defined on a complex open connected set $D$. Am I right in saying that: $f$ is analytic on $D$ if and only if $f$ is $n$ times continuously differentiable on $D$?

Or does this only apply if $f$ is complex valued?

thanks

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    Thanks Willie, that is very helpful.2011-04-17

1 Answers 1

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Let $D$ be a region in $\mathbb{C}$. Then $f : D \to \mathbb{R}$ is analytic on $D$ if and only if $f$ is constant. This is easily verified if you consider the Cauchy-Riemann equation.

If we consider a complex-valued function $f : D \to \mathbb{C}$ instead, we obtain the theorem that epitomizes the true nature of complex analysis: $f$ is analytic if and only if $f$ is complex differentiable on $D$. No further assumption on the differentiability of $f$, including $C^1$ condition, is needed. Notice that this drasically constrasts with the situation in the real case, where $C^0 (\Omega) \supsetneq C^1 (\Omega) \supsetneq C^2 (\Omega) \supsetneq \cdots \supsetneq C^{\infty}(\Omega) \supsetneq C^{\omega}(\Omega)$.

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    Thank you all for the help.2011-04-17