Pointwise Cauchy-Riemann equations suffice?

complex-analysis cauchy-riemann-equations

Another question just now reminded me of something I realized a while ago I didn't know how to do:

Say $V\subset\Bbb C$ is open, $f:V\to\Bbb C$, $f=u+iv$, and at every point of $V$ the partials of $u$ and $v$ exist and satisfy the Cauchy-Riemann equations. How do we show that $f$ is holomorphic?

I mean it seems it "must" follow. But note if you think this is totally trivial it's possible you're wrong; for instance it's not clear how the hypothesis implies that $f'(z)$ exists.


Solution 1:

The paper (pdf linked) When is a Function That Satisfies the Cauchy-Riemann Equations Analytic by J.D. Gray and S. A. Morris is quite useful as it offers various results about this issue and has a proof of the Looman-Menchoff theorem included as well as references for related results.

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