Can I get a few examples of complex functions being complex differentiable at a point but not holomorphic in their domain?

  • 2
    $\begingroup$ Do you mean real differentiable, or even $C^{\infty}$, but not holomorphic? I think you need to specify what you mean. $\endgroup$ – Moya Aug 17 '15 at 17:30
  • $\begingroup$ What exactly do you mean? Differentiable in the complex sense at some point? $\endgroup$ – Robert Israel Aug 17 '15 at 17:30
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    $\begingroup$ I assume you mean differentiable interpreted as a function $\Bbb R^2\to\Bbb R^2$. Take any polynomial in ${\rm Re}(z)$ and ${\rm Im}(z)$ that is not a polynomial in $z$. For example, ${\rm Re}(z)$ and ${\rm Im}(z)$ themselves. If you mean complex-differentiable in a nbhd of a point but not holomorphic at the point, then you'll run into a problem: there aren't any. Not sure what can be said pointwise. $\endgroup$ – whacka Aug 17 '15 at 17:31

$$ f(z) = |z|^2 $$ This is differentiable at just one point, $z=0$. If "holomorophic at $0$" were construed as meaning differentiable everywhere in some open neighborhood of $0$, then this would not be holomorphic at $0$. But I'm not sure that conventional definitions are sufficiently standard that you can say "holomorphic but not differentiable" and have everybody understand it in the same way.


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