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I was trying to solve some problems in Complex Analysis, and was encountering some trouble in problems that tried to show some relations between $f(z)$ and $f(\bar{z})$. I was trying to see some examples online and I tend to encounter this description of the derivatives that I'm not able to see where it comes from.

The mentioned description of the derivatives are: $\frac{du}{dz} = \frac{1}{2}\left(\frac{du}{dx} - i\frac{du}{dy}\right)$ and $\frac{du}{d\bar{z}} = \frac{1}{2}\left(\frac{du}{dx} + i\frac{du}{dy}\right)$

Arctic Char
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pdaranda661
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  • So, you're looking for a proof of the equation $\frac{\partial u}{\partial z} = \frac{1}{2}\left( \frac{\partial u}{\partial x} - i\frac{\partial u}{\partial y}\right)$, and similarly for $\frac{\partial u}{\partial \overline{z}}$? – Jesse Madnick Jun 05 '21 at 16:00
  • @JesseMadnick yes. I guess that is related to the fact that if you took the whole derivative in one direction, and added to the other, you would be doubling the derivative, but I cannot find a formal solution. – pdaranda661 Jun 05 '21 at 16:09
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    Jack's answer to another question may help you understand the Wirtinger derivatives better on a basic level, and depending on your background, most of my answer to an unrelated question might help you understand where they come from. – Mark S. Jun 05 '21 at 16:32
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    @MarkS. actually this was quite useful to see a direct analytical demonstration of the equation. Also, where should I have studied the Writinger equations? I'm self-learning in maths and I thought that with y previous background complex analysis could be suitable, but I'm with the Ahlford's book and that hasn't appear until now, and the exercises I'm trying seem to be required to be solved that way. – pdaranda661 Jun 05 '21 at 16:39
  • I didn't see the Wirtinger derivatives in an entire introductory semester of complex analysis. I imagine they're a little niche topic. – Mark S. Jun 05 '21 at 16:42
  • @MarkS. that is then problematic, because then there must be some other way to solve that problem. – pdaranda661 Jun 06 '21 at 06:24
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    Yes, probably. The wirtinger derivatives make things more convenient, but if you have a problem (not containing them in the statement) that's intended to use them: in the worst case you could at least write everything out in terms of $\partial/\partial x$ and $\partial/\partial y$ and use Cauchy-Riemann as applicable. But there may very well.be a nicer workaround, depending on the problem. – Mark S. Jun 06 '21 at 11:10

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