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Deriving the Cauchy-Riemann from the derivative of the complex conjugate.

Mathematics Asked by mojojojo on November 3, 2020

I am trying to prove the Cauchy-Riemann conditions naturally arise from the condition that:

$$frac{partial f(z,z^{*})}{partial z^{*}} = 0$$

But I’m having trouble starting from the definition of a derivative and get stuck:

$$frac{partial f(z,z^{*})}{partial z^{*}} = lim_{Delta zto 0}frac{f(z,z^{*}_{0}+Delta z)-f(z,z^{*}_{0})}{Delta z}$$

Am I going about this the right way? Is there a better way to do this?

Thanks!

One Answer

HINT:

You've written the correct definition of the derivative. But instead, note that

$$ frac{partial f}{partial z^*}=frac12left(frac{partial f}{partial x}+ifrac{partial f}{partial y} right)tag1 $$

Let $f=u+iv$ and set the right-hand of $(1)$ to $0$ . Can you finish now?

Correct answer by Mark Viola on November 3, 2020

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