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!

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

0 Asked on July 27, 2020 by strictly_increasing

numerical calculus numerical methods ordinary differential equations probability theory stochastic differential equations

1 Asked on July 27, 2020 by gulzar

2 Asked on July 26, 2020 by outlier

Get help from others!

Recent Answers

- Jon Church on Why fry rice before boiling?
- Lex on Does Google Analytics track 404 page responses as valid page views?
- Peter Machado on Why fry rice before boiling?
- Joshua Engel on Why fry rice before boiling?
- haakon.io on Why fry rice before boiling?

© 2022 AnswerBun.com. All rights reserved. Sites we Love: PCI Database, MenuIva, UKBizDB, Menu Kuliner, Sharing RPP