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Green's Theorem has the form: $$\oint P(x,y)dx = - \iint \frac{\partial P}{\partial x}dxdy , \oint Q(x,y)dy = \iint \frac{\partial Q}{\partial y}dxdy $$ The Cauchy-Riemann equations have the following form:(Assuming $z = P(x,y) + iQ(x,y)$) $$\frac{\partial P}{\partial x} = \frac{\partial Q}{\partial y}, \frac{\partial P}{\partial y} = - \frac{\partial Q}{\partial x}$$

Is there any connection between this two equations?

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    $\begingroup$ +1. This is a good question. You should read the book by Churchill and Brown on Complex Variables and Applications. $\endgroup$
    – user17762
    Nov 29, 2010 at 2:17
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    $\begingroup$ This is a great question. Keep finding connections and analogies between things you've already learned and you'll be well on your way to mathematical research. $\endgroup$ Nov 29, 2010 at 3:45

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Absolutely yes!

In Section 7 of these notes on Green's Theorem, I explain how Green's Theorem plus the Cauchy-Riemann equations immediately yields the Cauchy Integral Theorem.

Many serious students of mathematics realize this on their own at some point, but it is surprising how few standard texts make this connection. In (especially American) undergraduate texts on Subject X, there is a distressing tendency to politely ignore the existence of Subject Y, even when any student of Subject X will almost surely have already have studied / be concurrently studying / soon be studying Subject Y.

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    $\begingroup$ I think the X and Y thing is not just an American thing. Btw... as a contrast - suddenly when you come to graduate school you are supposed to know this, and that etc. :) $\endgroup$ Nov 29, 2010 at 14:05
  • $\begingroup$ @AD.: Well, I have a lot of experience with "the" American university system, having been involved with it for my entire adult life. I don't want to speak for the rest of the world -- what little I know there comes from a few brief visits and anecdotes from colleagues. I have heard tell, though, that there is a bit more unity in the European (let's say French, in particular) education system. $\endgroup$ Nov 29, 2010 at 19:23
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If you consider $P(x,y)$ and $Q(x,y)$ as the part real and imaginary of an analitic function you can get the connection.

Then $P(x,y)$ and $Q(x,y$) verify the Cauchy-Riemann equations, thus :

$$ \frac{\partial P}{\partial x} = \frac{\partial Q}{\partial y} , \frac{\partial P}{\partial x} = -\frac{\partial Q}{\partial y} $$

Depending of oriented get that the integral is Zero.

This satisfies the Cauchy's integral theorem that an analytic function on a closed curve is zero.

Edit:

You can see it here, where the proof of Cauchy's integral theorem uses Green's Theorem .

A little deeper you can see, Complex Analysis by Lars Ahlfors, section 4.6 page 144.

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That's Green's Theorem can be used to prove Cauchy's Theorem or a corollary of it, depending on your text, is in A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Exer 4.25

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Note that that $f'$ assumption is superfluous because $f$ is holomorphic, but this is not proven until Ch5.


Appendix

Green's Theorem is stated as:

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Cor 4.20 is a corollary of Cauchy's Thm 4.18 for the authors and is stated as:

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Cauchy's Thm 4.18 is stated as:

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The authors acknowledge that Cauchy's Theorem is sometimes designated to be the statement in Cor 4.20 instead of their Thm 4.18

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