Is there any connection between Green's Theorem and the Cauchy-Riemann equations?

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?

Solution 1:

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.

Solution 2:

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.

Solution 3:

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

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:

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

Cauchy's Thm 4.18 is stated as:

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