Morera's Theorem.

In two variable Calculus, you learnt the following result: let $ P(x,y)$ and $ Q(x,y)$ be two functions defined on the same simply connected domain $ \mathcal{D}$ . Suppose that $ \partial P \partial y$ and $ \partial Q \partial x$ are continuous on the interior of $ \mathcal{D}$ and that, for every Jordan curve in $ \mathcal{D}$ , the following equation holds:

$\displaystyle \oint_C P \; dx + Q \; dy =0$    

Then $ \partial P \partial y =\partial Q \partial x$ in $ \mathcal{D}$ .

Using this result we can prove the following converse to Cauchy-Goursat theorem:

Proposition 5.6.1   Let $ f(z)=u(x,y)+iv(x,y)$ be a function such that $ u_x,u_y,v_x,v_y$ are continuous in a simply connected domain $ \mathcal{D}$ . Suppose that, for every Jordan curve in $ \mathcal{D}$ , the integral $ \oint_C f(z) \; dz $ is equal to 0. Then $ f$ is analytic on $ \mathcal{D}$ .

Another converse of Cauchy-Goursat theorem, stronger than Proposition 6.1 is Morera's theorem:

Theorem 5.6.2 (Morera)   Let $ f$ be a continuous function on an open simply connected domain $ U$ . Assume that for every loop $ C$ in $ U$ , the integral $ \int_C f(z)
\; dz$ is equal to 0. Then $ f$ is analytic on $ U$ .

Example 5.6.3  

Noah Dana-Picard 2007-12-24