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:

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

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