Let be the closed region consisting of all points interior to and on the simple closed contour .
If is analytic in and is continuous in ,
(This is the precursor of Cauchy-Goursat Theorem, which allows us to drop the condition that is continuous.)
Proof Using Green’s Theorem:
Let denote a positively oriented simple closed contour , .
where and . Thus
Next, we need Green’s Theorem:
By assumption is continuous in , thus the first-order partial derivatives of and are also continous. This is exactly what we need for Green’s Theorem.
Continuing from above, we get which is exactly zero in view of the Cauchy-Riemann equations , !