If a function F is analytic at all points interior to and on a simple closed contour C, then:

(the integral under C of)f(z)dz=0.

In the early 1800's Augustin Cauchy obtained this result with the condition that f is analytic in R and f' is continuous there. Edouad Goursat was the first to prove that the condition of continuity on f' can be omitted. This allowed mathematicians to show that the derivative of an analytic function(in the complex plane) is analytic, of the nth derivative of an analytic function exists for all n.