The

Cauchy-

Goursat theorum states that:

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.