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.