Cauchy's theorem state if $f$ is analytic in a region $R$ and on its boundary $C$ then, $f(a)=1/2πi∮f(z)/(z-a)dz$. It was first proved by Green's theorem with added restriction that $f$ has a continuous derivative. However, Goursat gave a proof that which remove the constriction.
I want to ask according to Cauchy's integral formula, if a function of a complex variable has a first derivatives, i.e. analytic, in a simply connected region, than all its higher derivatives exist in that region.
So what is the significance of Goursat's proof? Is there a function which has an analytic region but the derivatives on the region is not continuous? Since if so, the higher derivatives will not exist and contradict with the Cauchy's integral formula.
Best Answer
The issue is one of different meanings of analytic function. There are several different definitions which, rather remarkably, turn out to be equivalent.
In the result of Cauchy you state, "analytic" means "complex differentiable". It is certainly not a priori clear that such a function has a continuous derivative. On the contrary it's something of a miracle, since this is false for real functions. And then once you know this, indeed the Cauchy-Goursat Integral Formula tells you that complex differentiable functions are infinitely differentiable and equal to their Taylor series expansion in a neighborhood around each point.
See any introductory complex analysis text for more information on this sequence of ideas. Or for that matter, see this wikipedia article.