Existence of antiderivative without Cauchy-Goursat's theorem
I wonder if anybody has tried the following kind of direct proof for the existence of an antiderivative of an analytic function on a star-shaped domain.
Theorem: Let be an analytic function on a star-shaped domain D. Then f has an antiderivative F on D.
"proof": For simplicity, assume that every point in D is connected to by a line segment. Define
It can be checked that as , the integrand converges to .
Therefore, , provided that the limit and integral are interchangeable. qed.
Of course, a limit and integral cannot always be interchanged. But I wonder if anybody seriously considered the above line of proof.
Thanks. As far as I can search from several textbooks in complex variables, the above theorem is proved by using Cauchy-Goursat's theorem. More concretely, they use the equality , which can be justified by Cauchy-Goursat's theorem. The point of my question is: Is it possible to directly invoke to the computation as above, without using Cauchy-Goursat's theorem? If this is possible, we get another proof of Cauchy-Goursat's theorem, at least for star-convex domains.