Difference between Cauchy theorem and Cauchy goursat theorem In some books it is given that Cauchy theorem is equivalent to Cauchy Goursat theorem.And in some other it is given to be different. In my class , our teacher has given it differently. Cauchy theorem states that if $~f(z)~$ is an analytic function over a domain $D$ and $~f'(z)~$ is continuous in $D$ ,then $~\int f(z)dz~$ over a simple closed contour $C$, which lies entirely in $D$, is zero. Whereas 
Cauchy Goursat theorem states that if $~f(z)~$ is analytic at all points on and inside of a simple closed contour $C$, then
$~∫f(z)dz~$ over $C$ is zero. So my doubt is whether in Cauchy Goursat theorem  is continuity of $~f'(z)~$ over $~z~$, not a required condition. Which of them is a stronger theorem. How? please elaborate.
 A: Cauchy's Theorem was earlier, and less refined. Cauchy's Theorem assumed the function was continuously differentiable in a simply-connected region, and it was then proved that all integrals $\oint_{C}f(z)dz$ over simple closed paths $C$ must be $0$. The proof basically relied on Green's Theorem. Goursat's version only required the derivative to exist at each point of the region, without any requirement of continuity of the derivative, and yet the conclusion was the same.  Goursat's version allows one to prove that the function has a continuous derivative, without assuming it.
A: Let $~f(z)~$ be analytic in a region $R$ and on its boundary $C$. Then $$\oint_C~f(z)~dz~=~0~.$$
This fundamental theorem, often called Cauchy's integral theorem or briefly Cauchy's theorem, is valid for both simply- and multiply-connected regions. It was first proved by use of Green's theorem with the added restriction that $~f'(z)~$ be continuous in $R$. However, Goursat gave a proof which removed this restriction. For this reason the theorem is sometimes called the Cauchy-Goursat theorem when one desires to emphasize the removal of this restriction.
So Cauchy-Goursat theorem is the most important theorem in complex analysis, from which all the other results on integration and differentiation follow.

Ref.: "Schaum's Outline of Complex Variables" by Murray Spiegel, Seymour Lipschutz, John Schiller, Dennis Spellman (Chapter $4$) (McGraw-Hill Education)
