1) Let $f$ have an antiderivative $F$ in a region $D$ containing a $\underline{closed \,\,\,contour}$ $C$. Then
\begin{equation}
\int_Cf(z)dz = 0
\end{equation}
2) Also, Cauchy's theorem says: If f is $\underline{holomorphic}$ inside and on a $\underline{closed \,\,\,contour}$ $C$ then
\begin{equation}
\int_Cf(z)dz = 0
\end{equation}
Then why is Cauchy's theorem so important and what is the difference between this one and the previous one? It looks to me that the previous one required less conditions than this.
3) Finally, another theorem is:
If f is $\underline{holomorphic}$ on a region $D$ and $C_1$, $C_2$ are $\underline{closed \,\,\,contours}$ in $D$, which can be continuously deformed into each other within $D$, then:
\begin{equation}
\int_{C_1}f(z)dz = \int_{C_2}f(z)dz
\end{equation}
Is it now obvious? Of course they are, they are zero!
4) Deformation Theorem:
Let $f$ be $\underline{holomorphic}$ in a simply connected region $D$. Then:
\begin{equation}
\int_Cf(z)dz = 0
\end{equation}
for every closed contour $C$ in $D$.
Where simply connected means: that any closed contour in $D$ can be continously deformed within $D$ to a point.
Again, is this not saying the same thing as above??
I really don't get the difference between all these theorems.
Best Answer
(2) Holomorphic means simply that $\exists f'$. Why $\exists F$ s.t. $F' = f$ is "less"? Even more, what condition is easier to check?
(3) In this case, interesting things can happen out of the region. Check what happens to $f(z) = 1/z$ in $\Bbb C\setminus\{0\}$. And the integrals can be $\ne 0$!
(4) is a consequence of (3).