I am having some trouble understanding the definition of Cauchy Integral from my textbook.
My problem is $\partial D$. I understand $\partial D$ is the boundary of $D$. Why would you take the integral $\partial D$ instead of $D$. Isn't $D$ inside of $\partial D$?
The theorem I found online states that: Suppose $f$ is analytic on domain $D$. Let $\gamma$ be a piece-wise simple closed curved in $D$ whose instead Omega is also in $D$. Then the integral from $\gamma$ is $0$.
I am confused on the boundary of integral part. Thank you!!
Since $dz$ is a line element, $$\int_D f(z) \ dz$$ doesn't make sense. So, I suppose the general answer would be that we do so because this is a theorem about line integrals. This theorem tells you that if you pick any piecewise smooth closed curve $\gamma$ and $f$ is holomorphic in the region bounded by $\gamma$, then $$\int_\gamma f(z) \ dz = 0.$$
Yes, in a sense, $D$ is "inside" $\partial D$.