I'm currently going through complex analysis, and I'm trying to grasp the concept of the whole residue theorem and so on.
I followed the derivation of the residue theorem from the Cauchy integral theorem, and I think I kind of understand what is going on there.
I thought about whether it's possible to derive the Cauchy integral formula from the residue theorem since I read somewhere that the integral formula is just a special case of the residue theorem.
I tried looking it up somewhere but didn't find anything.
Sure it's a special case, since the residue of $g(z):=\dfrac{f(z)}{z-a}$ at $z=a$ is $f(a)$.
This is fairly trivial, but to elaborate a little, recall that $f$ is holomorphic. So we can write $f(z)=f(a)+f'(a)(z-a)+f''(a)/2(z-a)^2+\dots$, and the residue is easy to get.