In the books I looked and also in the script of my former Complex Analysis Prof. we proved the Cauchy Integral Formula for simply connected domains and closed curves in the following order:
- Goursat's Theorem for Triangles for functions continuous everywhere and holomorphic everywhere except at a single point
- Cauchy's Theorem for convex domains for functions continuous everywhere and holomorphic everywhere except at a single point
- Cauchy's Integral Formula for convex domains
- Cauchy's Integral Theorem for null homologous cycles
- Introduction of homotopic curves and proof of Cauchy's Integral Theorem for those curves.
What I really don't like is two things: First this artificial single point in (1) and (2). It is of course nice to have such a weak version of Goursat and Cauchy Integral Theorem, because then the Integral Formula is easy to derive. But as a student is seems very artificial at first to exclude a single point out of the domain of holomorphy, and then see later that this makes sense and is handy. Second, I don't like to go first over nice domains like convex or starlike domains and afterwards end up in arbitrary domains.
So, what I know is the following version of Cauchy's Integral Theorem:
Let $\Omega$ be a simply connected domain, let $f$ be holomorphic in $\Omega$ and let $\gamma$ be a closed curve (piecewise $C^1$) in $\Omega$. Then \begin{align*} \int_\gamma f(z) dz=0. \end{align*}
I would like to deduce from this the following version of the Cauchy Integral Formula:
Let $\Omega$ be a simply connected domain, let $f$ be holomorphic in $\Omega$ and let $\gamma$ be a closed curve (piecewise $C^1$) in $\Omega$. Then \begin{align*} n(\gamma, z)f(z)=\frac{1}{2\pi i}\int_\gamma \frac{f(w)}{w-z} dw,\qquad z\in\Omega\backslash\gamma. \end{align*} Here, $n(\gamma,z)$ is the winding number of $\gamma$ at $z$.
The only way i can think of to achieve my goal is to fix some $z\in\Omega\backslash\gamma$ and cut a line connecting $z$ and the boundary of $\Omega$ out of $\Omega$ to obtain a simply connected domain. Then I need to adjust $\gamma$ so that it does not run trough this slit anymore and goes around $z$. After that I must show that this detour goes to zero if I move along the slit sufficiently close. This is certainly a way to go, but I really don't like how technical this gets if one wants to write down a rigorous proof.
My question is: Is there a more elegant way?
Once again, I don't know that holomorphic functions have a continuous derivative, that they can be developed in a power series expension and so on. I only have the abovementioned version of Cauchy's Theorem.