More direct proof of Cauchy's Integral Formula for simply connected domains 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.
 A: Every proof I have every seen of Cauchy's formula from Cauchy's theorem applies a version of Cauchy's theorem to the function $g$ defined by $g(w) = \dfrac{f(w) - f(z)}{w-z}$ for $w \neq z$ and $g(z) = f'(z)$. Notice $g$ is continuous wherever $f$ is continuous, and $g$ is holomorphic wherever $f$ is holomorphic, expect possibly at $z$. 
As you noticed, you can't apply your version of Cauchy's theorem to $g$ because of lack of holomorphic-ness at $z$, so people establish versions of Cauchy's theorem that allow for the lack of holomorphic-ness at a single point. 
I agree that this seems a bit tedious. I don't know if there is a fundamentally different method, but I think there is a way to make it look less tedious.
I suggest you have a look at the book Complex Variables and Applications by Churchill and Brown. You will find:


*

*A direct proof of Cauchy's theorem that does not first go through special regions like triangles or convex sets. Section title: Cauchy-Goursat Theorem.

*The statement of Cauchy's theorem in simply connected domains. Section title: Simply Connected Domains (or Simply and Mulitply Connected Domains if you have an older edition).

*Cauchy's theorem for multiply connected domains. The proof is just to draw some lines and use cancellation of contour integrals in opposite directions. Section title: Multiply Connected Domains (or Simply and Multiply Connected Domains if you have an older edition)

*Cauchy's formula in simply connected domains. Section title: Cauchy Integral Formula.


You can find this book online easily. Try googling Library Genesis.
