Complex Analysis/Cauchy's theorem for star-shaped domains, Cauchy's integral formula, Montel's theorem

In the last section, we learned about contour integrals. A fundamental theorem of complex analysis concerns contour integrals, and this is Cauchy's theorem, namely that if $$f: O \to \mathbb C$$ is holomorphic, and the domain of definition $$O$$ of $$f$$ has somehow the right shape, then
 * $$\int_\gamma f(z) dz = 0$$

for any contour $$\gamma$$ which is closed, that is, $$\gamma(a) = \gamma(b)$$ (the closed contours look a bit like a loop). For this theorem to hold, surprisingly, the shape of the domain of definition $$O$$ is supremely important; for some $$O$$ it does hold, for some it doesn't. In this chapter, we will prove that the theorem holds for certain $$O$$ which are so-called star-shaped domains. Later on in the book, we will see that it even holds for a larger class of domains, namely the simply connected ones, which will require advanced tools which we will build up along the course of this book.

Primitives on star-shaped domains
The basis for the following considerations (and thus for almost every theorem of the remainder of the book, except for some stuff that has to do with cycles) is the following technical lemma.

Cauchy's integral formula
Another important theorem by Cauchy, called Cauchy's integral formula, is almost as fundamental as Cauchy's integral theorem. We begin with the following lemma.

Lemma 5.5

Montel's theorem
Proof:

Let $$(z_n)_{n \in \mathbb N}$$ be an enumeration of the set $$[a, b] \cap \mathbb Q$$. The set $$\{f_n(z_1)|n \in \mathbb N\}$$ is bounded, and hence has a convergent subsequence $$(f_{k_{1, n}}(z_1))_{n \in \mathbb N}$$ due to the Heine–Borel theorem. Now the sequence $$(f_{k_{1, n}}(z_2))_{n \in \mathbb N}$$ also has a convergent subsequence $$(f_{k_{2, n}}(z_2))_{n \in \mathbb N}$$, and successively we may define $$f_{k_{m, n}}$$ in that way.

Set $$f_{l_m} := f_{k_{m, m}}$$ for all $$m \in \mathbb N$$. We claim that the sequence $$(f_{l_m})_{m \in \mathbb N}$$ is uniformly convergent. Indeed, let $$\epsilon > 0$$ be arbitrary and let $$\delta$$ such that $$|x - y| < \delta \Rightarrow \forall n \in \mathbb N: |f_n(x) - f_n(y)| < \epsilon/3$$.

Let $$N_1 \in \mathbb N$$ be sufficiently large that if we order $$a, x_1, \ldots, x_{N_1}, b$$ ascendingly, the maximum difference between successive elements is less than $$\delta$$ (possible since $$\mathbb Q$$ is dense in $$\mathbb R$$).

Let $$N_2 \in \mathbb N$$ be sufficiently large that for all $$n \in \{1, \ldots, N_1\}$$ and $$k \ge 1$$ $$\left|f_{l_{N_2 + k}}(x_n) - f_{l_{N_2}}(x_n)\right| < \epsilon/3$$.

Set $$N:= \max\{N_1, N_2\}$$, and let $$k \ge N$$. Let $$y \in [a, b]$$ be arbitrary. Choose $$x_n$$ such that $$|x_n - y| < \delta$$ (possible due to the choice of $$N_1$$). Due to the choice of $$\delta$$, the choice of $$N_2$$ and the triangle inequality we get
 * $$\left| f_{l_{N+k}}(y) - f_{l_N}(y) \right| \le \left| f_{l_{N+k}}(y) - f_{l_{N+k}}(x_n) \right| + \left| f_{l_{N+k}}(x_n) - f_{l_N}(x_n) \right| + \left| f_{l_N}(x_n) - f_{l_N}(y) \right| < \epsilon/3 + \epsilon/3 + \epsilon/3 = \epsilon$$.

Hence, we have a Cauchy sequence, which converges due to the completeness of $$\mathcal C([a, b])$$.