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.

Star-shaped domains

Definition 5.1:

A set $S\subseteq \mathbb {C}$ is called star-shaped if and only if there exists a $z^{*}\in S$ such that for all other $z\in S$, the line connecting $z$ and $z^{*}$ lies completely in $S$; this line can be written down in set notation for instance as follows:

$\{(1-t)z+tz^{*}|t\in [0,1]\}$,

which is why the condition of star-shapedness may be phrased in precise mathematical terms as follows:

$\exists z^{*}\in S:\forall z\in S:\{(1-t)z+tz^{*}|t\in [0,1]\}\subseteq S$.

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.

Lemma 5.2 (Goursat, Pringsheim variant):

Let $f:O\to \mathbb {C}$ be holomorphic, and let $\Delta$ be a triangle contained within $O$. Then

$\int _{\Delta }f(z)dz=0$,

where by $\Delta$ we also denote the contour that arises when traversing the triangle sides successively, as indicated in the following picture:

[[File:]]

Corollary 5.3:

Let $f:O\to \mathbb {C}$ be holomorphic, where $O$ is star-shaped. Then $f$ has a primitive (which we shall call $F$) on $O$, and it is given by

$\int _{[z^{*},z]}f(z)dz$,

where

${\displaystyle }$

is the straight line from $z^{*}$ to $z$.

Cauchy's theorem on star-shaped domains

Theorem 5.4:

Let $f:O\to \mathbb {C}$ be holomorphic, where $O$ is a star-shaped domain. Then for any closed contour $\gamma$ whose image is contained within $O$

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

Theorem 2.3 (Arzelà–Ascoli):

Let $(f_{n})_{n\in \mathbb {N} }$ be a sequence of functions defined on an interval $[a,b],a<b\in \mathbb {R}$ which is

• equicontinuous (that is, for any $\epsilon >0$ there exists $\delta >0$ such that $|x-y|<\delta \Rightarrow \forall n\in \mathbb {N} :|f_{n}(x)-f_{n}(y)|<\epsilon$) and
• uniformly bounded (that is, there exists $M>0$ such that $\forall n\in \mathbb {N} :\forall x\in [a,b]:|f_{n}(x)|<M$).

Then $(f_{n})_{n\in \mathbb {N} }$ contains a uniformly convergent subsequence.

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 $k\geq 1$ $\left|f_{l_{N_{2}+k}}(x_{n})-f_{l_{N_{2}}}(x_{n})\right|<\epsilon /3$.

Set $N:=\max\{N_{1},N_{2}\}$, and let $k\geq N$. 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

$\left|f_{l_{N+k}}(y)-f_{l_{N}}(y)\right|\leq \left|f_{l_{N+k}}(y)-f_{l_{N+k}}(x_{n})\right|+\left|f_{l_{N+k}}(x_{n})-f_{l_{N}}(x_{n})\right|+\left|f_{l_{N}}(x_{n})-f_{l_{N}}(y)\right|<\epsilon /3+\epsilon /3+\epsilon /3=\epsilon$.

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