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 is holomorphic, and the domain of definition of has somehow the right shape, then

for any contour which is closed, that is, (the closed contours look a bit like a loop). For this theorem to hold, surprisingly, the shape of the domain of definition is supremely important; for some it does hold, for some it doesn't. In this chapter, we will prove that the theorem holds for certain 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 domainsEdit

Definition 5.1:

A set   is called star-shaped if and only if there exists a   such that for all other  , the line connecting   and   lies completely in  ; this line can be written down in set notation for instance as follows:

 ,

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

 .

Primitives on star-shaped domainsEdit

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   be holomorphic, and let   be a triangle contained within  . Then

 ,

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

[[File:]]

Corollary 5.3:

Let   be holomorphic, where   is star-shaped. Then   has a primitive (which we shall call  ) on  , and it is given by

 ,

where

is the straight line from   to  .

Cauchy's theorem on star-shaped domainsEdit

Theorem 5.4:

Let   be holomorphic, where   is a star-shaped domain. Then for any closed contour   whose image is contained within  

Cauchy's integral formulaEdit

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 theoremEdit

Theorem 2.3 (Arzelà–Ascoli):

Let   be a sequence of functions defined on an interval   which is

  • equicontinuous (that is, for any   there exists   such that  ) and
  • uniformly bounded (that is, there exists   such that  ).

Then   contains a uniformly convergent subsequence.

Proof:

Let   be an enumeration of the set  . The set   is bounded, and hence has a convergent subsequence   due to the Heine–Borel theorem. Now the sequence   also has a convergent subsequence  , and successively we may define   in that way.

Set   for all  . We claim that the sequence   is uniformly convergent. Indeed, let   be arbitrary and let   such that  .

Let   be sufficiently large that if we order   ascendingly, the maximum difference between successive elements is less than   (possible since   is dense in  ).

Let   be sufficiently large that for all   and    .

Set  , and let  . Let   be arbitrary. Choose   such that   (possible due to the choice of  ). Due to the choice of  , the choice of   and the triangle inequality we get

 .

Hence, we have a Cauchy sequence, which converges due to the completeness of  .