Complex Analysis/Cauchy's theorem and Cauchy's integral formula
The Goursat–Pringsheim lemma edit
Cauchy's theorem states that if is holomorphic on ( a star-shaped domain; we'll make precise in the next section what that means) and is a closed contour with values in , then
As a first step of proving the theorem, we will prove it in the special case where is a triangle; this is the Goursat–Pringsheim lemma. Goursat had the idea first, but Pringsheim then provided the idea to use triangles (instead of squares, as Goursat had done).
Any triangle may be divided into four small triangles of equal side length as indicated in the picture. Each of the small triangles will have half the side length of the original triangle; this is clear from the formulae one would assign to the smaller triangle if one were to make things (even more) rigorous. We shall denote the four triangles as , , and . But one of the four non-negative numbers
will have to be the largest. We then set , the first of the four smaller triangles arising from such that the absolute value as above is largest, and so on; ie. once is defined for general , we decompose into triangles as above and then define the one of where the absolute value of the four ones indicated above is largest.
In this fashion, we obtain a sequence of ever smaller triangles . We now have the following lemma:
Lemma 6.2 (Cantor's intersection theorem):
Let be a decreasing sequence of compact sets situated in a Hausdorff topological space . Then
In a Hausdorff topological space, compact sets are closed; indeed, let be such a compact set, and assume that the neighbourhood filter of a certain is such that all its elements have nonempty intersection with .
Hence, if we assume that
we get by deMorgan's rules, taking complements in ,
Hence, if we regard as a topological space with subspace topology, the sets form an open cover of . By compactness of , there is a finite subcover , where without loss we assume that . Due to monoticity, we then obtain that in fact, and thus , which is a contradiction.
Thus, we can obtain a point
where shall denote the filled triangle (i.e. the convex hull) of .
Now by assumption, was differentiable, in particular at . Therefore, we may write
where as ; indeed, we may set
Furthermore, if we take any triangle in the complex plane and form the curve integral of a polynomial over it, we get zero; indeed, every polynomial has a primitive which is formed in the obvious way, according to chapter 2; namely it's exactly the same formula as in the real case, and the differentiation formulae confirm that. We may then decompose the triangle into three lines; say that are the corners of that triangle, then ( denoting concatenation, even though we will later, when considering chains, allow for more general curves as arguments on the left and right of ), and we get
as desired. In particular, we get
and by the fundamental estimate
However, this value is actually greater than the absolute value of the original integral, since we will have
since the integrals cancel out, as indicated in the picture, and hence by the triangle inequality
then use induction to get this "greaterness" for each . But this shows the theorem, since if the absolute value is less or equal than any positive value, it is zero.
Cauchy's theorem on starshaped domains edit
Now we are ready to prove Cauchy's theorem on starshaped domains. This theorem and Cauchy's integral formula (which follows from it) are the working horses of the theory; from these two we will deduce the local theory of holomorphic functions, and the global theory will then follow as well.
Informally, will "look like a star" with center .
Regarding these domains, we have the following lemma:
Let be holomorphic, where is star-shaped. Then will have a primitive in , i.e. a holomorphic function such that at every point .
We begin by just defining
this will be well-defined just by the star-shapedness of , i.e. must be and is defined on all of the path . Our claim now is that thus defined is indeed a primitive of .
Indeed, pick any fixed . Since is open, we find a (possibly small) radius such that is completely contained in .
Cauchy's integral formula edit
Cauchy's integral formula is a formula which looks at first glance a bit strange:
Now this formula may seem strange at first, but considering the special case and inserting the definition of curve integrals, we obtain that a special case of it is actually a mean value formula:
In what follows, we shall always assume that the path is traversed counterclockwise.
We first note that the function
is continuous; this follows from dominated convergence by the boundedness of on compact sets. Then we note that for any , the above expression coincides with
Furthermore, given , we have
where is the curve
where we define
denoting concatenation of curves. The curve is depicted in the following picture:
However, one finds star-shaped domains (eg. half-planes) which contain each of the cycles of which is composed, whence, in fact
by Cauchy's theorem on star-shaped domains. Hence, is constant on and by continuity and , this constant must be exactly .
Remark: Using a similar construction like the curve above, we in fact obtain the following much more general version of Cauchy's integral formula:
Now note that in our situation, we may differentiate under the integral sign, which gives us formulae for (the higher derivatives) for in the respective ball. Indeed, we get
Later, we will deduce the very same formulae from the Taylor expansion which we will compute; in fact, all holomorphic functions are equal to their Taylor series, as we will show.