Complex Analysis/Integration over chains

Definition (continuously differentiable 1-chain):

A continuously differentiable 1-chain is an element of the free -module over the set of all continuously differentiable curves .

Definition (image):

Let be a continuously differentiable 1-chain. Then the image of is defined to be

,

where are precisely the continuous differentiable curves such that .

Transfer of the theorems on integrationEdit

Argument and winding numbersEdit

Assume we are given a closed contour supported in  , and suppose that we are an observer located at the origin. Suppose we want to measure how often a moving object rotates about us (ie. passes through a point which is chosen fixed and has a fixed angle with respect to us). The resulting number is called the winding number of the given closed contour. Note though that it is signed; that is, if we contour were to travel (regarding angular distance) first round the circle, and then again but in reverse direction, the winding number is supposed to be zero.

To make this precise,

argument definition to circle and lift to standard covering

homotopy invariance of the latter


 

To do:

  1. the exchange theorem of integration and differentiation will be needed for this
  2. link to liouville thm
  3. clarification on the function being holo on an open cover and well-defined


Theorem (Cauchy's formula):

Let   be open, and let   be a cycle which is contained within  , and which is nullhomologous in  . Let also   be a holomorphic function. Then we have

 

for all   that are not in the image of  

Proof: Define a function on   by

 

When one variable is kept fixed, this function is holomorphic in the other. Hence, upon considering the function

 ,

we find that   is holomorphic in   by exchanging integration and differentiation. But it is in fact holomorphic on  , because we assumed the cycle   to be nullhomologous in  . By shrinking   if necessary, we may assume that   is bounded, since the image of a curve is compact and finite unions of compact sets are compact. Then   becomes a bounded function by a Weierstraß-type theorem and by Liouville's theorem it is then constant, and hence equal to zero. In particular, inserting  , we get

 ,

that is,

 .  

Definition (chain integral):

Let   be open and let   be holomorphic. Let   be a continuously differentiable 1-chain whose image is contained within  . Then the integral over   is defined to be

 ,

where

 .

Proposition (homologous chains induce equal integrals):

Let   be open, and let   be a holomorphic function. Suppose that   are continuously differentiable 1-chains, whose image is contained within  , such that   for some 2-chain   in the [[singular chain complex over  ]]. Then

 .

Proof: By definition of integration over a chain, it suffices to prove that whenever   is a 2-chain, then

 .

Moreover, by linearity we may restrict to the case where   is a simplex. But   is nullhomologous, so that

 

by Cauchy's theorem.  

Theorem (Residue theorem):

Let   be an open, bounded subset. Let   be a cycle whose image is contained within  , and let   be meromorphic, so that no singularity of   is contained within the image of  . Then

 ,

where   are the singularities of  .

Proof: Note that the image of any continuous 1-chain of   is compact, hence closed since   is Hausdorff. Hence, for each singularity   of  , choose a radius   such that the image of   does not intersect  , and the latter set shall also be contained in   (which is open, after all). Moreover, set  , where the latter boundary path is traversed once and counterclockwise (so that its winding number is one). Then define a new continuously differentiable 1-chain by

 .

Then   will be nullhomologous, so by Cauchy's theorem and Cauchy's formula

 .