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 .
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 integration[edit | edit source]
Argument and winding numbers[edit | edit source]
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
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
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
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