Cauchy's Integral Theorem - Discussion

Discussion

As was shown by Goursat, Cauchy's integral theorem can be proven assuming only that the complex derivative f '(z) exists everywhere in U. This is significant, because one can then prove Cauchy's integral formula for these functions, and from that deduce these functions are in fact infinitely differentiable.

The condition that U be simply connected means that U has no "holes" or, in homotopy terms, that the fundamental group of U is trivial; for instance, every open disk qualifies. The condition is crucial; consider

which traces out the unit circle, and then the path integral

is non-zero; the Cauchy integral theorem does not apply here since is not defined (and certainly not holomorphic) at .

One important consequence of the theorem is that path integrals of holomorphic functions on simply connected domains can be computed in a manner familiar from the fundamental theorem of real calculus: let U be a simply connected open subset of C, let f : UC be a holomorphic function, and let γ be a piecewise continuously differentiable path in U with start point a and end point b. If F is a complex antiderivative of f, then

The Cauchy integral theorem is valid in slightly stronger forms than given above. e.g. Let U be a simply connected open subset of C and f a function which is holomorphic on U and continuous on . Let be a loop in which is uniform limit of a sequence of rectifiable loops in U with bounded length. Then, applying the Cauchy theorem to the, and passing to the limit one has

See e.g. (Kodaira 2007, Theorem 2.3) for a more general result.

The Cauchy integral theorem leads to Cauchy's integral formula and the residue theorem.

Read more about this topic:  Cauchy's Integral Theorem

Famous quotes containing the word discussion:

    We should seek by all means in our power to avoid war, by analysing possible causes, by trying to remove them, by discussion in a spirit of collaboration and good will. I cannot believe that such a programme would be rejected by the people of this country, even if it does mean the establishment of personal contact with the dictators.
    Neville Chamberlain (1869–1940)

    Americans, unhappily, have the most remarkable ability to alchemize all bitter truths into an innocuous but piquant confection and to transform their moral contradictions, or public discussion of such contradictions, into a proud decoration, such as are given for heroism on the battle field.
    James Baldwin (1924–1987)

    If the abstract rights of man will bear discussion and explanation, those of women, by a parity of reasoning, will not shrink from the same test: though a different opinion prevails in this country.
    Mary Wollstonecraft (1759–1797)