Cauchy-Goursat Theorem Thursday, October 17, 2013 1:59 PM A technical variation on the considerations of antiderivatives from last time. This by itself is not so important, but the idea becomes important when developed in later contexts. Let's return to calculation of Where C is a contour consisting of a circle of radius R centered at the origin, traversed counterclockwise. Here the contour crosses the branch cut. We saw last time that this is no technical problem, and one can do the contour integral by parameterization, and the calculation is simplest if one chooses a parameterization that starts and stops the contour at either side of the branch cut. But how can we approach this with antiderivatives? Here it will be useful to be careful. Both of these are multivalued so this only becomes meaningful if we specify branches. Generally speaking a multivalued function and its antiderivative are related branch by branch. That is, if we take the principal branch for f(z) = z 1/n then the antiderivative is the principal branch of (n/n+1) z (n+1)/n ComplexAnalysis Page 1
This can be verified by a straightforward boring calculation. Now how can we use the antiderivative for evaluating a contour integral crossing a branch cut? We approximate the contour integral by integrating over an approximating contour which avoids the branch cut and for which the contour integral can be easily evaluated using the antiderivative. This approximating contour is not closed; no problem. When using approximating contours, one should always make a clean argument for how they relate to the actual contour. Here the approximating contour will converge to the desired contour integral, but this is not always the case, and one should make an argument to indicate whether or not the approximation requires a correction term. Provided that: We can check that the first statement is clearly true, and we can verify the second for our function: ComplexAnalysis Page 2
Now we will formulate a fundamental result in complex variable theory. Cauchy-Goursat Theorem: Let C be a closed contour which encloses a region R in its interior. Then if f(z) is analytic over the interior region R and is continuous on the closure of the interior region R, then: ComplexAnalysis Page 3
(The fussing in the technical condition, which is actually useful, is that the result only requires analyticity on the interior (not on) the contour, but you do need continuity over the contour and its interior.) The proof of the "Cauchy" part of this theorem is just an application of Green's theorem (similar to showing that certain closed loop line integrals of curl-free vector fields are 0): To do this simple proof, one needs to assume somewhat stronger technical conditions, namely that f(z) is continuously differentiable over the closure of the region R. Then we can express the contour integral in terms of real line integrals and use Green's theorem. ComplexAnalysis Page 4
So Cauchy's theorem is relatively straightforward application of Green's theorem. What Goursat did was to extend the argument to the weaker technical conditions stated in the theorem. Goursat's argument avoids the use of Green's theorem because Green's theorem requires the stronger condition. The essence of the proof is a sort of adaptive mesh refinement Want to argue that with a suitably fine mesh and a prescribed error tolerance that over any particular box in the mesh One establishes such a statement through an adaptive mesh refinement procedure, by showing that if the refinement doesn't eventually lead to such a result, then there must be a breakdown of analyticity in the interior, contrary to the assumption of the theorem. ComplexAnalysis Page 5
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