Abstract
Abstract
This chapter explores conditions under which the value of an integral does not depend on the choice of contour. One such result is an analogue of the Fundamental Theorem of real analysis, and in deference to that subject it bears the same name. However, there exists a still deeper result which has no counterpart in the world of the reals. It is called Cauchy’s Theorem. It is not only possible, but sometimes useful to integrate non-analytic functions. However, new phenomena arise if one concentrates on the integrals of mappings that are analytic. Cauchy’s Theorem is the essence of these new phenomena. Essentially, it says that any two integrals from a to b will agree, provided that the mapping is analytic everywhere in the region lying between the two contours. Almost all the fundamental results of the subject flow from this single horn of plenty.
Publisher
Oxford University PressOxford
Reference125 articles.
1. Ahlfors, L. V. 1979. Complex analysis: an introduction to the theory of analytic functions of one complex variable, International series in pure and applied mathematics, 3d ed edition. New York: McGraw-Hill.
2. Hamilton, Rodrigues, and the quaternion scandal.;Mathematics Magazine,1989
3. Arnol’d, V. I. 1989. Mathematical Methods of Classical Mechanics, volume 60 of Graduate Texts in Mathematics, 2nd edition. New York: Springer-Verlag.