Consequences of the Cauchy—Goursat Theorem—Maximum Principles and the Local Theory

Abstract

Of great utility in the differential calculus of functions on ℝ are primitives or antiderivatives. The Fundamental Theorem here affirms that integration always produces such primitives. In particular, every continuous function on R has a primitive. [We will see that this is not so for continuous functions on regions in ℂ.] Naturally we look to integration to produce primitives in the plane too. But now we must face the fact that integration can be carried out over many paths from z₀ to z, whereas if the integrand f has a primitive F, then the integral over any such path must equal F(z) - F(z₀) (see 2.10) and be consequently independent of the path chosen. This independence of path is equivalent to the integral over any closed path being 0. Conversely, when the integral is path independent, a primitive can be manufactured by using an “indefinite” integral (see the proof of 2.11). Thus the corresponding fundamental theorem(s) in the plane affirm that certain integrals over closed paths are 0.

An erratum to this chapter is available at http://dx.doi.org/10.1007/978-3-0348-9374-9_15