The Cauchy theorem for functions on closed sets
Philip T. Maker · Bulletin of the American Mathematical Society · 1942
The object of this paper is to extend the theorem of Cauchy to functions of a complex variable defined on any bounded closed set, £, by determining conditions on ƒ (z) in order that for certain coverings of E, C n , and an extension of f(z), /*(z), lim,*.^f Cn f*(z)dz = 0.It was suggested partly by the notion of a general monogenic function due to Trjitzinsky 1 and partly by the measure theory methods of Menchoff 2 and others, which succeed so well in lightening the restrictions on the real and imaginary parts of a complex function in order that f{z) be regular.Throughout this paper we shall consider only rectangles with sides parallel to the real and imaginary axes.A C-covering of a plane set F, denoted by C, will be a set of closed rectangles, possibly abutting, but nonoverlapping, which contain F. c will denote the boundary of C. The covering C n is to be composed of rectangles R mn so that