ABSTRACT
The theory of complex integration is founded on the fundamental Cauchy integral theorem and its extension to the Cauchy integral formulas. The first form of the integral theorem was proved by the French mathematician Augustin-Louis Cauchy in 1825 under the hypothesis that the derivative of a function f(z), analytic in a domain D, is continuous on the boundary of D. This proof of the theorem restricts its application to domains D with smooth boundaries, and so would seem to imply that the result of the theorem does not apply to domains with piecewise smooth boundaries, such as, semicircles and rectangles. Later in 1883, this restriction was lifted by Edouard Goursat who, using a different method of proof, showed the theorem to be applicable to domains with piecewise smooth boundaries. This extension of the validity of the theorem is extremely important because it enables it to be applied in a great variety of situations, most of which arise in applications. Because of Goursat’s contribution, the theorem is now called the Cauchy-Goursat theorem. Both versions of the theorem will be proved, as well as the Cauchy integral formulas and many of their consequences, but although this generalization of the Cauchy formula will be used throughout this section, its proof will be postponed until the end of the section. The proof can be omitted at a first reading by those mainly interested in applications of the theorem, many of which were first made by Cauchy.
