ABSTRACT
A holomorphic function f(z) is defined by the existence of a first-order derivative, and its real and imaginary parts necessarily satisfy the Cauchy-Riemann conditions. It can be shown that (Section 13.1) the latter imply the existence of a primitive F , defined to be a function whose derivative is f =F ′. Hence a holomorphic functions f always has a primitive F such that F ′ = f , and since the latter is also holomorphic, F too has a primitive . . . , that is, given a holomorphic function an infinite number of successive primitives can be found. A real function can integrated only along the real axis (Section 13.2), whereas a complex function can be integrated (Sections 13.3-13.5) along any contour in the complex plane, where contour means a suitable curve. The integral of a complex function along a contour generally depends on the function itself, the end-points and the contour or path joining them. If the function is holomorphic, it has a primitive, so the integral is the difference of the primitive at the end-points, and is independent of the path joining them; the converse (Section 13.6) can also be proved: if an integral does not depend on the contour joining the end-points, the integrand is a holomorphic function. The operations derivation and integration have several useful properties when applied to holomorphic functions, for example (Section 13.7) the rule of integration by parts and the chain rule of derivation. The derivative and primitive are almost inverse operations, for example, the derivate of an integral with a free upper bound coincides with the integrand. This result may be generalized to cases in which (i) the integrand depends not only on the variable of integration but also on a parameter (Section 13.8); (ii) the limits of integration are functions rather than constants, that is, the end-points that are not fixed but may lie on given curves (Section 13.9). There will be frequent subsequent use of the integration along curved paths, and of the derivation of an integral with regard to a parameter appearing in the integrand, in the limits of integration or in both; the latter occur in problems with variable limits, where the end-points are not fixed, but can “slide” along a curve or surface. These appear in optimization and other problems, for example, the shortest bridge between two banks of a river, a constrained path around an obstacle, etc.
