ABSTRACT
It was shown (Section 13.5) that if a function is holomorphic in a region, its loop integral in that region is zero. It follows that if the function is holomorphic in a doubly-connected (or ring-shaped) region, the integrals along the inner and outer loops are equal. This implies that a loop may be contracted (Section 15.2), without changing the value of the integral, as long as it crosses only regions where the function is holomorphic. Thus a loop can be shrunk to zero if the function is holomorphic in its interior, implying that the integral is zero (Section 15.1). A point where the function is not holomorphic is called a singularity. If a function is holomorphic in the interior of a loop, except at an isolated singularity, the loop can be shrunk to a small circle around it. If the singularity is of the type called simple pole, the value of the integral will be determined by the value at the pole of an associated function (Section 15.3), called residue (Section 15.7); the derivate of any order of the function can be obtained if the singularity is (Section 15.8) a pole of order N . It follows that if a function is holomorphic in a region, its value, and that of all its derivates, at any interior point, may be determined by loop integrals around the boundary; thus the values of a holomorphic function on the boundary determine the values of the function and all its derivatives at all interior points. This implies that a holomorphic function that is defined as having a first-order derivate also has derivates of all orders (Section 15.4), as well as primitives of all orders (Section 13.1). The singularities of a function, either in the interior (Section 15.6) or boundary (Section 15.5) of a region, if they are of the type called pole, provide a convenient way of evaluating loop integrals (Section 15.9) using residues. The loop integrals with poles will be used to calculate forces on bodies (Chapter 28), for example, hydrodynamic and electromagnetic; the integrals along loops (the real axis) integrals also specify the fields due to source distributions (Chapters 18,24,26), along curves (straight segments).
