ABSTRACT
A real integral is a particular case of a contour integral, in that the path of integration lies along the real axis, that is, it consists of one or several straight segments. Complex integrals along open contours in general, and real integrals in particular, can be calculated by means of residues, by two methods: (i) using a change of variable to transform the open contour into a closed loop (Section 17.2); (ii) extension of the open loop until it becomes a closed loop, in such a way (Section 17.3) that the contribution to the integral of the additional path can be calculated (or made to vanish). In both cases (i) and (ii), a loop integral results that may be calculated by residues (Chapter 15). The Riemann integral defined before (Chapter 13) is a proper integral, as it applies to a bounded function over a finite path (Section 17.1). The integral is improper of the first (second) kind if the path is infinite (function is unbounded), and of mixed kind is both features occur. A improper integral of first kind (Sections 17.4 and 17.5) is unilateral (bilateral) if only one (both) of the endpoints are at infinity; an improper integral of the second kind (Section 17.6) is unilateral (bilateral) if the singularity of the integrand occurs at the end-points (in the interior of the path of integration). Improper integrals are calculated as limits of proper integrals, for example, as the end-point(s) tend to infinity (the integral approaches its singularity) for the first (second) kind. If the integrand is a many-valued function, with branch-point on the path of integration, the integral may become indeterminate, unless a principal value is chosen (Section 17.7). Even for a single-valued integrand, the primitive may be multivalued, leading to an improper integral of the second kind whose value is indeterminate (Section 17.8), and can be specified uniquely as the Cauchy principal value (Section 17.9). The improper integrals of first (second) kind apply to source distributions that have infinite extent (singularities); an improper integral of third kind appears if the source distribution has infinite extent and contains singularities. A particular kind of singularity occurs when calculating the field due to a source distribution at point not outside the source distribution but within it; in this case a source element should not create a field upon itself, leading to the Cauchy principal value of the integral.
