An unwinding number pair for continuous expressions of integrals

Abstract

We consider the problem of obtaining expressions for integrals that are continuous over the entire domain of integration where the true mathematical integral is continuous, which has been called the problem of obtaining integrals on domains of maximum extent (Jeffrey, 1993). We develop a method for correcting discontinuous integrals using an extension of the concept of unwinding numbers for complex functions to treat passage of paths of integration through branch points. The approach amounts to treating the codomain of a complex function as a pair of 2-dimensional real manifolds (the real and imaginary parts) and computing the intersection of paths and branch boundaries. These intersection points determine where discontinuities appear, which are where the unwinding numbers change value. We demonstrate the approach, and its computability, by considering how to compute continuous codomain manifolds for the logarithm and arctangent functions, which appear frequently in the integration problem on account of Liouville's theorem. Treating both real integrals and complex contour integrals, explicit computable formulas for the jump conditions of unwinding numbers are provided in the case of integrands that can be expressed as a rational function.