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 the case of integrands that can be expressed as a rational function.

我们考虑的问题是，在真正的数学积分是连续的整个积分域中，获得积分表达式的连续性，这被称为在最大范围域中获得积分的问题（Jeffrey，1993）。我们开发了一种对不连续积分进行校正的方法，该方法使用展开数概念的扩展来处理复杂函数，以处理通过分支点的积分路径。该方法相当于将复杂函数的共域视为一对二维实数流形（实部和虚部），并计算路径和分支边界的交集。这些交点确定不连续出现的位置，即展开编号更改值的位置。我们演示了该方法及其可计算性，通过考虑如何计算对数和反正切函数的连续共域流形，由于Liouville定理，它们经常出现在积分问题中。在处理实数积分和复数轮廓积分时，对于可表示为有理函数的被积数，提供了展开数跳跃条件的显式可计算公式。