structural:reformatting, stylistic:tone, structural:summarization
Contour integrals

The contour integral of a complex function f: C → C {\displaystyle f:\mathbb {C} \to \mathbb {C} } is a generalization of the integral for real-valued functions. For continuous functions in the complex plane, the contour integral can be defined in analogy to the line integral by first defining the integral along a directed smooth curve in terms of an integral over a real valued parameter. A more general definition can be given in terms of partitions of the contour in analogy with the partition of an interval and the Riemann integral. In both cases the integral over a contour is defined as the sum of the integrals over the directed smooth curves that make up the contour.