Bruce's theorem

Bruce's theorem is a principle in complex analysis that relates the sum of certain types of integrals taken around parts of a closed curve to the integral around the entire curve. Specifically, it states that if you have a closed loop and two functions that are analytic (smooth and well-behaved) inside and outside parts of that loop, then the combined integral of these functions over the loop can be expressed as the sum of integrals over the separate regions, accounting for the functions' behavior. This helps in calculating complex integrals by breaking them down into manageable sections.