Rouché's Theorem

Rouché's Theorem is a concept from complex analysis that helps determine the number of roots a complex function has inside a certain area, typically a circle. It states that if, on the boundary of that area, one function's magnitude is always greater than the other’s, then both functions have the same number of roots (solutions) inside the area. Essentially, it allows you to simplify complicated problems by replacing part of the function with a more manageable one, ensuring the count of solutions remains unchanged within the boundary.