Cauchy (Riemann)

The Cauchy-Riemann equations are a set of mathematical conditions in complex analysis that help determine if a function of a complex variable is differentiable, meaning it behaves nicely and has a complex derivative. These equations relate the partial derivatives of the function’s real and imaginary parts. If a function meets these criteria, it is considered "holomorphic," which implies it's smooth and has important properties, such as being integrable. This concept is foundational in fields like fluid dynamics and electrical engineering, where complex functions are used to model various phenomena.