Cauchy–Riemann Equations

The Cauchy–Riemann equations are a set of conditions that a complex function must satisfy to be considered "holomorphic," meaning it is complex differentiable at every point in its domain. They relate the partial derivatives of the function's real part and imaginary part, ensuring these parts change in a coordinated way. This coordination guarantees the function behaves smoothly and preserves angles, making it important in complex analysis, fluid dynamics, and engineering. In essence, these equations ensure the function has a well-defined, smooth behavior similar to differentiability in real calculus, but in the complex plane.