The Cauchy-Riemann equations

The Cauchy-Riemann equations are a set of conditions that a complex function must satisfy to be considered "holomorphic" or complex differentiable. Think of a complex function as a rule assigning a point in the plane to each input point, with both real and imaginary parts. These equations ensure that the function behaves smoothly, without jumps or kinks, similar to how derivative rules work in real calculus. Specifically, they relate the ways the output's real and imaginary parts change with respect to the input's real and imaginary components, guaranteeing the function's differentiability in the complex sense.