The Cauchy-Riemann equations provide a necessary and sufficient test for whether a complex function f = u + iv, viewed via its real-valued component functions u(x,y) and v(x,y), is analytic on a domain: f is analytic if and only if u and v have continuous first partial derivatives satisfying ∂u/∂x = ∂v/∂y and ∂u/∂y = −∂v/∂x. This belongs to complex analysis and connects complex differentiability to multivariable real calculus, reducing the question of analyticity to a checkable system of partial differential equations, and it yields immediate corollaries about the rigidity of analytic functions (e.g., constant-derivative or purely real analytic functions on connected domains must be constant).
The Cauchy-Riemann equations provide a necessary and sufficient test for whether a complex function f = u + iv, viewed via its real-valued component functions u(x,y) and v(x,y), is analytic on a doma…