Cauchy–Riemann equations

The Cauchy–Riemann equations are necessary and sufficient for a function f(z) = u(x, y) + iv(x, y) to be holomorphic, provided the partial derivatives are continuous. In complex form, they can be written as ∂f/∂x̄ = 0, where ∂/∂x̄ = (1/2)(∂/∂x + i∂/∂y). This formulation shows that holomorphic functions are independent of the conjugate variable, a property that leads to the powerful result that such functions are analytic, meaning they can be expanded in Taylor series around every point in their domain.¹

The equations imply that holomorphic functions are conformal, preserving angles between curves, except at points where the derivative is zero. This geometric property arises because the Jacobian matrix of a holomorphic function represents a rotation and scaling, not a shear. Consequently, the Cauchy–Riemann equations are central to applications in fluid dynamics and electrostatics, where they describe irrotational and incompressible flows, and in cartography, where conformal maps are used to project the Earth's surface.²

The equations generalize to several complex variables, where they become the Cauchy–Riemann system, and to Clifford analysis, where they take the form of the Dirac equation. In physics, they appear in the theory of two-dimensional potential flow and in the study of electromagnetic fields. The equations also underpin the Riemann mapping theorem, which states that any simply connected domain can be conformally mapped to the unit disk, a result with deep implications in complex analysis and differential geometry.

Beyond the standard formulation, the Cauchy–Riemann equations have subtle variants. For instance, they can be expressed in polar coordinates, yielding u_r = (1/r)v_θ and v_r = -(1/r)u_θ, which are useful for functions with radial symmetry. The equations also fail to hold for functions that are merely real-differentiable but not complex-differentiable, such as f(z) = |z|^2. Historically, the equations were known to d'Alembert and Euler before Cauchy and Riemann, but their systematic use in complex analysis is credited to the latter two. In numerical analysis, the equations are used to construct conformal maps via the solution of elliptic boundary value problems.³

Holomorphic function

A complex function that is complex differentiable in a neighborhood of every point in its domain.

Conformal map

A function that preserves angles locally, often used in complex analysis and cartography.

Partial differential equation

An equation involving partial derivatives of a function of several variables.