complexdifferentiable

A function f defined on an open subset Ω of the complex plane is said to be complex differentiable at a point z0 ∈ Ω if the limit

(f(z0 + h) − f(z0)) / h as h → 0

exists in the complex sense. That limit, when it exists, is the complex derivative f′(z0). If f is complex differentiable at every point of Ω, then f is holomorphic on Ω.

In practice, one often writes f in terms of real and imaginary parts: f(z) = u(x,y) + i v(x,y) with z = x + i y. A necessary and, under suitable regularity conditions, sufficient set of criteria is given by the Cauchy–Riemann equations: ∂u/∂x = ∂v/∂y and ∂u/∂y = −∂v/∂x, where the partial derivatives exist. If these partial derivatives exist in a neighborhood of z0 and are continuous at z0, and the Cauchy–Riemann equations hold there, then f is complex differentiable at z0. If f is differentiable at every point of an open set, it is holomorphic on that set.

holomorphic functions are analytic: locally they admit a power series expansion and are infinitely differentiable. They are conformal at points where f′(z) ≠ 0, preserving angles locally.

Examples of complex differentiable functions include polynomials in z, e^z, sin z, and rational functions with no poles in the domain. Notable non-examples are the complex conjugate function z̄, which is nowhere differentiable, and |z|^2, which is differentiable only at z = 0. In many contexts, “complex differentiable” on a domain is synonymous with “holomorphic.”