Holomorphy

Holomorphy is the property of a function being holomorphic. Given an open set U in the complex plane and a function f: U → C, f is holomorphic at a point z0 ∈ U if the complex derivative f′(z0) = lim_{h→0} (f(z0 + h) − f(z0))/h exists. If f is holomorphic at every point of U, it is holomorphic on U. Equivalently, f is holomorphic iff around every z0 there is a neighborhood on which f admits a convergent power series expansion f(z) = ∑_{n=0}^∞ a_n (z − z0)^n.

Holomorphic functions are infinitely differentiable and analytic; they satisfy the Cauchy–Riemann equations when expressed in terms of real and imaginary parts. They enjoy strong local and global properties: the Cauchy integral formula expresses f in terms of its values on a nearby contour, and many powerful results follow, including Liouville’s theorem, the maximum modulus principle, the identity theorem, and the open mapping theorem. The derivative of a holomorphic function is itself holomorphic on its domain. Common examples include polynomials, the exponential function, and trigonometric functions when viewed in the complex setting; entire functions are holomorphic on the whole complex plane.

In more general contexts, holomorphy extends to several complex variables and to maps between complex manifolds. There, a function is holomorphic if it is complex differentiable in all variables and satisfies appropriate higher-dimensional Cauchy–Riemann conditions. Holomorphy underpins much of complex analysis, providing a robust analytic framework with broad theoretical and applied implications.