One complex derivative forces infinite smoothness
A holomorphic function takes complex values and has a complex derivative throughout a neighbourhood of every point of its domain. That condition is so strong that the function is automatically differentiable infinitely often and matches its own Taylor series near each point, making holomorphic functions the central objects of complex analysis.
Charles Briot and Jean-Claude Bouquet, both pupils of Augustin-Louis Cauchy, coined the name in 1875 from Greek words for whole and form. It contrasts with meromorphic, built on the word for part: a meromorphic function is holomorphic apart from isolated poles and behaves like a ratio of entire functions. Cauchy himself had used the word synectic. Such functions are also called regular, and one defined on the whole complex plane is called entire.
The derivative uses the same difference quotient as real calculus, except that the limit must come out the same however z approaches the point from any direction in the plane. Because this derivative is linear and follows the familiar chain, product, and quotient rules, sums, products, and compositions of holomorphic functions stay holomorphic, and a quotient keeps the property wherever its denominator is nonzero.
Differentiability at one isolated point is not enough. The function |z|², equal to z times its conjugate, has a complex derivative at the origin and nowhere else, so it fails to be holomorphic there. Writing f as u + iv, complex differentiability corresponds to the Cauchy–Riemann equations, which require ∂u/∂x = ∂v/∂y and ∂u/∂y = −∂v/∂x. Continuous partial derivatives satisfying those equations guarantee a holomorphic function, and the far harder Looman–Menchoff theorem relaxes the continuity demand. The complex derivative can then be read off directly from the partials.
Strictly, analytic means expressible as a convergent power series near each point, a notion that applies to real functions too. That every holomorphic function is complex analytic, and conversely, is a major theorem that does not follow obviously from the definitions, which is why some writers now prefer holomorphic while analytic stays in wide use.
Source: Holomorphic function
