Holomorphic function
The existence of a complex derivative in a neighbourhood is a very strong condition: It implies that a holomorphic function is infinitely differentiable and locally equal to its own Taylor series (is analytic).
Holomorphic functions are the central objects of study in complex analysis.
Though the term analytic function is often used interchangeably with "holomorphic function", the word "analytic" is defined in a broader sense to denote any function (real, complex, or of more general type) that can be written as a convergent power series in a neighbourhood of each point in its domain.
That all holomorphic functions are complex analytic functions, and vice versa, is a major theorem in complex analysis.
in its domain is defined as the limit[3] This is the same definition as for the derivative of a real function, except that all quantities are complex.
, and satisfy the Cauchy–Riemann equations:[6] or, equivalently, the Wirtinger derivative of
have continuous first partial derivatives and satisfy the Cauchy–Riemann equations, then
A more satisfying converse, which is much harder to prove, is the Looman–Menchoff theorem: if
have first partial derivatives (but not necessarily continuous), and they satisfy the Cauchy–Riemann equations, then
[8] The term holomorphic was introduced in 1875 by Charles Briot and Jean-Claude Bouquet, two of Augustin-Louis Cauchy's students, and derives from the Greek ὅλος (hólos) meaning "whole", and μορφή (morphḗ) meaning "form" or "appearance" or "type", in contrast to the term meromorphic derived from μέρος (méros) meaning "part".
, then the holomorphic functions coincide with those functions of two real variables with continuous first derivatives which solve the Cauchy–Riemann equations, a set of two partial differential equations.
[6] Every holomorphic function can be separated into its real and imaginary parts
is a rectifiable path in a simply connected complex domain
Cauchy's integral formula states that every function holomorphic inside a disk is completely determined by its values on the disk's boundary.
can be written as a contour integral[13] using Cauchy's differentiation formula: for any simple loop positively winding once around
In regions where the first derivative is not zero, holomorphic functions are conformal: they preserve angles and the shape (but not size) of small figures.
in any disk centred at that point and lying within the domain of the function.
From an algebraic point of view, the set of holomorphic functions on an open set is a commutative ring and a complex vector space.
[7] In fact, it is a locally convex topological vector space, with the seminorms being the suprema on compact subsets.
The principal branch of the complex logarithm function
As a consequence of the Cauchy–Riemann equations, any real-valued holomorphic function must be constant.
Another typical example of a continuous function which is not holomorphic is the complex conjugate
The definition of a holomorphic function generalizes to several complex variables in a straightforward way.
Osgood's lemma shows (using the multivariate Cauchy integral formula) that, for a continuous function
The much deeper Hartogs' theorem proves that the continuity assumption is unnecessary:
More generally, a function of several complex variables that is square integrable over every compact subset of its domain is analytic if and only if it satisfies the Cauchy–Riemann equations in the sense of distributions.
For example, the region of convergence of a power series is not necessarily an open ball; these regions are logarithmically-convex Reinhardt domains, the simplest example of which is a polydisk.
Unlike functions of a single complex variable, the possible domains on which there are holomorphic functions that cannot be extended to larger domains are highly limited.
is holomorphic if and only if its antiholomorphic Dolbeault derivative is zero:
For instance, the Fréchet or Gateaux derivative can be used to define a notion of a holomorphic function on a Banach space over the field of complex numbers.
