Holomorphic function

In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is, at every point of its domain, complex differentiable in a neighbourhood of the point. The existence of a complex derivative in a neighbourhood is a very strong condition, for it implies that any holomorphic function is actually infinitely differentiable and equal, locally, to its own Taylor series (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. The fact that all holomorphic functions are complex analytic functions, and vice versa, is a major theorem in complex analysis. Holomorphic functions are also sometimes referred to as regular functions. A holomorphic function whose domain is the whole complex plane is called an entire function. The phrase 'holomorphic at a point z0' means not just differentiable at z0, but differentiable everywhere within some neighbourhood of z0 in the complex plane. Given a complex-valued function f of a single complex variable, the derivative of f at a point z0 in its domain is defined by the limit This is the same as the definition of the derivative for real functions, except that all of the quantities are complex. In particular, the limit is taken as the complex number z approaches z0, and must have the same value for any sequence of complex values for z that approach z0 on the complex plane. If the limit exists, we say that f is complex-differentiable at the point z0. This concept of complex differentiability shares several properties with real differentiability: it is linear and obeys the product rule, quotient rule, and chain rule. If f is complex differentiable at every point z0 in an open set U, we say that f is holomorphic on U. We say that f is holomorphic at the point z0 if f is complex differentiable on some neighbourhood of z0. We say that f is holomorphic on some non-open set A if it is holomorphic in an open set containing A. As a pathological non-example, the function given by f(z) = |z|2 is complex differentiable at exactly one point (z0 = 0), and for this reason, it is not holomorphic at 0 because there is no open set around 0 on which f is complex differentiable. The relationship between real differentiability and complex differentiability is the following. If a complex function f(x + i y) = u(x, y) + i v(x, y) is holomorphic, then u and v have first partial derivatives with respect to x and y, and satisfy the Cauchy–Riemann equations: or, equivalently, the Wirtinger derivative of f with respect to the complex conjugate of z is zero: