In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space ⁠${\displaystyle \mathbb {C} ^{n}}$⁠. 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.

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 ⁠${\displaystyle z_{0}}$⁠" means not just differentiable at ⁠${\displaystyle z_{0}}$⁠, but differentiable everywhere within some close neighbourhood of ⁠${\displaystyle z_{0}}$⁠ in the complex plane.

Given a complex-valued function ⁠${\displaystyle f}$⁠ of a single complex variable, the derivative of ⁠${\displaystyle f}$⁠ at a point ⁠${\displaystyle z_{0}}$⁠ in its domain is defined as the limit

This is the same definition as for the derivative of a real function, except that all quantities are complex. In particular, the limit is taken as the complex number ⁠${\displaystyle z}$⁠ tends to ⁠${\displaystyle z_{0}}$⁠, and this means that the same value is obtained for any sequence of complex values for ⁠${\displaystyle z}$⁠ that tends to ⁠${\displaystyle z_{0}}$⁠. If the limit exists, ⁠${\displaystyle f}$⁠ is said to be complex differentiable at ⁠${\displaystyle z_{0}}$⁠. This concept of complex differentiability shares several properties with real differentiability: It is linear and obeys the product rule, quotient rule, and chain rule.

A function is holomorphic on an open set ⁠${\displaystyle U}$⁠ if it is complex differentiable at every point of ⁠${\displaystyle U}$⁠. A function ⁠${\displaystyle f}$⁠ is holomorphic at a point ⁠${\displaystyle z_{0}}$⁠ if it is holomorphic on some neighbourhood of ⁠${\displaystyle z_{0}}$⁠. A function is holomorphic on some non-open set ⁠${\displaystyle A}$⁠ if it is holomorphic at every point of ⁠${\displaystyle A}$⁠.

A function may be complex differentiable at a point but not holomorphic at this point. For example, the function ${\displaystyle \textstyle f(z)=|z|{\vphantom {l}}^{2}=z{\bar {z}}}$ is complex differentiable at ⁠${\displaystyle 0}$⁠, but is not complex differentiable anywhere else, esp. including in no place close to ⁠${\displaystyle 0}$⁠ (see the Cauchy–Riemann equations, below). So, it is not holomorphic at ⁠${\displaystyle 0}$⁠.

The relationship between real differentiability and complex differentiability is the following: If a complex function ⁠${\displaystyle f(x+iy)=u(x,y)+i\,v(x,y)}$⁠ is holomorphic, then ⁠${\displaystyle u}$⁠ and ⁠${\displaystyle v}$⁠ have first partial derivatives with respect to ⁠${\displaystyle x}$⁠ and ⁠${\displaystyle y}$⁠, and satisfy the Cauchy–Riemann equations: