Holomorphic Function - Definition

Definition

Given a complex-valued function ƒ of a single complex variable, the derivative of ƒ 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 ƒ 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 ƒ is complex differentiable at every point z₀ in an open set U, we say that ƒ is holomorphic on U. We say that ƒ is holomorphic at the point z₀ if it is holomorphic on some neighborhood of z₀. We say that ƒ is holomorphic on some non-open set A if it is holomorphic in an open set containing A.

The relationship between real differentiability and complex differentiability is the following. If a complex function ƒ(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 ƒ with respect to the complex conjugate of z is zero: which is to say that, roughly, ƒ is functionally independent from the complex conjugate of z.

If continuity is not a given, the converse is not necessarily true. A simple converse is that if u and v have continuous first partial derivatives and satisfy the Cauchy–Riemann equations, then ƒ is holomorphic. A more satisfying converse, which is much harder to prove, is the Looman–Menchoff theorem: if ƒ is continuous, u and v have first partial derivatives (but not necessarily continuous), and they satisfy the Cauchy–Riemann equations, then ƒ is holomorphic.