Holomorphic function

From Companal

Definition

Definition for one-variable function

Let be an open subset (not necessarily connected, though we can for practical purposes restrict ourselves to domains -- connected open subsets) of . A function is termed a holomorphic function if it satisfies the following equivalent conditions:

  1. is complex-differentiable at every point of
  2. is complex-differentiable at every point of , and the function we obtain as the derivative, is a continuous function.
  3. is infinitely often complex-differentiable. In other words, we can take the derivative of for any
  4. For any point , and any disc centered at of radius , that lies completely inside , can be expressed using a power series in
  5. is a continuous function and satisfies the Cauchy-Riemann differential equations at every point in
  6. is a continuous function and its integral along the boundary of any triangle contained completely inside is zero.

Equivalence of definitions

Definition on Riemann surfaces

Let be a Riemann surface. A holomorphic function on is a map from to whose restriction to any coordinate chart is a holomorphic function in the usual sense.