Holomorphic function
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:
- is complex-differentiable at every point of
- is complex-differentiable at every point of , and the function we obtain as the derivative, is a continuous function.
- is infinitely often complex-differentiable. In other words, we can take the derivative of for any
- For any point , and any disc centered at of radius , that lies completely inside , can be expressed using a power series in
- is a continuous function and satisfies the Cauchy-Riemann differential equations at every point in
- is a continuous function and its integral along the boundary of any triangle contained completely inside is zero.
Equivalence of definitions
- For equivalence of (2), (3), (4), refer complex-analytic implies holomorphic and holomorphic implies complex-analytic
- Goursat's lemma and Morera's theorem establish the equivalence of (1), (2) and (6)
- Looman-Menchoff theorem establishes the equivalence with (5).
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.