nLab holomorphic function

Definitions

A function $f$ between complex manifolds is holomorphic if it is complex-differentiable, or equivalently complex-analytic. One may also see definitions referring to the intermediate notions of continuous differentiability or infinite differentiability.

The theorem that every continuously complex-differentiable function is analytic is soft and due to Augustin Cauchy; the theorem that every complex-differentiable function is continuously differentiable is hard and due essentially to Édouard Goursat (which may be seen as filling in a gap in Cauchy’s proof). See Cauchy integral formula and Goursat theorem.

On infinite-dimensional manifolds, we have several notions of holomorphic function; see Wikipeda.

Related concepts

• analytic function

• meromorphic function

• domain of holomorphy

• biholomorphic function

• Dolbeault complex

References

• Claire Voisin, section 1 of Hodge theory and Complex algebraic geometry I,II, Cambridge Stud. in Adv. Math. 76, 77, 2002/3

In relation to the homotopy-theoretic Oka principle:

• Franc Forstnerič, Stein manifolds and holomorphic mappings – The homotopy principle in complex analysis, Springer 2011 (doi:10.1007/978-3-642-22250-4)