Several complex variables and analytic spaces Holomorphic functions of several variables Concept
Holomorphic function of several variables
Summary
A holomorphic function of several variables is a function on an open subset of that is complex differentiable at every point. It extends the notion of a holomorphic function of one variable. Such a function is holomorphic in each variable separately and infinitely differentiable. It is locally the sum of a convergent power series in several variables. These functions are the main objects of the theory of several complex variables.
Where this leads
62 pages build on this one.
- Riemann domain over C^n
- Holomorphic map
- Cartan–Thullen theorem
- Ohsawa–Takegoshi extension theorem
- Germ of a holomorphic function
- Demailly's regularization of currents
- Remmert–Stein theorem
- Isolated hypersurface singularity
- First Cousin problem
- Zeros of holomorphic functions are not isolated
- Cartan's coherence theorem for ideal sheaves
- Riemann extension theorem in several variables
- Oka's solution of the Levi problem
- Chow's theorem
- Domain of holomorphy
- Oka–Weil theorem
- Identity theorem in several variables
- Bishop's theorem on analytic sets
- Siegel modular form
- Hartogs–Bochner extension theorem
- Maximum modulus principle in several variables
- Thom–Sebastiani theorem
- Germ of an analytic set
- Peak function
- Edge-of-the-wedge theorem
- Poincaré–Lelong formula
- Dimension of an analytic set
- Hartogs triangle
- Second Cousin problem
- Domain of convergence of a power series
- Monodromy theorem for singularities
- Hartogs's theorem on separate holomorphicity
- Stein manifold
- Cartan's uniqueness theorem
- Irreducible analytic germ
- Cauchy integral formula on a polydisc
- Hartogs's extension theorem
- Hartogs figure
- Szegő kernel
- Regular and singular points of an analytic set
- Local parametrization theorem
- Milnor fibration theorem
- Kontinuitätssatz
- Oka manifold
- Envelope of holomorphy
- Oka coherence theorem
- Runge domain
- Grauert's solution of the Levi problem for manifolds
- Vanishing cycle
- Cauchy–Riemann equations in several variables
- Holomorphically convex hull
- Poincaré problem
- Complex analytic space
- Cauchy–Fantappiè formula
- Analytic continuation in several variables
- Meromorphic function of several variables
- Lewy extension theorem
- Complex analytic variety
- Milnor fibre is a bouquet of spheres
- Automorphism group of a bounded domain is a Lie group
- Bochner–Martinelli formula
- Bochner's tube theorem
Further reading
About this page
| Tag | 507C34W0ZT |
|---|---|
| URI | m2://507C34W0ZT |
| Kind | concept |
| Writing | W0 stub |
| MSC | 32A10 |
| Specializes | Several complex variables |
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Used by
- Riemann domain over C^n
- Holomorphic map
- Cartan–Thullen theorem
- Ohsawa–Takegoshi extension theorem
- Germ of a holomorphic function
- Isolated hypersurface singularity
- First Cousin problem
- Zeros of holomorphic functions are not isolated
- Cartan's coherence theorem for ideal sheaves
- Riemann extension theorem in several variables
- Domain of holomorphy
- Oka–Weil theorem
- Identity theorem in several variables
- Siegel modular form
- Hartogs–Bochner extension theorem
- Maximum modulus principle in several variables
- Peak function
- Edge-of-the-wedge theorem
- Poincaré–Lelong formula
- Second Cousin problem
- Domain of convergence of a power series
- Hartogs's theorem on separate holomorphicity
- Stein manifold
- Cartan's uniqueness theorem
- Cauchy integral formula on a polydisc
- Hartogs's extension theorem
- Hartogs figure
- Szegő kernel
- Milnor fibration theorem
- Envelope of holomorphy
- Oka coherence theorem
- Runge domain
- Vanishing cycle
- Cauchy–Riemann equations in several variables
- Holomorphically convex hull
- Poincaré problem
- Complex analytic space
- Cauchy–Fantappiè formula
- Analytic continuation in several variables
- Meromorphic function of several variables
- Lewy extension theorem
- Complex analytic variety
- Bochner–Martinelli formula
- Bochner's tube theorem
Definitions
Results about this
- Ohsawa–Takegoshi extension theorem
- Zeros of holomorphic functions are not isolated
- Riemann extension theorem in several variables
- Oka–Weil theorem
- Identity theorem in several variables
- Hartogs–Bochner extension theorem
- Maximum modulus principle in several variables
- Edge-of-the-wedge theorem
- Hartogs's theorem on separate holomorphicity
- Cauchy integral formula on a polydisc
- Hartogs's extension theorem
- Cauchy–Fantappiè formula
- Bochner–Martinelli formula
- Bochner's tube theorem