Complex Analysis
Several Complex Variables
Hartogs in 1906 discovered that a function holomorphic outside the origin in C^2 extends to all of C^2. With one observation he destroyed the analogy with one variable and created the theory of several complex variables.
- Algebraic geometry: D-modules and microlocal analysis in several variables
- Quantum physics: Fock spaces and holomorphic functions of infinitely many variables
- Signal theory: multidimensional analytic signals via C^n-holomorphy
Предварительные знания
Hartogs Phenomenon and the Levi Problem
Hartogs in 1906 proved that any holomorphic function on C^2 minus the origin extends to all of C^2. This result overturned intuition: isolated singularities that exist in one variable simply disappear in several variables. The Levi problem (solved by Oka, 1942): a pseudoconvex domain is a domain of holomorphy.
How many Cauchy-Riemann equations must a holomorphic function on C^n satisfy?
Bochner-Martinelli Kernel
Integral representation in C^n: the Bochner-Martinelli kernel generalizes the Cauchy integral. For holomorphic f on domain D with smooth boundary: f(z) = integral over dD of f(zeta) BM(zeta,z) dsigma(zeta). Unlike the Cauchy kernel in C^1, BM is not a projection operator, complicating the theory.
What does Hartogs' theorem state for C^n with n >= 2?
Key Ideas
- n CR equations: df/dz_bar_j = 0 -- holomorphy in C^n