Complex analysis in C^n for n ≥ 2, where the phenomena diverge fundamentally from the single-variable case. Hartogs 1906 — for n ≥ 2, a holomorphic function on the 'Hartogs figure' (a polydisk with a smaller polydisk removed) extends…
several-complex-variables
Polydisk D^n and its distinguished boundary T^n
The polydisk D^n = {z ∈ C^n : |zᵢ| < 1 for all i} is the product of n copies of the unit disk. Its topological boundary ∂D^n = {z : max…
Hartogs extension (codim-2 singularities auto-extend for n ≥ 2)
Hartogs 1906 (Math. Ann. 62). For n ≥ 2, a holomorphic function on the Hartogs figure H_{n,ε} = {z : |z₁| < 1, |z_j| < ε for j ≥ 2} ∪ {z :…
Domain of holomorphy: holomorphically convex ⇔ pseudoconvex
A domain Ω ⊂ C^n is a 'domain of holomorphy' if there is a holomorphic f on Ω with no holomorphic extension past any boundary point. …
Wirtinger ∂̄-residual ≡ 0 on f(z₁,z₂) = z₁²z₂³
Exact sympy verification of the Cauchy-Riemann system in two complex variables for the monomial f(z₁, z₂) = z₁² z₂³ on C² with Wirtinger…
Bergman kernel K_D(z,w̄) = 1/(π(1-zw̄)²); K_D(0,0̄) = 1/π
The Bergman reproducing kernel of the open unit disk D = {|z| < 1} on the Hilbert space A²(D) = L²(D) ∩ Hol(D) is K_D(z, w̄) = 1/(π(1 −…
Σ z₁ʲz₂ᵏ/4^{j+k} = 16/((z₁-4)(z₂-4)); pins at (1,1), (0,2)
Double geometric series in two complex variables — the canonical Hartogs-series example that converges on the full bidisk D²(4) = {|z₁|,…