Quadrature Domain Solver

Domain type

Classical QD

Bounded Unbounded

Log-weighted QD (LQD)

Bounded Bounded singular

Unbounded Unbounded singular

Log-weighted QD: ∫_Ω f(w)/|w|² dA = ∮_∂Ω f·h dw. Non-singular only: requires 0 ∉ Ω̄, so w₀ = φ(0) must be nonzero.

Singular log-weighted QD: 0 ∈ Ω; test functions f ∈ L¹_a(Ω; ρ₀) automatically vanish at 0. Riemann map factors as φ(z) = γ · b_z₀(z) · exp(r#(z)). The pole of h at the origin contributes residue q (complex); q = 0 is the degenerate case with no log-charge.

Unbounded non-singular log-weighted QD: 0 ∉ Ω̄, ∞ ∈ Ω; test functions f ∈ L¹_a(Ω; ρ₀) vanish at ∞. Riemann map: φ(z) = c·z · exp(r#(z) − r#(∞)) on 𝔻*, with conformal radius c = φ′(∞). The − r#(∞) subtraction pins the leading coefficient at ∞ to exactly c.

Unbounded singular log-weighted QD: 0 ∈ Ω AND ∞ ∈ Ω; test functions f vanish at BOTH 0 and ∞ (e.g. w/(w−b)^k for k ≥ 2, b ∈ K). Riemann map: φ(z) = c·|z₀|·z·b_z₀(z) · exp(r#(z) − r#(∞)) with z₀ ∈ 𝔻* the preimage of 0, and q (complex) the residue of h at the origin. q = 0 is the degenerate case with no log-charge.

Quadrature function h(w)

h(w) = Σ_j Σ_{s=1..m_j} C_j,s / (w − a_j)^s. Enter complex values as e.g. 1+2i.

Preset:

Riemann map center φ(0)

Centroid of poles

Manual:

Solver settings

Boundary samples:

Aggressiveness:

Auto-fit view on solve

Vector field:

Status

Idle

Algebraic Quadrature Domains

AQDs are Ω with weight ρ = |α|² where α = R′ for some rational primitive R. The inverse problem reduces to solving for the rational function R∘φ on 𝔻̄ via (R∘φ)(z) = R(w₀) + r#(z) (Thm 6.4.1), then numerically inverting R to recover φ.

Stage 0: tab scaffolding only. R-input UI and solver land in Stages 1–2.