Analytic Warp Map

📐 Definition

Let $(a_n)_{n \ge 1} \subset \mathbb{C}$ with radius of convergence $R > 0$ . The analytic warp is

w(z) = z + \sum_{n=1}^{\infty} a_n z^{n+1}, \quad |z| < R

Domain and Codomain

Domain: $z \in \mathbb{C}$ with $|z| < R$ . Codomain: $w(z) \in \mathbb{C}$ ; if $w'(z) \ne 0$ the map is locally conformal and orientation-preserving.

⚙️ Key Properties

Derivative

w'(z) = 1 + \sum_{n=1}^{\infty} (n+1)a_n z^{n}

remains analytic inside $|z|<R$ . Composition of two analytic warps is analytic where radii overlap.

Truncating to $N$ terms yields a polynomial warp $w_N(z)$ ; setting $a_1 \ne 0$ alone gives a quadratic perturbation $w(z) = z + a_1 z^{2}$ .

🎯 Special Cases and Limits

Truncation yields a polynomial warp.
Small coefficients $a_n$ produce small conformal perturbations near the origin until critical points appear where $w'(z)=0$ .

Power-law warps provide specific monomial perturbations; Fourier transforms of warped fields encode phase-modulated spectra.

Usage in Oakfield

Oakfield implements analytic “warp maps” as a first-class operator that deforms field values step-by-step:

analytic_warp operator applies profile-specific analytic responses (digamma, trigamma, power, tanh, hyperexp, q-hyperexp) by sampling profile gradients and integrating a response term.
Continuity + safety: warp sampling is guarded by per-operator continuity modes (none|strict|clamped|limited) with clamp ranges and tolerance windows to handle singularities/poles.
Complex handling: complex fields can be processed component-wise or in a polar mode (compute a scalar response at $|z|$ and apply along $z/|z|$ ).
Runtime/UI integration: the warp profile is exposed via schemas and visual bridge mappings, so the UI and Lua can configure and inspect it consistently.

Historical Foundations

📜 Power Series and Holomorphic Maps

Power series expansions provide canonical local representations of holomorphic functions, making them a natural way to build analytic warps with controllable behavior near a point.

🌍 Modern Perspective

Analytic warps are used when smoothness and conformal structure matter, especially in 2D complex-coordinate formulations.

📚 References

Ahlfors, Complex Analysis
Conway, Functions of One Complex Variable