Complex Analytic Warps

📐 Definition

An analytic warp is a holomorphic map $w : \mathbb{C} \to \mathbb{C}$ with complex derivative $w'(z)$. In Cartesian coordinates $z = x + iy$, $w$ satisfies the Cauchy–Riemann equations.

Domain and Codomain

Defined on open subsets of $\mathbb{C}$ where $w$ is holomorphic. Output lies in $\mathbb{C}$; conformal wherever $w'(z) \neq 0$.

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⚙️ Key Properties

Locally angle-preserving and orientation-preserving when $w'(z) \neq 0$. Jacobian determinant equals $|w'(z)|^{2}$. Composition of analytic warps remains analytic.

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Affine maps $w(z) = az + b$ are globally conformal. Möbius transforms $w(z) = \frac{az+b}{cz+d}$ are analytic except at poles. Points where $w'(z)=0$ introduce critical points and lose conformality.

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🎯 Special Cases and Limits

• Affine maps are globally conformal.
• Möbius transforms are conformal except at poles.
• Critical points ($w'(z)=0$) lose conformality.

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🔗 Related Functions

Warp gradients reduce to $w'$ and its complex conjugate in this setting; phase-space warps specialize to symplectic maps in real coordinates.

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Usage in Oakfield

Oakfield’s “complex analytic warps” are currently about complex-valued fields, not conformal remapping of 2D geometry:

• Analytic warp complex modes: analytic_warp supports complex fields either component-wise (Re/Im) or in a phase-preserving polar mode (update magnitude response along the current phase direction).
• Complex special functions: Oakfield includes complex digamma/trigamma/tetragamma paths (analytic continuation) used when warp profiles require complex evaluation.
• Measurement loop compatibility: complex polar warps pair naturally with the remainder operator’s polar mode when measuring “magnitude-only” discrepancies without scrambling phase.

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Historical Foundations

📜 Conformal Mapping

Conformal maps and holomorphic functions form the backbone of complex analysis, connecting analytic structure with geometric angle preservation.

🌍 Modern Perspective

Complex analytic warps are used to remap domains and reshape fields while preserving local angles, which can simplify geometry-dependent problems.

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📚 References

• Ahlfors, Complex Analysis
• Needham, Visual Complex Analysis