Hyperexponential Derivatives

📐 Definition

With $\phi(\lambda,\varepsilon;K) = \lambda[\psi(\lambda+\varepsilon+K) - \psi(\lambda+\varepsilon)]$, derivatives are

$$\partial_{\lambda}\phi = \psi(\lambda+\varepsilon+K) - \psi(\lambda+\varepsilon) + \lambda\big[\psi_{1}(\lambda+\varepsilon+K) - \psi_{1}(\lambda+\varepsilon)\big]$$

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$$\partial_{\varepsilon}\phi = \lambda\big[\psi_{1}(\lambda+\varepsilon+K) - \psi_{1}(\lambda+\varepsilon)\big]$$

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where $\psi_{1}$ is the trigamma function.

Domain and Codomain

Defined for $\lambda,\varepsilon > 0$ and integer $K \ge 1$ (extendable to complex parameters away from polygamma poles). Outputs are real for real inputs.

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⚙️ Key Properties

Smooth in parameters; $\partial_{\lambda}\phi > 0$ and $\partial_{\varepsilon}\phi < 0$ for positive arguments since $\psi_{1}>0$ on $(0,\infty)$. Higher derivatives involve higher-order polygammas.

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🎯 Special Cases and Limits

As $\lambda \to 0^{+}$, $\partial_{\lambda}\phi \to \psi(\varepsilon+K) - \psi(\varepsilon)$. For large $\lambda$ (with fixed $K$), one has $\partial_{\lambda}\phi=\mathcal{O}(K^{2}/\lambda^{2})$ and $\partial_{\varepsilon}\phi=\mathcal{O}(K/\lambda)$.

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🔗 Related Functions

Polygammas provide the necessary derivatives; warp gradients propagate these sensitivities through spatial mappings.

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Usage in Oakfield

Oakfield uses hyperexponential derivatives to support warp dynamics and parameter sensitivity:

• Warp gradients: core/operators/analytic_warp.c calls sim_hyperexp_phi_deriv_safe() (and sim_qhyperexp_phi_deriv() for QHYPEREXP) from core/math/hyperexponential.* to compute a stable local gradient.
• Complex support exists via complex-valued derivative helpers (e.g. sim_hyperexp_phi_deriv_complex_opt) for complex-field experimentation.
• Error handling: the “safe” APIs return diagnostic reports and route invalid inputs through fallback handlers (important near poles and branch issues).

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Historical Foundations

📜 Oakfield Construction

These derivatives are an Oakfield-specific companion to the hyperexponential warp, written in terms of classical polygamma functions to enable stable analytic gradients.

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📚 References

• NIST Digital Library of Mathematical Functions (DLMF), §5.15