Hyperexponential Function

📐 Definition

For $\lambda > 0$ , $\varepsilon > 0$ , and integer $K \ge 1$ ,

\phi(\lambda,\varepsilon;K) = \sum_{k=0}^{K-1} \frac{\lambda}{\lambda + \varepsilon + k} = \lambda\big[\psi(\lambda+\varepsilon+K) - \psi(\lambda+\varepsilon)\big]

where $\psi$ is the digamma function.

Domain and Codomain

Defined for positive $\lambda$ , $\varepsilon$ and integer $K \ge 1$ (extendable to complex parameters avoiding poles of $\psi$ ). Output is real for real inputs.

⚙️ Key Properties

Monotone increasing in $\lambda$ and $K$ ; decreasing in $\varepsilon$ . Smooth in parameters away from poles of the digamma. Bounded above by $K$ .

🎯 Special Cases and Limits

As $K \to \infty$ with fixed $\lambda,\varepsilon$ , the sum diverges logarithmically; as $\lambda \to 0^{+}$ , $\phi \to 0$ ; for large $\lambda$ ,

\phi(\lambda,\varepsilon;K) = K - \frac{K\varepsilon + \tfrac{K(K-1)}{2}}{\lambda} + \mathcal{O}\!\left(\frac{K^3}{\lambda^2}\right)

Hyperexponential derivatives differentiate $\phi$ with respect to parameters; warp gradients propagate $\phi$ through spatial warps.

Usage in Oakfield

Oakfield uses hyperexponential φ primarily as a warp profile helper and exposes it to scripting:

Analytic warp profile: analytic_warp includes a HYPEREXP profile that evaluates hyperexponential φ and its derivative to drive warp responses.
Stable evaluation: φ is implemented via digamma/trigamma differences (avoids explicit inner sums) with safe-domain handling and fallback hooks.
Scripting API: φ and φ′ (including complex variants) are exposed through the Lua API for experimentation and parameter sweeps.

Historical Foundations

📜 Oakfield Construction

This hyperexponential warp is an Oakfield-specific construction expressed in terms of classical special functions (digamma and related polygamma derivatives), chosen for analytic tractability and stable parameter differentiation.

📚 References

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