Inverse spherical Bessel functions generalize Lambert W and solve similar equations containing trigonometric or hyperbolic subexpressions or their inverses
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Bibliographic record
Abstract
A strict integer Laurent polynomial in a variable x is 0 or a sum of one or more terms having integer coefficients times x raised to a negative integer exponent. Equations that can be transformed to certain such polynomials times exp(-x) = constant are exactly solvable by inverses of modified spherical Bessel functions of the second kind k_{n}(x) where n is the order, generalizing the Lambert W function when n > 0. Equations that can be converted to certain such polynomials times cos(x) or such polynomials times sin(x) or a sum thereof = constant are exactly solvable by inverses of spherical Bessel functions y_{n}(x) or j_{n}(x). Such equations include cos(x)/x = constant, for which the solution inverse_{1}(y_{0})(-constant) is the Dottie number when constant = 1, where subscript 1 is the branch number. Equations that can be converted to certain strict integer Laurent polynomials times sinh(x) and possibly also plus such a polynomial times cosh(x) are exactly solvable by inverses of modified spherical Bessel functions of the first kind i_{n}(x). Abstract These discoveries arose from the AskConstants program surprisingly proposing the explicit exact closed form solution inverse_{1}(y_{0})(-1) for the approximate input 0.739085133215160642, because no explicit exact closed form representation was known for this Dottie number from approximately 1865 to 2022. This article includes descriptions of how to implement such multi-branched real inverses.
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Full frame distilled prediction
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Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.002 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
Machine scores (provisional)
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it