Inverse spherical Bessel functions generalize Lambert W and solve similar equations containing trigonometric or hyperbolic subexpressions or their inverses

TL;DR

This paper introduces a class of inverse spherical Bessel functions that generalize Lambert W and solve equations involving trigonometric, hyperbolic functions, or their inverses, providing explicit solutions for equations previously lacking closed-form solutions.

Contribution

The paper extends the Lambert W function concept to inverse spherical Bessel functions, enabling solutions to a broader class of equations involving trigonometric and hyperbolic functions.

Findings

Inverse spherical Bessel functions generalize Lambert W.

Explicit solutions for equations involving trigonometric/hyperbolic functions.

Implementation methods for these functions and their inverses.

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 $\exp(-x)=\mathit{constant}$ 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 $=\mathit{constant}$ are exactly solvable by inverses of spherical Bessel functions $y_{n}(x)$ or $j_{n}(x)$. Such equations include $\cos(x)/x=\mathit{constant}$, for which the solution $\mathrm{inverse}_{1}(y_{0})(\mathit{-constant})$ is Dottie's number when $\mathit{constant}=1$, 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 $i_{n}(x)$. These discoveries arose from the AskConstants program surprisingly proposing the explicit exact closed form $\mathrm{inverse}_{1}(y_{0})(-1)$ for the approximate input 0.739085133215160642, because no explicit exact closed form representation was known for Dottie's number from approximately 1865 to 2022. This article includes descriptions of how to implement these spherical Bessel functions and their multi-branched real inverses.