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A strict integer Laurent polynomial in a variable $x$ is 0 or a sum of one ormore terms having integer coefficients times $x$ raised to a negative integerexponent. Equations that can be transformed to certain such polynomials times$\exp(-x)=\mathit{constant}$ are exactly solvable by inverses of modifiedspherical Bessel functions of the second kind $k_{n}(x)$ where $n$ is theorder, generalizing the Lambert $W$ function when $n>0.$ Equations that can beconverted to certain such polynomials times $\cos(x)$ or such polynomials times$\sin(x)$ or a sum thereof $=\mathit{constant}$ are exactly solvable byinverses of spherical Bessel functions $y_{n}(x)$ or $j_{n}(x)$. Such equationsinclude $\cos(x)/x=\mathit{constant}$, 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 thatcan be converted to certain strict integer Laurent polynomials times $\sinh(x)$and possibly also plus such a polynomial times $\cosh(x)$ are exactly solvableby inverses of modified spherical Bessel functions of the first kind$i_{n}(x)$.  These discoveries arose from the AskConstants program surprisingly proposingthe explicit exact closed form $\mathrm{inverse}_{1}(y_{0})(-1)$ for theapproximate input 0.739085133215160642, because no explicit exact closed formrepresentation was known for Dottie's number from approximately 1865 to 2022.This article includes descriptions of how to implement these spherical Besselfunctions and their multi-branched real inverses.

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