2022/07/02 by David R. Stoutemyer, Stoutemyer, David R.
Engineering · Medicine · #33C10 #41 #65 #FOS: Mathematics #G.1 #G.4 #General Mathematics (math.GM) #I.1 #Sports Dynamics and Biomechanics #Sports Performance and Training
paper · pdf · doi:10.48550/arxiv.2207.00707
openalex publication_date 2022/07/02 · openalex created_date 2022/07/07 · openalex updated_date 2026/07/28
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 kn(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 yn(x) or jn(x). Such equations include cos(x)/x=constant, for which the solution inverse1(y0)(-constant) is Dottie's 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 in(x). These discoveries arose from the AskConstants program surprisingly proposing the explicit exact closed form inverse1(y0)(-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.