论文标题
逆球形贝塞尔函数概括了兰伯特W，并求解了包含三角或双曲线表的相似方程
Inverse spherical Bessel functions generalize Lambert W and solve similar equations containing trigonometric or hyperbolic subexpressions or their inverses
论文作者
论文摘要
可变$ x $中的严格整数laurent多项式为0或一个或多个术语的总和，其整数系数时间$ x $升级给负整数指数。可以转换为某些此类多项式时间$ \ exp（-x）= \ mathit {content} $的方程式可以通过第二种$ k_ {n} $ k_ {n}（x）$ $ n $的修改的球形贝塞尔函数的倒置可以解决。 $ \ cos（x）$或这样的多项式时间$ \ sin（x）$或其总和$ = \ mathit {constant} $可以通过球形贝塞尔函数的倒置$ y_ {n}（x）（x）$或$ j_ {n}（x）$来解决。这样的方程包括$ \ cos（x）/x = \ mathit {constant} $，在其中，解决方案$ \ m athrm {inverse} _ {1}（y_ _ {0}）（\ sathit {-constant}）$是dottie的数量，当$ \ m artiit {constant} = 1 $时，订阅率是订阅数字。可以转换为某些严格整数laurent多项式时间$ \ sinh（x）$以及可能加上这样的多项式时间$ \ cosh（x）$的方程式可以通过第一个$ i_ {n}（x）$的经过修改的球形贝塞尔函数来解决。 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关于如何实施这些球形贝塞尔函数及其多分支的真实倒置。
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.
