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On some applications of the generalized hyper-Lambert functions. (English) Zbl 1151.30019
The Lambert’s $W$ function is defined as the inverse function of $z↦z\phantom{\rule{0.166667em}{0ex}}{e}^{z}$ which plays an important role for solving equations containing exponentials or logarithms. Therefore this function is available in computer algebra systems like Maple (LambertW) and Mathematica (ProductLog). In the current paper the author considers a hierarchy of hyper-Lambert functions, and shows how these functions can be used to solve algebraically an infinite class of unsolvable transcendental equations, an infinite class of unsolvable differential equations, the equation ${z}^{z{\cdots }^{z}}=y,$ and Kepler’s equation.
##### MSC:
 30D05 Functional equations in the complex domain, iteration and composition of analytic functions 30D10 Representations of entire functions by series and integrals 30D20 General theory of entire functions
Maple