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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 $z\mapsto z\,e^z$ which plays an important role for solving equations containing exponentials or logarithms. Therefore this function is available in computer algebra systems like Maple ({\tt LambertW}) and Mathematica ({\tt 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.

30D05Functional equations in the complex domain, iteration and composition of analytic functions
30D10Representations of entire functions by series and integrals
30D20General theory of entire functions
Maple; Mathematica
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