I want to solve this equation for $x$, but I'm stuck. Is it possible to solve this equation analytically?
$$-(\beta+1) + \lambda \beta x^{-\beta} (1 + e^{-\lambda x^{-\beta}} \ln \alpha) = 0$$ where $\alpha > 0$, $\alpha \neq 1$, $\lambda > 0$, $\beta > 0$.
Thank you.
On
The equation is an equation of Elementary functions. It can be transformed to the form that is given in the answer of Bobby Laspy. Applying Ritt's theorem on the existence of elementary inverses of elementary functions ([Ritt 1925]) and Schanuel's conjecture yields that the elementary functions $z\to z+pze^{-z}-q$ over non-discrete domains cannot have elementary partial inverses over non-discrete domains. Therefore the equation cannot be solved by simply rearranging it by applying only finite numbers of elementary functions (elementary operations) readable from the equation. It is not known if the equation has solutions that are elementary numbers. See [Chow 1999] for elementary numbers as solutions of equations.
As the answer of Bobby Laspy shows, the equation is not in a form which can be solved in terms of Lambert W. But the equation can be solved in terms of Generalized Lambert W:
$$z=-W(^{\pm 0}_{-q};-\frac{1}{p})=W(^q_0;-p)$$
$-$ see [Mezö 2017], [Mezö/Baricz 2017], [Castle 2018].
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[Chow 1999] Chow, T.: What is a closed-form number. Am. Math. Monthly 106 (1999) (5) 440-448
[Mezö 2017] Mezö, I.: On the structure of the solution set of a generalized Euler-Lambert equation. J. Math. Anal. Appl. 455 (2017) (1) 538-553
[Mezö/Baricz 2017] Mezö, I.; Baricz, A.: On the generalization of the Lambert W function. Transact. Amer. Math. Soc. 369 (2017) (11) 7917–7934 (On the generalization of the Lambert W function with applications in theoretical physics. 2015)
[Castle 2018] Castle, P.: Taylor series for generalized Lambert W functions. 2018
With changes of variables, the equation can be recast as $$z (1 + pe^{-z}) =q.$$
Compared to Lambert's equation, there is an extra term, and no analytical solution is possible.