The solution can be expressed also through the related Lambert W function.

(see Lambert W function and references therein for more details).

Lambert W function, also known as the omega function.

The transcendental equations above can be solved exactly in terms of the Lambert W function.

In particular, has a solution in terms of the Lambert W Function.

The Lambert W function was "re-discovered" every decade or so in specialized applications.

The analytical solutions for the energy eigenvalues are a generalization of the Lambert W function.

Generalizations of the Lambert W function include:

Closure of complex exponentials and the Lambert W functions, preprint October 2002. 36 pages.

These can be obtained to within arbitrary accuracy using computer algebra from the generalized Lambert W function (see eq.

Recently, the Lambert W function has been employed to obtain explicit reformulation of the Colebrook equation.

The Lambert W function has been recently (2013) shown to be the optimal solution for the required magnetic field of a Zeeman slower.

The quantile function can be expressed in a closed-form expressions using the Lambert W function:

The standard Lambert W function expresses exact solutions to transcendental algebraic equations (in x) of the form:

More recently, as explained further in the quantum-mechanical version, analytical solutions to the eigenenergies have been obtained: these are a generalization of the Lambert W function.

The Lambert W function was employed in the field of epitaxial film growth for the determination of the critical dislocation onset film thickness.

Two standard physics problems are solved in terms of the Lambert W function, to show the applicability of this recently defined function to physics.

Note that although the generalized Lambert W function eigenvalue solutions supersede these asymptotic expansions, in practice, they are most useful near the bond length.

Monographs on the Lambert W function, its numerical approximation and generalizations for W-like inverses of transcendental forms with repeated exponential towers.

W(a) if a is algebraic and nonzero, for any branch of the Lambert W Function (by the Lindemann-Weierstrass theorem).

Except for z along the branch cut (where the integral does not converge), the principal branch of the Lambert W function can be computed by the following integral:

In mathematics, the Wright omega function or Wright function, denoted ω, is defined in terms of the Lambert W function as:

The Lambert W function was employed in the field of Chemical Engineering for modelling the porous electrode film thickness in a glassy carbon based supercapacitor for electrochemical energy storage.

In biochemistry, and in particular enzyme kinetics, a closed-form solution for the time course kinetics analysis of Michaelis-Menten kinetics is described in terms of the Lambert W function.

The analytical results are expressed in terms of the Lambert W function, and from those a number of mathematical and numerical results obtained previously with the implicit relation for the range are obtained more easily.