Lambert's Transcendental Equation

An equation proposed by Lambert (1758) and studied by Euler in 1779 (Euler 1921).

$$x^\alpha-x^\beta=(\alpha-\beta)vx^{\alpha+\beta}.$$

When $\alpha\to\beta$, the equation becomes

$$\ln x=vx^\beta,$$

which has the solution

$$x=\mathop{\rm exp}\nolimits \left[{-{W(-\beta v)\over\beta}}\right],$$

where $W(x)$ is Lambert's W-Function.

References

de Bruijn, N. G. Asymptotic Methods in Analysis. Amsterdam, Netherlands: North-Holland, pp. 27-28, 1961.

Euler, L. ``De Serie Lambertina Plurismique Eius Insignibus Proprietatibus.'' Leonhardi Euleri Opera Omnia, Ser. 1. Opera Mathematica, Bd. 6, 1921.

Lambert, J. H. ``Observations variae in Mathesin Puram.'' Acta Helvitica, physico-mathematico-anatomico-botanico-medica 3, 128-168, 1758.