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Lambert's Transcendental Equation

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

\begin{displaymath}
x^\alpha-x^\beta=(\alpha-\beta)vx^{\alpha+\beta}.
\end{displaymath}

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

\begin{displaymath}
\ln x=vx^\beta,
\end{displaymath}

which has the solution

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

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.




© 1996-9 Eric W. Weisstein
1999-05-26