An equation proposed by Lambert (1758) and studied by Euler in 1779.
 |
(1)
|
When 
,
the equation becomes
 |
(2)
|
which has the solution
 |  | ![exp[-(W(-betav))/beta]](/images/equations/LambertsTranscendentalEquation/Inline4.svg) |
(3)
|
 |  | ![[-(W(-vbeta))/(vbeta)]^(1/beta)](/images/equations/LambertsTranscendentalEquation/Inline7.svg) |
(4)
|
 |  | ![v[[-(W(-betav))/(betav)]^(1/beta)]^beta](/images/equations/LambertsTranscendentalEquation/Inline10.svg) |
(5)
|
where 
is the Lambert W-function.
Lambert's Transcendental Equation
See also
Lambert W-FunctionExplore with Wolfram|Alpha
References
Corless, R. M.; Gonnet, G. H.; Hare, D. E. G.; Jeffrey, D. J.; and Knuth, D. E. "On the Lambert
Function." Adv. Comput. Math. 5, 329-359,
1996.de Bruijn, N. G. Asymptotic
Methods in Analysis. New York: Dover pp. 27-28, 1981.Euler,
L. "De serie Lambertina Plurimisque eius insignibus proprietatibus." Acta
Acad. Scient. Petropol. 2, 29-51, 1783. Reprinted in Euler, L. Opera
Omnia, Series Prima, Vol. 6: Commentationes Algebraicae. Leipzig, Germany:
Teubner, pp. 350-369, 1921.Lambert, J. H. "Observations
variae in Mathesin Puram." Acta Helvitica, physico-mathematico-anatomico-botanico-medica 3,
128-168, 1758.Referenced on Wolfram|Alpha
Lambert's Transcendental EquationCite this as:
Weisstein, Eric W. "Lambert's Transcendental Equation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LambertsTranscendentalEquation.html