The Lambert 
-function,
also called the omega function, is the inverse function
of
 |
(1)
|
The plot above shows the function along the real axis. The principal value of the Lambert 
-function is implemented in the Wolfram
Language as ProductLog[z].
Different branches of the function are available in the Wolfram
Language as ProductLog[k,
z], where 
is any integer and 
corresponds to the principal value. Although undocumented, LambertW[k,
z] autoevaluates to ProductLog[k,
z] in the Wolfram Language.
Lambert (1758) considered the solution to
 |
(2)
|
now known as Lambert's transcendental equation. Euler heard about Lambert's paper in 1764 when Lambert moved from Zurich
to Berlin. After some private disputes about the priorities of some related series
expansions in 1770/1771, Euler (1783) wrote a paper about Lambert's
transcendental equation in which he introduced a special case which reduces to
,
which is nearly the definition of 
, although Euler proposed defining a function more like
.
Euler considered series solutions in this paper and, in the first paragraph, explicitly
quotes Lambert as the one who first considered this equation.
Eisenstein (1844) considered the series of the infinite power tower
 |
(3)
|
which can be expressed in closed form as
 |
(4)
|
Pólya and Szegö (1925) were the first to use the symbol 
for the Lambert function.
Banwell and Jayakumar (2000) showed that a 
-function describes the relation between voltage, current,
and resistance in a diode, and Packel and Yuen (2004) applied the 
-function to a ballistic projectile in the presence of air
resistance. Other applications have been discovered in statistical mechanics, quantum
chemistry, combinatorics, enzyme kinetics, the physiology of vision, the engineering
of thin films, hydrology, and the analysis of algorithms (Hayes 2005).
 |
The Lambert 
-function
is illustrated above in the complex plane.
The real (left) and imaginary (right) parts of the analytic continuation of 
over the complex plane are illustrated
above (M. Trott, pers. comm.).
is real for 
.
It has the special values
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  |  |
(8)
|
(OEIS A030178) is called the omega
constant and can be considered a sort of "golden
ratio" of exponentials since
![exp[-W(1)]=W(1),](/images/equations/LambertW-Function/NumberedEquation5.svg) |
(9)
|
giving
![ln[1/(W(1))]=W(1).](/images/equations/LambertW-Function/NumberedEquation6.svg) |
(10)
|
The Lambert 
-function
obeys the identity
 |
(11)
|
(pers. comm. from R. Corless to O. Marichev, Sep. 29, 2015).
 |
The function 
has a very complicated structure in the complex plane, but is simply equal to 1 for
![R[z]>=1](/images/equations/LambertW-Function/Inline30.svg)
and a slightly extended region above and below the real axis.
The Lambert 
-function
has the series expansion
 |  |  |
(12)
|
 |  |  |
(13)
|
The Lagrange inversion theorem gives the equivalent series expansion
 |
(14)
|
where 
is a factorial. However, this series oscillates between
ever larger positive and negative
values for real 
, and so cannot be used for practical numerical computation.
An asymptotic formula which yields reasonably accurate results for 
is
 |  |  |
(15)
|
 |  | ![L_1-L_2+(L_2)/(L_1)+(L_2(-2+L_2))/(2L_1^2)+(L_2(6-9L_2+2L_2^2))/(6L_1^3)+(L_2(-12+36L_2-22L_2^2+3L_2^3))/(12L_1^4)+(L_2(60-300L_2+350L_2^2-125L_2^3+12L_2^4))/(60L_1^5)+O[((L_2)/(L_1))^6],](/images/equations/LambertW-Function/Inline46.svg) |
(16)
|
where
 |  |  |
(17)
|
 |  |  |
(18)
|
(Corless et al. 1996), correcting a typographical error in de Bruijn (1981). Another expansion due to Gosper (pers. comm., July 22, 1996) is the double series
![W(x)=a+sum_(n=0)^infty{sum_(k=0)^n(S_1(n,k))/([ln(x/a)-a]^(k-1)(n-k+1)!)}[1-(ln(x/a))/a]^n,](/images/equations/LambertW-Function/NumberedEquation9.svg) |
(19)
|
where 
is a nonnegative Stirling number
of the first kind and 
is a first approximation which can be used to select between
branches. The Lambert 
-function is two-valued for 
. For 
, the function is denoted 
or simply 
, and this is called the principal
branch. For 
,
the function is denoted 
. The derivative of
is
 |  | ![1/([1+W(x)]exp[W(x)])](/images/equations/LambertW-Function/Inline65.svg) |
(20)
|
 |  | ![(W(x))/(x[1+W(x)])](/images/equations/LambertW-Function/Inline68.svg) |
(21)
|
for 
.
For the principal branch when 
,
![ln[W(z)]=lnz-W(z).](/images/equations/LambertW-Function/NumberedEquation10.svg) |
(22)
|
The 
th
derivatives of the Lambert 
-function are given by
![W^((n))(z)=(W^(n-1)(z))/(z^n[1+W(z)]^(2n-1))sum_(k=1)^na_(kn)W^k(z),](/images/equations/LambertW-Function/NumberedEquation11.svg) |
(23)
|
where 
is the number triangle
 |
(24)
|
(OEIS A042977). This has exponential generating function
 |  |  |
(25)
|
 |  |  |
(26)
|
Function." ACM Trans. Math. Software 21,
161-171, 1995.Borwein, J. M. and Corless, R. M. "Emerging
Tools for Experimental Mathematics." Amer. Math. Monthly 106,
899-909, 1999.Briggs, K. "
-ology, or, Some Exactly Solvable Growth Models." 
function." 
Function." (Ed. W. W. Küchlin). New York:
ACM, pp. 197-204, 1997. 
Function." Adv. Comput. Math. 5, 329-359,
1996.Corless, R. M.; Gonnet, G. H.; Hare, D. E. G.;
and Jeffrey, D. J. "Lambert's 
Function in Maple." Maple Technical Newsletter 9,
12-22, Spring 1993.Corless, R. M. and Jeffrey, D. J. "The
Wright 
Function." In 
Function." In Proceedings of
the 1997 International Symposium on Symbolic and Algebraic Computation, Maui, Hawaii.
New York: ACM Press, pp. 197-204, 1997.de Bruijn, N. G. 
." J. reine angew. Math. 28,
49-52, 1844.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.Fritsch,
F. N.; Shafer, R. E.; and Crowley, W. P. "Algorithm 443: Solution
of the Transcendental Equation 
." Comm. ACM 16, 123-124, 1973.Gosper,
R. W. Jr. "The Solutions of 
and 
." ACM SIGSAM Bull. 32, 8-10, 1998.Gosper,
R. W. "Re: 2nd Order Eulerians." math-fun@cs.arizona.edu
posting, July 22, 1996.Gray, J. J. and Tilling, L. "Johann
Heinrich Lambert, Mathematician and Scientist 1728-1777." Historia Math. 5,
13-41, 1978.Hayes, B. "Why 
?" Amer. Sci. 93, 104-108, 2005.Jeffrey,
D. J.; Hare, D. E. G.; and Corless, R. M. "Unwinding the
Branches of the Lambert 
Function." Math. Scientist 21, 1-7, 1996.Jeffrey,
D. J.; Hare, D. E. G.; and Corless, R. M. "Exact Rational
Solutions of a Transcendental Equation." C. R. Math. Acad. Sci. Canada 20,
71-76, 1998.Jeffrey, D. J.; Corless, R. M.; Hare, D. E. G.;
and Knuth, D. E. "Sur l'inversion de 
au moyen des nombres de Stirling associes." Comptes
Rendus Acad. Sci. Paris 320, 1449-1452, 1995.Kalman, D. "A
Generalized Logarithm for Exponential-Linear Equations." College Math. J. 32,
2-14, 2001.Lambert, J. H. "Observations variae in Mathesin
Puram." Acta Helvitica, physico-mathematico-anatomico-botanico-medica 3,
128-168, 1758.ORCCA. "The Lambert W Function." 
Function." College Math. J. 35, 337-350,
2004.Pólya, G. and Szegö, G. Aufgaben und Lehrsätze
der Analysis. Berlin, 1925. Reprinted as 
Function to Physics." Canad. J. Phys. 78, 823-831, 2000.Wright,
E. M. "Solution of the Equation 
." Bull. Amer. Math. Soc. 65, 89-93,
1959.