The Lambert 
-function,
also called the omega function, is the multivalued
function obtained by inverting
 |
(1)
|
Unless a branch is specified, 
denotes the principal branch 
. 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. The synonym 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, for real 
and 
, can be expressed in closed
form as
 |
(4)
|
At 
,
the removable value is 
.
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 part (left) and imaginary part (right) 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 exponential functions
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)
|
For real 
with 
and 
, the principal
branch obeys the identity
 |
(11)
|
(R. Corless, pers. comm. to O. Marichev, Sep. 29, 2015).
The function 
has a complicated structure in the complex plane.
It equals 1 for real 
with 
, and its removable value at 
is also 1.
The Lambert 
-function
has the series expansion
 |  |  |
(12)
| ||
 |  |  |  |  |
(13)
|
The Lagrange inversion theorem gives the equivalent series expansion
 |
(14)
|
where 
is a factorial. This Taylor
series converges absolutely for 
and diverges for 
(Corless et al. 1996).
Along the positive real axis, an asymptotic expansion for the principal branch of the
Lambert 
-function
as 
is
 |  |  |
(15)
| |||
 |  |  |  |  | ![(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/Inline63.svg) |
(16)
|
where 
,
,
and
![c_(km)=((-1)^k)/((m+1)!)[k+m+1; k+1].](/images/equations/LambertW-Function/NumberedEquation9.svg) |
(17)
|
Here ![[a; b]](/images/equations/LambertW-Function/Inline66.svg)
denotes an unsigned Stirling number
of the first kind. The accuracy of a finite approximation depends on 
and on the number and grouping of terms retained. Truncating
the expansion in powers of 
is different from applying independent cutoffs to 
and 
. This expansion (Corless et al. 1996) corrects 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/NumberedEquation10.svg) |
(18)
|
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 has two distinct real
branches for 
,
which coincide at 
.
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/Inline84.svg) |
(19)
|
 |  | ![(W(x))/(x[1+W(x)])](/images/equations/LambertW-Function/Inline87.svg) |
(20)
|
away from 
,
with the second form also requiring 
. On the principal branch,
.
For the principal branch when 
,
![ln[W(z)]=lnz-W(z).](/images/equations/LambertW-Function/NumberedEquation11.svg) |
(21)
|
For 
,
,
and 
,
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/NumberedEquation12.svg) |
(22)
|
where 
is the number triangle
 |
(23)
|
(OEIS A042977). For real 
, this has exponential
generating function in 
about 
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(24)
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(25)
|
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