The inverse of the function

(1) 
also called the Omega Function. The function is implemented as the Mathematica
(Wolfram
Research, Champaign, IL) function ProductLog[z]. is called the Omega Constant and can be considered a
sort of ``Golden Ratio'' of exponentials since

(2) 
giving

(3) 
Lambert's Function has the series expansion

(4) 
The Lagrange Inversion Theorem gives the equivalent series expansion

(5) 
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
where
(Corless et al.), correcting a typographical error in de Bruijn (1961). Another expansion due to Gosper is the
Double Sum

(9) 
where is a nonnegative Stirling Number of the First Kind and is a first approximation which can be
used to select between branches. Lambert's function is twovalued for . For , the function is
denoted or simply , and this is called the principal branch. For , the function is denoted
. The Derivative of is

(10) 
for . For the principal branch when ,

(11) 
See also Iterated Exponential Constants, Omega Constant
References
de Bruijn, N. G. Asymptotic Methods in Analysis. Amsterdam, Netherlands: NorthHolland, pp. 2728, 1961.
© 19969 Eric W. Weisstein
19990526