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 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
|
(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: North-Holland, pp. 27-28, 1961.
© 1996-9 Eric W. Weisstein
1999-05-26