## Lambert's W-Function

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

 (6)

where
 (7) (8)

(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)