The Lambert -function is the solution of the equation
On the -interval there is one real solution, and it is nonnegative and increasing. On the -interval there are two real solutions, one increasing and the other decreasing. We call the solution for which the principal branch and denote it by . The other solution is denoted by . See Figure 4.13.1.
where for , for ,
where is defined in §5.11(i).
where . As
For integral representations of all branches of the Lambert -function see Kheyfits (2004).