# §4.13 Lambert -Function

The Lambert -function is the solution of the equation

4.13.1

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.

Figure 4.13.1: Branches and of the Lambert -function. and denote the points and , respectively, on the -axis.

For the foregoing results and further information see Borwein and Corless (1999), Corless et al. (1996), de Bruijn (1961, pp. 25–28), Olver (1997b, pp. 12–13), and Siewert and Burniston (1973).

For integral representations of all branches of the Lambert -function see Kheyfits (2004).