**Name:**

LambertW0 - principal branch of the Lambert-W function

**Synopsis:**

double LambertW0 -> double

**Examples:**

The Lambert-W function has applications in many areas as described in [1] and [3].

For example, the solution of

exp(s) = 1 + a*s

with respect to s can be written in closed form as

s=1/a * (-aW(-exp(-1/a)/a) -1 )

Version: 090818

**Description:**

The Lambert-W function [1] is the inverse function of x=W*exp(W). For real values of

x and W, the function W(x) is defined on [-1/e,\infty). On the interval [-1/e,0)

it is double valued. The two branches coincide at W(-1/e)=-1. The so called

principal branch LambertW0 continuously grows (W>=-1) and crosses the origin (0,0).

The non-principal branch LambertWm1 is defined on [-1/e,0) and declines to -\infty for

growing x.

NEST uses the GSL [2] implementations of LambertW0 and LambertWm1 if available and

falls back to to the iterative scheme LambertW suggested in [1] if not.

The GSL interfaces for LambertW0 and LambertWm1 are in the SpecialFunctionsModule

of SLI.

**References:**

[1] Corless, R. M., Gonnet, G. H., Hare, D. E. G., Jeffrey, D. J., & Knuth, D. E.

(1996). On the lambert w function. Advances in Computational Mathematics 5,

329--359.

[2] Galassi, M., Davies, J., Theiler, J., Gough, B., Jungman, G., Booth, M.,

& Rossi, F. (2006). GNU Scientific Library Reference Manual (2nd Ed.).

Network Theory Limited.

[3] Wikipedia (2009). Lambert W function ---wikipedia, the free encyclopedia.

**Author:**

Diesmann

**SeeAlso:**

**Source:**

/home/nest/work/nest-2.14.0/lib/sli/mathematica.sli