# LambertWm1

Name:
LambertWm1 - non-principal branch of the Lambert-W function
Synopsis:
   double LambertWm1 -> double

Examples:
      The Lambert-W function has applications in many areas as described in [1] and [3].    For example, the problem of finding the location of the maximum of the postsynaptic    potential generated by an alpha-function shaped current can be reduced to the    equation    exp(s) = 1 + a*s .    Here, s is the location of the maximum in scaled time s=b*t, where    b= 1/tau_alpha	- 1/tau_m and a is the ratio of the time constants tau_m/tau_alpha.    In terms of the Lambda_W function the solution is    s=1/a * (-aW(-exp(-1/a)/a) -1 ) .    The solution is guaranteed to live on the non-principal branch because the scaled time    needs to be positive. This requires W < -1/a which is trivially fulfilled for the    non-principal branch and there is no solution on the principal branch.          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