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

Synopsis:

double LambertWm1 -&gt; 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 &lt; -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&gt;=-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., &amp; 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., 
&amp; 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:

LambertW0
LambertW
testsuite::test_lambertw

Source:

/var/www/debian/nest/nest-simulator-2.20.0/lib/sli/mathematica.sli
