# Command: LambertWm1

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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:

- LambertW0
- LambertW
- testsuite::test_lambertw

Source:

    /opt/conda/conda-bld/nest_1512418852064/work/nest-2.12.0/lib/sli/mathematica.sli

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