Lambert W function [R157].
The Lambert W function W(z) is defined as the inverse function of w * exp(w). In other words, the value of W(z) is such that z = W(z) * exp(W(z)) for any complex number z.
The Lambert W function is a multivalued function with infinitely many branches. Each branch gives a separate solution of the equation z = w exp(w). Here, the branches are indexed by the integer k.
Parameters :  z : array_like
k : int, optional
tol : float, optional


Returns :  w : array

Notes
All branches are supported by lambertw:
The Lambert W function has two partially real branches: the principal branch (k = 0) is real for real z > 1/e, and the k = 1 branch is real for 1/e < z < 0. All branches except k = 0 have a logarithmic singularity at z = 0.
Possible issues
The evaluation can become inaccurate very close to the branch point at 1/e. In some corner cases, lambertw might currently fail to converge, or can end up on the wrong branch.
Algorithm
Halley’s iteration is used to invert w * exp(w), using a firstorder asymptotic approximation (O(log(w)) or O(w)) as the initial estimate.
The definition, implementation and choice of branches is based on [R158].
References
[R157]  (1, 2) http://en.wikipedia.org/wiki/Lambert_W_function 
[R158]  (1, 2) Corless et al, “On the Lambert W function”, Adv. Comp. Math. 5 (1996) 329359. http://www.apmaths.uwo.ca/~djeffrey/Offprints/Wadvcm.pdf 
Examples
The Lambert W function is the inverse of w exp(w):
>>> from scipy.special import lambertw
>>> w = lambertw(1)
>>> w
(0.56714329040978384+0j)
>>> w*exp(w)
(1.0+0j)
Any branch gives a valid inverse:
>>> w = lambertw(1, k=3)
>>> w
(2.8535817554090377+17.113535539412148j)
>>> w*np.exp(w)
(1.0000000000000002+1.609823385706477e15j)
Applications to equationsolving
The Lambert W function may be used to solve various kinds of equations, such as finding the value of the infinite power tower :
>>> def tower(z, n):
... if n == 0:
... return z
... return z ** tower(z, n1)
...
>>> tower(0.5, 100)
0.641185744504986
>>> lambertw(np.log(0.5)) / np.log(0.5)
(0.64118574450498589+0j)