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r""" lambertw(z, k=0, tol=1e-8)
Lambert W function.
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 Input argument. k : int, optional Branch index. tol : float, optional Evaluation tolerance.
Returns ------- w : array `w` will have the same shape as `z`.
Notes ----- All branches are supported by `lambertw`:
* ``lambertw(z)`` gives the principal solution (branch 0) * ``lambertw(z, k)`` gives the solution on branch `k`
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 first-order asymptotic approximation (O(log(w)) or `O(w)`) as the initial estimate.
The definition, implementation and choice of branches is based on [2]_.
See Also -------- wrightomega : the Wright Omega function
References ---------- .. [1] http://en.wikipedia.org/wiki/Lambert_W_function .. [2] Corless et al, "On the Lambert W function", Adv. Comp. Math. 5 (1996) 329-359. http://www.apmaths.uwo.ca/~djeffrey/Offprints/W-adv-cm.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 * np.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+1.609823385706477e-15j)
**Applications to equation-solving**
The Lambert W function may be used to solve various kinds of equations, such as finding the value of the infinite power tower :math:`z^{z^{z^{\ldots}}}`:
>>> def tower(z, n): ... if n == 0: ... return z ... return z ** tower(z, n-1) ... >>> tower(0.5, 100) 0.641185744504986 >>> -lambertw(-np.log(0.5)) / np.log(0.5) (0.64118574450498589+0j) """ return _lambertw(z, k, tol) |