lambertw(z, k=0, tol=1e-8)
The Lambert W function :None:None:`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.
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 (:None:None:`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 :None:None:`O(w)`) as the initial estimate.
The definition, implementation and choice of branches is based on .
Input argument.
Branch index.
Evaluation tolerance.
Lambert W function.
wrightomega
the Wright Omega function
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 $z^{z^{z^{\ldots}}}$ :
>>> 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)See :
The following pages refer to to this document either explicitly or contain code examples using this.
scipy.special._lambertw.lambertw
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