(phz"SSKJr SSKrSSjrg)) _lambertwNcl[R"U[R"S5S9n[XU5$)a 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`. See Also -------- wrightomega : the Wright Omega function 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]_. References ---------- .. [1] https://en.wikipedia.org/wiki/Lambert_W_function .. [2] Corless et al, "On the Lambert W function", Adv. Comp. Math. 5 (1996) 329-359. https://cs.uwaterloo.ca/research/tr/1993/03/W.pdf Examples -------- The Lambert W function is the inverse of ``w exp(w)``: >>> import numpy as np >>> 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. We give two examples here. First, the function can be used to solve implicit equations of the form :math:`x = a + b e^{c x}` for :math:`x`. We assume :math:`c` is not zero. After a little algebra, the equation may be written :math:`z e^z = -b c e^{a c}` where :math:`z = c (a - x)`. :math:`z` may then be expressed using the Lambert W function :math:`z = W(-b c e^{a c})` giving :math:`x = a - W(-b c e^{a c})/c` For example, >>> a = 3 >>> b = 2 >>> c = -0.5 The solution to :math:`x = a + b e^{c x}` is: >>> x = a - lambertw(-b*c*np.exp(a*c))/c >>> x (3.3707498368978794+0j) Verify that it solves the equation: >>> a + b*np.exp(c*x) (3.37074983689788+0j) The Lambert W function may also be used find 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) long)dtype)npasarrayrr)zktols J/var/www/html/venv/lib/python3.13/site-packages/scipy/special/_lambertw.pylambertwrs+\ 1BHHV,-A Q3 )rg:0yE>)_ufuncsrnumpyrrrr rsO r