.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/special_functions/lambert_w/plot_01_lambert_w_branches.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_special_functions_lambert_w_plot_01_lambert_w_branches.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_special_functions_lambert_w_plot_01_lambert_w_branches.py:


The two real branches of the Lambert W function
====================================================

Plots W_0 and W_{-1}, the two real inverses of w e^w, which meet at the
branch point z = -1/e, and uses W to solve x^x = 27.

.. GENERATED FROM PYTHON SOURCE LINES 10-17

.. code-block:: Python

    import math

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.special_functions import lambert_w








.. GENERATED FROM PYTHON SOURCE LINES 18-20

Both real branches
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 20-34

.. code-block:: Python


    z0 = np.linspace(-1 / math.e + 1e-9, 6.0, 400)
    zm1 = np.linspace(-1 / math.e + 1e-9, -1e-4, 400)

    fig, ax = plt.subplots()
    ax.plot(z0, lambert_w(z0), label=r"$W_0$ (principal)")
    ax.plot(zm1, lambert_w(zm1, branch=-1), label=r"$W_{-1}$")
    ax.plot(-1 / math.e, -1, "ko", label=r"branch point $(-1/e, -1)$")
    ax.set_ylim(-6, 2)
    ax.set_xlabel("z")
    ax.set_ylabel("W(z)")
    ax.set_title(r"Solutions of $we^w = z$")
    ax.legend()




.. image-sg:: /api/gallery/special_functions/lambert_w/images/sphx_glr_plot_01_lambert_w_branches_001.png
   :alt: Solutions of $we^w = z$
   :srcset: /api/gallery/special_functions/lambert_w/images/sphx_glr_plot_01_lambert_w_branches_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x11c4ffa10>



.. GENERATED FROM PYTHON SOURCE LINES 35-37

The omega constant and a transcendental equation
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 37-44

.. code-block:: Python


    omega = lambert_w(1.0)
    print(f"Omega = W(1) = {omega:.12f}, exp(-Omega) = {math.exp(-omega):.12f}")

    y = 27.0
    x = math.log(y) / lambert_w(math.log(y))
    print(f"x^x = {y:g} is solved by x = ln y / W(ln y) = {x:.12f}")




.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    Omega = W(1) = 0.567143290410, exp(-Omega) = 0.567143290410
    x^x = 27 is solved by x = ln y / W(ln y) = 3.000000000000





.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.028 seconds)


.. _sphx_glr_download_api_gallery_special_functions_lambert_w_plot_01_lambert_w_branches.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_01_lambert_w_branches.ipynb <plot_01_lambert_w_branches.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_01_lambert_w_branches.py <plot_01_lambert_w_branches.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_01_lambert_w_branches.zip <plot_01_lambert_w_branches.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_