.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/special_functions/hypergeometric/plot_02_kummer_confluent.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_hypergeometric_plot_02_kummer_confluent.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_special_functions_hypergeometric_plot_02_kummer_confluent.py:


Kummer's confluent hypergeometric function
================================================

Checks Kummer's transformation M(a, b, z) = e^z M(b - a, b, -z) and
shows 2F1(a, c; b; z/c) merging into 1F1(a; b; z) as c grows.

.. GENERATED FROM PYTHON SOURCE LINES 10-15

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.special_functions import confluent_hypergeometric_1f1, hypergeometric_2f1








.. GENERATED FROM PYTHON SOURCE LINES 16-18

Kummer's transformation
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 18-25

.. code-block:: Python


    a, b = 0.6, 2.3
    z = np.linspace(-4, 4, 9)
    lhs = confluent_hypergeometric_1f1(a, b, z)
    rhs = np.exp(z) * confluent_hypergeometric_1f1(b - a, b, -z)
    print(f"max |M(a,b,z) - e^z M(b-a,b,-z)| = {np.max(np.abs(lhs - rhs)):.1e}")





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

 .. code-block:: none

    max |M(a,b,z) - e^z M(b-a,b,-z)| = 2.2e-16




.. GENERATED FROM PYTHON SOURCE LINES 26-28

The confluence of two singular points
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 28-38

.. code-block:: Python


    a, b = 0.5, 1.5
    z = np.linspace(-3, 3, 300)
    fig, ax = plt.subplots()
    for c in (2.0, 5.0, 20.0):
        ax.plot(z, hypergeometric_2f1(a, c, b, z / c), "--", label=f"${{}}_2F_1(a, {c:g}; b; z/{c:g})$")
    ax.plot(z, confluent_hypergeometric_1f1(a, b, z), "k", lw=2, label=r"${}_1F_1(a; b; z)$")
    ax.set_xlabel("z")
    ax.set_title(r"$a = 1/2$, $b = 3/2$: the confluent limit")
    ax.legend()



.. image-sg:: /api/gallery/special_functions/hypergeometric/images/sphx_glr_plot_02_kummer_confluent_001.png
   :alt: $a = 1/2$, $b = 3/2$: the confluent limit
   :srcset: /api/gallery/special_functions/hypergeometric/images/sphx_glr_plot_02_kummer_confluent_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x11c4fdbe0>




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

   **Total running time of the script:** (0 minutes 0.033 seconds)


.. _sphx_glr_download_api_gallery_special_functions_hypergeometric_plot_02_kummer_confluent.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_02_kummer_confluent.ipynb <plot_02_kummer_confluent.ipynb>`

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

      :download:`Download Python source code: plot_02_kummer_confluent.py <plot_02_kummer_confluent.py>`

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

      :download:`Download zipped: plot_02_kummer_confluent.zip <plot_02_kummer_confluent.zip>`


.. only:: html

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

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