.. 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_01_gauss_hypergeometric.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_01_gauss_hypergeometric.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_special_functions_hypergeometric_plot_01_gauss_hypergeometric.py:


Gauss's hypergeometric function contains the elementary functions
======================================================================

Recovers log(1 + z), arcsin z, and a Chebyshev polynomial from 2F1 with
special parameters, and checks Gauss's summation theorem at z = 1.

.. GENERATED FROM PYTHON SOURCE LINES 10-15

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.special_functions import chebyshev_polynomial, gamma_function, hypergeometric_2f1








.. GENERATED FROM PYTHON SOURCE LINES 16-18

Three familiar functions in one family
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 18-36

.. code-block:: Python


    z = np.linspace(-0.95, 0.95, 300)
    log_curve = z * hypergeometric_2f1(1.0, 1.0, 2.0, -z)
    arcsin_curve = z * hypergeometric_2f1(0.5, 0.5, 1.5, z**2)
    cheb_curve = hypergeometric_2f1(-4, 4, 0.5, (1 - z) / 2)

    fig, ax = plt.subplots()
    ax.plot(z, log_curve, label=r"$z\,{}_2F_1(1,1;2;-z) = \ln(1+z)$")
    ax.plot(z, arcsin_curve, label=r"$z\,{}_2F_1(\frac{1}{2},\frac{1}{2};\frac{3}{2};z^2) = \arcsin z$")
    ax.plot(z, cheb_curve, label=r"${}_2F_1(-4,4;\frac{1}{2};\frac{1-z}{2}) = T_4(z)$")
    ax.set_xlabel("z")
    ax.legend(fontsize="small")
    ax.set_title("Special cases of the hypergeometric function")

    print(f"max |log error|    = {np.max(np.abs(log_curve - np.log1p(z))):.1e}")
    print(f"max |arcsin error| = {np.max(np.abs(arcsin_curve - np.arcsin(z))):.1e}")
    print(f"max |T_4 error|    = {np.max(np.abs(cheb_curve - chebyshev_polynomial(4, z))):.1e}")




.. image-sg:: /api/gallery/special_functions/hypergeometric/images/sphx_glr_plot_01_gauss_hypergeometric_001.png
   :alt: Special cases of the hypergeometric function
   :srcset: /api/gallery/special_functions/hypergeometric/images/sphx_glr_plot_01_gauss_hypergeometric_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    max |log error|    = 1.9e-14
    max |arcsin error| = 2.4e-15
    max |T_4 error|    = 1.7e-15




.. GENERATED FROM PYTHON SOURCE LINES 37-39

Gauss's summation theorem at z = 1
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 39-44

.. code-block:: Python


    a, b, c = 0.5, 0.25, 2.0
    gauss = gamma_function(c) * gamma_function(c - a - b) / (gamma_function(c - a) * gamma_function(c - b))
    print(f"\n2F1({a}, {b}; {c}; 1) = {hypergeometric_2f1(a, b, c, 1.0):.12f}")
    print(f"Gamma ratio          = {gauss:.12f}")




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

 .. code-block:: none


    2F1(0.5, 0.25; 2.0; 1) = 1.112835788899
    Gamma ratio          = 1.112835788899





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

   **Total running time of the script:** (0 minutes 0.045 seconds)


.. _sphx_glr_download_api_gallery_special_functions_hypergeometric_plot_01_gauss_hypergeometric.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_gauss_hypergeometric.ipynb <plot_01_gauss_hypergeometric.ipynb>`

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

      :download:`Download Python source code: plot_01_gauss_hypergeometric.py <plot_01_gauss_hypergeometric.py>`

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

      :download:`Download zipped: plot_01_gauss_hypergeometric.zip <plot_01_gauss_hypergeometric.zip>`


.. only:: html

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

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