.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/special_functions/airy/plot_01_airy_functions.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_airy_plot_01_airy_functions.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_special_functions_airy_plot_01_airy_functions.py:


The Airy functions Ai and Bi
=================================

Plots Ai and Bi across the turning point x = 0 where y'' = xy switches
from oscillation to exponential behaviour, and checks the Wronskian
Ai Bi' - Ai' Bi = 1/pi.

.. GENERATED FROM PYTHON SOURCE LINES 11-16

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.special_functions import airy_functions








.. GENERATED FROM PYTHON SOURCE LINES 17-19

Oscillation on the left, growth or decay on the right
------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 19-31

.. code-block:: Python


    x = np.linspace(-15, 3, 18001)
    r = airy_functions(x)
    fig, ax = plt.subplots()
    ax.plot(x, r.ai, label="Ai(x)")
    ax.plot(x, r.bi, label="Bi(x)")
    ax.axvline(0, color="gray", ls=":")
    ax.set_ylim(-0.6, 1.2)
    ax.set_xlabel("x")
    ax.set_title(r"Solutions of $y'' = xy$")
    ax.legend()




.. image-sg:: /api/gallery/special_functions/airy/images/sphx_glr_plot_01_airy_functions_001.png
   :alt: Solutions of $y'' = xy$
   :srcset: /api/gallery/special_functions/airy/images/sphx_glr_plot_01_airy_functions_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x1196e30e0>



.. GENERATED FROM PYTHON SOURCE LINES 32-34

Wronskian and the first zeros of Ai
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 34-39

.. code-block:: Python


    wronskian = r.ai * r.bi_prime - r.ai_prime * r.bi
    print(f"Wronskian range: [{wronskian.min():.12f}, {wronskian.max():.12f}], 1/pi = {1 / np.pi:.12f}")
    zeros = x[:-1][np.diff(np.sign(r.ai)) != 0][::-1][:3]
    print(f"first zeros of Ai near {np.round(zeros, 3)} (known: -2.338, -4.088, -5.521)")




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

 .. code-block:: none

    Wronskian range: [0.318309886184, 0.318309886184], 1/pi = 0.318309886184
    first zeros of Ai near [-2.339 -4.088 -5.521] (known: -2.338, -4.088, -5.521)





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

   **Total running time of the script:** (0 minutes 0.051 seconds)


.. _sphx_glr_download_api_gallery_special_functions_airy_plot_01_airy_functions.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_airy_functions.ipynb <plot_01_airy_functions.ipynb>`

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

      :download:`Download Python source code: plot_01_airy_functions.py <plot_01_airy_functions.py>`

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

      :download:`Download zipped: plot_01_airy_functions.zip <plot_01_airy_functions.zip>`


.. only:: html

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

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