.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/special_functions/bessel/plot_01_bessel_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_bessel_plot_01_bessel_functions.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_special_functions_bessel_plot_01_bessel_functions.py:


Bessel functions of the first and second kind
====================================================

Plots J_0, J_1 (regular at the origin) and Y_0 (singular at the
origin) over a range of x, and finds J_0's first zero.

.. GENERATED FROM PYTHON SOURCE LINES 10-15

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.special_functions import bessel_first_kind, bessel_second_kind








.. GENERATED FROM PYTHON SOURCE LINES 16-18

Evaluate over a range
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 18-36

.. code-block:: Python


    x = np.linspace(0.1, 10.0, 200)
    j0 = bessel_first_kind(0.0, x)
    j1 = bessel_first_kind(1.0, x)
    y0 = bessel_second_kind(0.0, x)

    fig, ax = plt.subplots()
    ax.plot(x, j0, label=r"$J_0(x)$")
    ax.plot(x, j1, label=r"$J_1(x)$")
    ax.plot(x, y0, label=r"$Y_0(x)$")
    ax.axhline(0, color="gray", lw=0.5)
    ax.set_ylim(-1.5, 1.1)
    ax.set_xlabel("x")
    ax.set_title("Bessel functions of the first and second kind")
    ax.legend()

    print(f"J_0(0.1) = {j0[0]:.4f}, Y_0(0.1) = {y0[0]:.4f} (Y_0 diverges toward x=0)")




.. image-sg:: /api/gallery/special_functions/bessel/images/sphx_glr_plot_01_bessel_functions_001.png
   :alt: Bessel functions of the first and second kind
   :srcset: /api/gallery/special_functions/bessel/images/sphx_glr_plot_01_bessel_functions_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    J_0(0.1) = 0.9975, Y_0(0.1) = -1.5342 (Y_0 diverges toward x=0)




.. GENERATED FROM PYTHON SOURCE LINES 37-39

The first zero of J_0 (approximately 2.4048)
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 39-43

.. code-block:: Python


    sign_changes = np.where(np.diff(np.sign(j0)) != 0)[0]
    first_zero = x[sign_changes[0]]
    print(f"\nfirst zero of J_0 near x = {first_zero:.4f} (known value: 2.4048)")




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

 .. code-block:: none


    first zero of J_0 near x = 2.3884 (known value: 2.4048)





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

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


.. _sphx_glr_download_api_gallery_special_functions_bessel_plot_01_bessel_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_bessel_functions.ipynb <plot_01_bessel_functions.ipynb>`

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

      :download:`Download Python source code: plot_01_bessel_functions.py <plot_01_bessel_functions.py>`

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

      :download:`Download zipped: plot_01_bessel_functions.zip <plot_01_bessel_functions.zip>`


.. only:: html

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

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