.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/special_functions/mathieu/plot_01_mathieu_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_mathieu_plot_01_mathieu_functions.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_special_functions_mathieu_plot_01_mathieu_functions.py:


Mathieu functions and their characteristic values
=======================================================

Plots the first even Mathieu functions ce_m(x, q) and how the
characteristic values a_m(q), b_m(q) split away from m^2 as q grows --
the band structure behind the Mathieu stability chart.

.. GENERATED FROM PYTHON SOURCE LINES 11-16

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.special_functions import mathieu_characteristic_a, mathieu_characteristic_b, mathieu_even








.. GENERATED FROM PYTHON SOURCE LINES 17-19

Even Mathieu functions at q = 2
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 19-30

.. code-block:: Python


    x = np.linspace(0, np.pi, 300)
    q = 2.0
    fig, ax = plt.subplots()
    for m in range(3):
        ax.plot(x, mathieu_even(m, q, x), label=rf"$\mathrm{{ce}}_{m}(x, {q:g})$")
        ax.plot(x, mathieu_even(m, 0.0, x), ":", color=f"C{m}")
    ax.set_xlabel("x (radians)")
    ax.set_title("Even Mathieu functions (dotted: q = 0, i.e. cosines)")
    ax.legend()




.. image-sg:: /api/gallery/special_functions/mathieu/images/sphx_glr_plot_01_mathieu_functions_001.png
   :alt: Even Mathieu functions (dotted: q = 0, i.e. cosines)
   :srcset: /api/gallery/special_functions/mathieu/images/sphx_glr_plot_01_mathieu_functions_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x11b88f380>



.. GENERATED FROM PYTHON SOURCE LINES 31-33

Characteristic values versus q
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 33-45

.. code-block:: Python


    qs = np.linspace(0, 10, 200)
    fig, ax = plt.subplots()
    for m in range(4):
        ax.plot(qs, [mathieu_characteristic_a(m, qq) for qq in qs], f"C{m}", label=f"$a_{m}$")
        if m >= 1:
            ax.plot(qs, [mathieu_characteristic_b(m, qq) for qq in qs], f"C{m}--", label=f"$b_{m}$")
    ax.set_xlabel("q")
    ax.set_ylabel("characteristic value")
    ax.legend(ncol=2)
    for m in range(4):
        print(f"a_{m}(0) = {mathieu_characteristic_a(m, 0.0) + 0.0:.1f}, a_{m}(10) = {mathieu_characteristic_a(m, 10.0):.4f}")



.. image-sg:: /api/gallery/special_functions/mathieu/images/sphx_glr_plot_01_mathieu_functions_002.png
   :alt: plot 01 mathieu functions
   :srcset: /api/gallery/special_functions/mathieu/images/sphx_glr_plot_01_mathieu_functions_002.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    a_0(0) = 0.0, a_0(10) = -13.9370
    a_1(0) = 1.0, a_1(10) = -2.3991
    a_2(0) = 4.0, a_2(10) = 7.7174
    a_3(0) = 9.0, a_3(10) = 15.5028





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

   **Total running time of the script:** (0 minutes 0.058 seconds)


.. _sphx_glr_download_api_gallery_special_functions_mathieu_plot_01_mathieu_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_mathieu_functions.ipynb <plot_01_mathieu_functions.ipynb>`

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

      :download:`Download Python source code: plot_01_mathieu_functions.py <plot_01_mathieu_functions.py>`

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

      :download:`Download zipped: plot_01_mathieu_functions.zip <plot_01_mathieu_functions.zip>`


.. only:: html

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

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