.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/complex_analysis/residues/plot_05_rouche_theorem.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_complex_analysis_residues_plot_05_rouche_theorem.py>`
        to download the full example code or to run this example in your browser via JupyterLite.

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

.. _sphx_glr_api_gallery_complex_analysis_residues_plot_05_rouche_theorem.py:


Rouché's theorem: counting zeros by domination
==============================================

Eugène Rouché (1862): if :math:`|g| < |f|` on a closed contour, then
:math:`f` and :math:`f + g` have the same number of zeros inside it.
Picture :math:`f(\gamma)` as a person walking around a lamppost at 0
and :math:`g` as the leash to a dog at :math:`f + g`: a leash shorter
than the distance to the post means the dog circles it the same number
of times. Here :math:`z^5 + 3z + 1` has all five zeros in
:math:`|z| < 2` (dominated by :math:`z^5`) and exactly one in
:math:`|z| < 1` (dominated by :math:`3z`).

.. GENERATED FROM PYTHON SOURCE LINES 16-28

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.complex_analysis import argument_principle, circle_contour


    def h(z):
        return z**5 + 3 * z + 1


    roots = np.roots([1, 0, 0, 0, 3, 1])








.. GENERATED FROM PYTHON SOURCE LINES 29-31

Two contours, two dominant terms
--------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 31-52

.. code-block:: Python


    fig, axes = plt.subplots(1, 2, figsize=(11, 5))
    for ax, (R, f, name) in zip(axes, [(2.0, lambda z: z**5, "z^5"), (1.0, lambda z: 3 * z, "3z")], strict=False):
        contour = circle_contour(0.0, R)
        z = contour.points(2000)
        g = h(z) - f(z)
        print(f"|z| = {R}: min |{name}| = {np.min(np.abs(f(z))):.2f} > max |rest| = {np.max(np.abs(g)):.2f}")
        print(
            f"   zeros of {name}: {argument_principle(f, contour)}, zeros of z^5+3z+1: {argument_principle(h, contour)}, "
            f"numpy.roots inside: {int(np.sum(np.abs(roots) < R))}"
        )
        ax.plot(f(z).real, f(z).imag, lw=1, label=f"walker: {name}")
        ax.plot(h(z).real, h(z).imag, lw=1, label="dog: z^5 + 3z + 1")
        ax.plot(0, 0, "k*", ms=12, label="lamppost 0")
        ax.set_aspect("equal")
        ax.set_title(f"|z| = {R}")
        ax.legend(fontsize=8)
    fig.suptitle("Rouché's theorem as dog walking")
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/complex_analysis/residues/images/sphx_glr_plot_05_rouche_theorem_001.png
   :alt: Rouché's theorem as dog walking, |z| = 2.0, |z| = 1.0
   :srcset: /api/gallery/complex_analysis/residues/images/sphx_glr_plot_05_rouche_theorem_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    |z| = 2.0: min |z^5| = 32.00 > max |rest| = 7.00
       zeros of z^5: 5, zeros of z^5+3z+1: 5, numpy.roots inside: 5
    |z| = 1.0: min |3z| = 3.00 > max |rest| = 2.00
       zeros of 3z: 1, zeros of z^5+3z+1: 1, numpy.roots inside: 1





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

   **Total running time of the script:** (0 minutes 0.250 seconds)


.. _sphx_glr_download_api_gallery_complex_analysis_residues_plot_05_rouche_theorem.py:

.. only:: html

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

    .. container:: lite-badge

      .. image:: images/jupyterlite_badge_logo.svg
        :target: ../../../../lite/lab/index.html?path=api/gallery/complex_analysis/residues/plot_05_rouche_theorem.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_05_rouche_theorem.ipynb <plot_05_rouche_theorem.ipynb>`

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

      :download:`Download Python source code: plot_05_rouche_theorem.py <plot_05_rouche_theorem.py>`

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

      :download:`Download zipped: plot_05_rouche_theorem.zip <plot_05_rouche_theorem.zip>`


.. only:: html

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

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