.. 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_02_argument_principle.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_02_argument_principle.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_02_argument_principle.py:


The argument principle: counting zeros minus poles by winding
=============================================================

:math:`\frac{1}{2\pi i}\oint_\gamma f'/f = N - P`, the number of zeros
minus poles inside :math:`\gamma`. Equivalently, the image curve
:math:`f(\gamma)` winds :math:`N - P` times around 0. For a quintic,
counting roots inside disks of growing radius needs no root finder at
all.

.. GENERATED FROM PYTHON SOURCE LINES 13-23

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.complex_analysis import argument_principle, circle_contour

    coefficients = [1, -0.5, 0.2, -1.3, 0.1, 0.4]
    p = lambda z: np.polyval(coefficients, z)
    dp = lambda z: np.polyval(np.polyder(coefficients), z)
    roots = np.roots(coefficients)








.. GENERATED FROM PYTHON SOURCE LINES 24-26

Root counts from the argument principle
---------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 26-35

.. code-block:: Python


    for radius in (0.3, 0.6, 0.9, 1.2, 1.5):
        circle = circle_contour(0.0, radius)
        print(
            f"|z| < {radius}: phase winding = {argument_principle(p, circle)}, "
            f"oint p'/p = {argument_principle(p, circle, fprime=dp)}, "
            f"numpy.roots = {int(np.sum(np.abs(roots) < radius))}"
        )





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

 .. code-block:: none

    |z| < 0.3: phase winding = 0, oint p'/p = 0, numpy.roots = 0
    |z| < 0.6: phase winding = 1, oint p'/p = 1, numpy.roots = 1
    |z| < 0.9: phase winding = 2, oint p'/p = 2, numpy.roots = 2
    |z| < 1.2: phase winding = 5, oint p'/p = 5, numpy.roots = 5
    |z| < 1.5: phase winding = 5, oint p'/p = 5, numpy.roots = 5




.. GENERATED FROM PYTHON SOURCE LINES 36-38

The image curve winds around 0 once per enclosed zero
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 38-52

.. code-block:: Python


    fig, axes = plt.subplots(1, 2, figsize=(10, 5))
    for radius in (0.6, 1.2):
        circle = circle_contour(0.0, radius)
        z = circle.points()
        axes[0].plot(z.real, z.imag, label=f"|z| = {radius}")
        w = p(z)
        axes[1].plot(w.real, w.imag, label=f"p(|z| = {radius}): winds {argument_principle(p, circle)}x")
    axes[0].plot(roots.real, roots.imag, "ko", label="roots")
    axes[1].plot(0, 0, "k+", ms=12)
    for ax, title in zip(axes, ("z-plane", "w = p(z)"), strict=False):
        ax.set_aspect("equal")
        ax.set_title(title)
        ax.legend(fontsize=8)



.. image-sg:: /api/gallery/complex_analysis/residues/images/sphx_glr_plot_02_argument_principle_001.png
   :alt: z-plane, w = p(z)
   :srcset: /api/gallery/complex_analysis/residues/images/sphx_glr_plot_02_argument_principle_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.184 seconds)


.. _sphx_glr_download_api_gallery_complex_analysis_residues_plot_02_argument_principle.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_02_argument_principle.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_argument_principle.ipynb <plot_02_argument_principle.ipynb>`

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

      :download:`Download Python source code: plot_02_argument_principle.py <plot_02_argument_principle.py>`

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

      :download:`Download zipped: plot_02_argument_principle.zip <plot_02_argument_principle.zip>`


.. only:: html

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

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