.. 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_04_laurent_series.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_04_laurent_series.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_04_laurent_series.py:


Laurent series: expansions with negative powers in an annulus
=============================================================

Pierre Alphonse Laurent (1843, after Weierstrass in 1841) expanded a
function holomorphic in an annulus :math:`r < |z - a| < R` as
:math:`\sum_{n=-\infty}^{\infty} a_n (z-a)^n`, with every coefficient a
contour integral :math:`a_n = \frac{1}{2\pi i}\oint f(z)(z-a)^{-n-1}dz`.
For :math:`f(z) = 1/((z-1)(z-2))` the expansion depends on the annulus:
in :math:`1 < |z| < 2` it is :math:`a_n = -2^{-n-1}` for :math:`n \ge 0`
and :math:`a_n = -1` for :math:`n < 0`.

.. GENERATED FROM PYTHON SOURCE LINES 15-29

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.complex_analysis import circle_contour, contour_integral


    def f(z):
        return 1.0 / ((z - 1.0) * (z - 2.0))


    def laurent_coefficient(n, radius):
        return contour_integral(lambda z: f(z) * z ** (-n - 1), circle_contour(0.0, radius)) / (2j * np.pi)









.. GENERATED FROM PYTHON SOURCE LINES 30-32

Coefficients by contour integration in three annuli
---------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 32-46

.. code-block:: Python


    orders = np.arange(-5, 6)
    closed_form = np.where(orders >= 0, -(2.0 ** (-orders - 1.0)), -1.0)
    fig, ax = plt.subplots()
    for radius, label in ((0.5, "|z| < 1 (Taylor)"), (1.5, "1 < |z| < 2"), (3.0, "|z| > 2")):
        coefficients = np.array([laurent_coefficient(n, radius) for n in orders]).real
        ax.plot(orders, coefficients, "o-", label=label)
        if radius == 1.5:
            print(f"max |numeric - closed form| in 1 < |z| < 2: {np.max(np.abs(coefficients - closed_form)):.1e}")
    ax.set_xlabel("n")
    ax.set_ylabel("$a_n$")
    ax.set_title("Laurent coefficients of 1/((z-1)(z-2)) depend on the annulus")
    ax.legend()




.. image-sg:: /api/gallery/complex_analysis/residues/images/sphx_glr_plot_04_laurent_series_001.png
   :alt: Laurent coefficients of 1/((z-1)(z-2)) depend on the annulus
   :srcset: /api/gallery/complex_analysis/residues/images/sphx_glr_plot_04_laurent_series_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    max |numeric - closed form| in 1 < |z| < 2: 1.4e-12

    <matplotlib.legend.Legend object at 0x7fd7e8ee3e60>



.. GENERATED FROM PYTHON SOURCE LINES 47-49

The truncated series converges only inside its annulus
------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 49-57

.. code-block:: Python


    z = 1.5 * np.exp(0.7j)
    for terms in (5, 10, 20, 40):
        n = np.arange(-terms, terms + 1)
        a = np.where(n >= 0, -(2.0 ** (-n - 1.0)), -1.0)
        print(f"{terms:>2} terms each side: |partial sum - f(z)| = {abs(np.sum(a * z**n) - f(z)):.2e}")

    plt.show()




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

 .. code-block:: none

     5 terms each side: |partial sum - f(z)| = 7.32e-02
    10 terms each side: |partial sum - f(z)| = 3.21e-02
    20 terms each side: |partial sum - f(z)| = 2.13e-03
    40 terms each side: |partial sum - f(z)| = 5.78e-06





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

   **Total running time of the script:** (0 minutes 0.095 seconds)


.. _sphx_glr_download_api_gallery_complex_analysis_residues_plot_04_laurent_series.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_04_laurent_series.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_04_laurent_series.ipynb <plot_04_laurent_series.ipynb>`

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

      :download:`Download Python source code: plot_04_laurent_series.py <plot_04_laurent_series.py>`

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

      :download:`Download zipped: plot_04_laurent_series.zip <plot_04_laurent_series.zip>`


.. only:: html

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

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