.. 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_01_residue_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_01_residue_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_01_residue_theorem.py:


Cauchy's residue theorem: the contour integral is 2 pi i times the enclosed residues
====================================================================================

For :math:`f` meromorphic inside :math:`\gamma`,
:math:`\oint_\gamma f = 2\pi i \sum_k n(\gamma, z_k)\operatorname{Res}(f, z_k)`.
Growing a circle past the poles of
:math:`f(z) = e^{z}/((z - 1)(z + 2)(z - 3i))` adds each residue in
turn, and the same theorem evaluates the real integral
:math:`\int_{-\infty}^\infty dx/(1 + x^4) = \pi/\sqrt 2`.

.. GENERATED FROM PYTHON SOURCE LINES 14-26

.. code-block:: Python

    import math

    import matplotlib.pyplot as plt
    import numpy as np
    from scipy import integrate

    from mathematicskit.complex_analysis import circle_contour, residue, residue_theorem
    from mathematicskit.complex_analysis.visualizers import plot_contour

    poles = [1, -2, 3j]
    f = lambda z: np.exp(z) / ((z - 1) * (z + 2) * (z - 3j))








.. GENERATED FROM PYTHON SOURCE LINES 27-29

Growing circles pick up one residue at a time
---------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 29-38

.. code-block:: Python


    fig, ax = plt.subplots()
    for radius in (0.5, 1.5, 2.5, 3.5):
        circle = circle_contour(0.0, radius)
        result = residue_theorem(f, circle, poles)
        plot_contour(circle, marked_points=poles, ax=ax)
        print(f"r={radius}: enclosed={result.winding_numbers.tolist()}  oint f = {result.integral:.8f}  2 pi i sum Res = {result.predicted:.8f}")
    ax.set_title("Circles of radius 0.5, 1.5, 2.5, 3.5 around three poles")




.. image-sg:: /api/gallery/complex_analysis/residues/images/sphx_glr_plot_01_residue_theorem_001.png
   :alt: Circles of radius 0.5, 1.5, 2.5, 3.5 around three poles
   :srcset: /api/gallery/complex_analysis/residues/images/sphx_glr_plot_01_residue_theorem_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    r=0.5: enclosed=[0, 0, 0]  oint f = -0.00000000-0.00000000j  2 pi i sum Res = 0.00000000+0.00000000j
    r=1.5: enclosed=[1, 0, 0]  oint f = -1.70794684+0.56931561j  2 pi i sum Res = -1.70794684+0.56931561j
    r=2.5: enclosed=[1, 1, 0]  oint f = -1.64253633+0.61292262j  2 pi i sum Res = -1.64253633+0.61292262j
    r=3.5: enclosed=[1, 1, 1]  oint f = -1.71105483+1.15971815j  2 pi i sum Res = -1.71105483+1.15971815j

    Text(0.5, 1.0, 'Circles of radius 0.5, 1.5, 2.5, 3.5 around three poles')



.. GENERATED FROM PYTHON SOURCE LINES 39-43

A real integral by residues
---------------------------
Closing the real line with a large upper semicircle encloses the poles
e^{i pi/4} and e^{3 i pi/4} of 1/(1 + z^4); the arc's contribution vanishes.

.. GENERATED FROM PYTHON SOURCE LINES 43-48

.. code-block:: Python


    g = lambda z: 1 / (1 + z**4)
    by_residues = 2j * math.pi * sum(residue(g, np.exp(1j * math.pi * k / 4)) for k in (1, 3))
    by_quadrature, _ = integrate.quad(lambda x: 1 / (1 + x**4), -np.inf, np.inf)
    print(f"residues: {by_residues.real:.12f}   quad: {by_quadrature:.12f}   pi/sqrt(2): {math.pi / math.sqrt(2):.12f}")




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

 .. code-block:: none

    residues: 2.221441469079   quad: 2.221441469079   pi/sqrt(2): 2.221441469079





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

   **Total running time of the script:** (0 minutes 0.093 seconds)


.. _sphx_glr_download_api_gallery_complex_analysis_residues_plot_01_residue_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_01_residue_theorem.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_residue_theorem.ipynb <plot_01_residue_theorem.ipynb>`

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

      :download:`Download Python source code: plot_01_residue_theorem.py <plot_01_residue_theorem.py>`

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

      :download:`Download zipped: plot_01_residue_theorem.zip <plot_01_residue_theorem.zip>`


.. only:: html

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

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