.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/complex_analysis/contour_integrals/plot_03_liouville_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_contour_integrals_plot_03_liouville_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_contour_integrals_plot_03_liouville_theorem.py:


Liouville's theorem: a bounded entire function is constant
==========================================================

Cauchy's integral formula on the circle :math:`|z| = R` gives the
estimate :math:`|f'(0)| \le M(R)/R`, where :math:`M(R)` is the maximum
of :math:`|f|` on the circle. If :math:`f` is entire and bounded, letting
:math:`R \to \infty` forces :math:`f' \equiv 0`. So every nonconstant
entire function, such as :math:`\sin z`, is unbounded. Applied to
:math:`1/p(z)`, the theorem proves the fundamental theorem of algebra.

.. GENERATED FROM PYTHON SOURCE LINES 14-19

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.complex_analysis import cauchy_integral_formula, circle_contour








.. GENERATED FROM PYTHON SOURCE LINES 20-22

Cauchy's estimate :math:`|f'(0)| \le M(R)/R`
---------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 22-36

.. code-block:: Python


    radii = np.geomspace(0.5, 20, 12)
    functions = {r"$\sin z$": np.sin, r"$z^3 - z$": lambda z: z**3 - z, r"$e^{-z^2}$": lambda z: np.exp(-(z**2))}
    fig, ax = plt.subplots()
    for name, f in functions.items():
        max_modulus = np.array([np.max(np.abs(f(circle_contour(0, R).points(2000)))) for R in radii])
        ax.loglog(radii, max_modulus, "o-", label=f"M(R) for {name}")
        derivative = abs(cauchy_integral_formula(f, circle_contour(0, 1.0), 0.0, n=1))
        print(f"{name:12s} |f'(0)| = {derivative:.4f},  min over R of M(R)/R = {np.min(max_modulus / radii):.4f}")
    ax.set_xlabel("R")
    ax.set_ylabel("max |f| on |z| = R")
    ax.set_title("Nonconstant entire functions are unbounded")
    ax.legend()




.. image-sg:: /api/gallery/complex_analysis/contour_integrals/images/sphx_glr_plot_03_liouville_theorem_001.png
   :alt: Nonconstant entire functions are unbounded
   :srcset: /api/gallery/complex_analysis/contour_integrals/images/sphx_glr_plot_03_liouville_theorem_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    $\sin z$     |f'(0)| = 1.0000,  min over R of M(R)/R = 1.0422
    $z^3 - z$    |f'(0)| = 1.0000,  min over R of M(R)/R = 1.2500
    $e^{-z^2}$   |f'(0)| = 0.0000,  min over R of M(R)/R = 2.3319

    <matplotlib.legend.Legend object at 0x7fd7c9244ef0>



.. GENERATED FROM PYTHON SOURCE LINES 37-41

The fundamental theorem of algebra
----------------------------------
If p had no roots, 1/p would be entire; it also tends to 0 as :math:`|z|` grows,
so it would be bounded, hence constant -- impossible for a nonconstant p.

.. GENERATED FROM PYTHON SOURCE LINES 41-46

.. code-block:: Python


    p = lambda z: z**4 + z + 1
    for R in (1, 10, 100):
        print(f"R = {R:3d}: max |1/p| on |z| = R: {np.max(np.abs(1 / p(circle_contour(0, R).points(2000)))):.2e}")
    print("roots of p (where 1/p fails to be entire):", np.round(np.roots([1, 0, 0, 1, 1]), 4))




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

 .. code-block:: none

    R =   1: max |1/p| on |z| = R: 1.93e+00
    R =  10: max |1/p| on |z| = R: 1.00e-04
    R = 100: max |1/p| on |z| = R: 1.00e-08
    roots of p (where 1/p fails to be entire): [ 0.7271+0.9341j  0.7271-0.9341j -0.7271+0.43j   -0.7271-0.43j  ]





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

   **Total running time of the script:** (0 minutes 0.162 seconds)


.. _sphx_glr_download_api_gallery_complex_analysis_contour_integrals_plot_03_liouville_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/contour_integrals/plot_03_liouville_theorem.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_03_liouville_theorem.ipynb <plot_03_liouville_theorem.ipynb>`

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

      :download:`Download Python source code: plot_03_liouville_theorem.py <plot_03_liouville_theorem.py>`

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

      :download:`Download zipped: plot_03_liouville_theorem.zip <plot_03_liouville_theorem.zip>`


.. only:: html

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

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