.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/complex_analysis/conformal_maps/plot_04_schwarz_lemma.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_conformal_maps_plot_04_schwarz_lemma.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_conformal_maps_plot_04_schwarz_lemma.py:


Schwarz's lemma: self-maps of the disk fixing 0 cannot expand
=============================================================

If :math:`f` maps the unit disk into itself with :math:`f(0) = 0`, then
:math:`|f(z)| \le |z|` and :math:`|f'(0)| \le 1`, with equality only
for rotations :math:`f(z) = e^{i\theta}z` (Schwarz 1869; Carathéodory
gave the general statement in 1912). Blaschke products, the model
self-maps of the disk, obey it with room to spare.

.. GENERATED FROM PYTHON SOURCE LINES 13-31

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.complex_analysis import complex_derivative, complex_grid, mobius_transform


    def blaschke_factor(a):
        """The disk automorphism (z - a) / (1 - conj(a) z), a Möbius transformation."""
        return lambda z: mobius_transform(z, 1, -a, -np.conj(a), 1)


    maps = {
        "rotation e^{0.7i} z": lambda z: np.exp(0.7j) * z,
        "z^2": lambda z: z**2,
        "z (z - 0.5)/(1 - 0.5 z)": lambda z: z * blaschke_factor(0.5)(z),
        "z (z - 0.3i)/(1 + 0.3i z)": lambda z: z * blaschke_factor(0.3j)(z),
    }








.. GENERATED FROM PYTHON SOURCE LINES 32-34

:math:`|f(z)| \le |z|` everywhere in the disk, :math:`|f'(0)| \le 1`
----------------------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 34-48

.. code-block:: Python


    z = complex_grid((-0.99, 0.99), (-0.99, 0.99), 201)
    z = z[(np.abs(z) < 0.99) & (z != 0)]
    fig, ax = plt.subplots()
    for name, f in maps.items():
        ratio = np.abs(f(z)) / np.abs(z)
        print(f"{name:26s} max |f(z)|/|z| = {ratio.max():.6f},  |f'(0)| = {abs(complex_derivative(f, 0.0)):.6f}")
        order = np.argsort(np.abs(z))
        ax.plot(np.abs(z)[order][::50], ratio[order][::50], ".", ms=3, label=name)
    ax.axhline(1, color="k", lw=1)
    ax.set_xlabel("|z|")
    ax.set_ylabel("|f(z)| / |z|")
    ax.set_title("Schwarz's lemma: the ratio never exceeds 1")
    ax.legend(fontsize=8)



.. image-sg:: /api/gallery/complex_analysis/conformal_maps/images/sphx_glr_plot_04_schwarz_lemma_001.png
   :alt: Schwarz's lemma: the ratio never exceeds 1
   :srcset: /api/gallery/complex_analysis/conformal_maps/images/sphx_glr_plot_04_schwarz_lemma_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    rotation e^{0.7i} z        max |f(z)|/|z| = 1.000000,  |f'(0)| = 1.000000
    z^2                        max |f(z)|/|z| = 0.990000,  |f'(0)| = 0.000000
    z (z - 0.5)/(1 - 0.5 z)    max |f(z)|/|z| = 0.996591,  |f'(0)| = 0.500000
    z (z - 0.3i)/(1 + 0.3i z)  max |f(z)|/|z| = 0.994503,  |f'(0)| = 0.300000

    <matplotlib.legend.Legend object at 0x7fd7c69a1550>




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

   **Total running time of the script:** (0 minutes 0.092 seconds)


.. _sphx_glr_download_api_gallery_complex_analysis_conformal_maps_plot_04_schwarz_lemma.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/conformal_maps/plot_04_schwarz_lemma.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_04_schwarz_lemma.ipynb <plot_04_schwarz_lemma.ipynb>`

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

      :download:`Download Python source code: plot_04_schwarz_lemma.py <plot_04_schwarz_lemma.py>`

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

      :download:`Download zipped: plot_04_schwarz_lemma.zip <plot_04_schwarz_lemma.zip>`


.. only:: html

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

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