.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/complex_analysis/holomorphic/plot_01_euler_formula.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_holomorphic_plot_01_euler_formula.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_holomorphic_plot_01_euler_formula.py:


Euler's formula: e^{i theta} = cos theta + i sin theta
======================================================

Euler's 1748 formula wraps the real line around the unit circle:
:math:`e^{i\theta}` is the point at angle :math:`\theta`, and
:math:`e^{i\pi} + 1 = 0`. The domain coloring of :math:`e^z` shows the
consequence for the whole plane: the hue (phase) depends only on
:math:`\operatorname{Im} z`, so :math:`e^z` repeats every :math:`2\pi i`.

.. GENERATED FROM PYTHON SOURCE LINES 13-19

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.complex_analysis import circle_contour, domain_coloring
    from mathematicskit.complex_analysis.visualizers import plot_domain_coloring








.. GENERATED FROM PYTHON SOURCE LINES 20-22

e^{i theta} traces the unit circle
----------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 22-38

.. code-block:: Python


    theta = np.linspace(0, 2 * np.pi, 9)
    unit_circle = circle_contour(0.0, 1.0)
    print("max |e^{i theta} - (cos theta + i sin theta)| =", np.max(np.abs(np.exp(1j * theta) - (np.cos(theta) + 1j * np.sin(theta)))))
    print("e^{i pi} + 1 =", np.exp(1j * np.pi) + 1)

    fig, ax = plt.subplots()
    circle = unit_circle.points()
    ax.plot(circle.real, circle.imag, "C0")
    for t in theta[:-1]:
        w = np.exp(1j * t)
        ax.plot([0, w.real], [0, w.imag], "C1", lw=0.8)
        ax.annotate(rf"$\theta={t / np.pi:.2g}\pi$", (w.real, w.imag), textcoords="offset points", xytext=(5, 5), fontsize=8)
    ax.set_aspect("equal")
    ax.set_title(r"$e^{i\theta} = \cos\theta + i\sin\theta$")




.. image-sg:: /api/gallery/complex_analysis/holomorphic/images/sphx_glr_plot_01_euler_formula_001.png
   :alt: $e^{i\theta} = \cos\theta + i\sin\theta$
   :srcset: /api/gallery/complex_analysis/holomorphic/images/sphx_glr_plot_01_euler_formula_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    max |e^{i theta} - (cos theta + i sin theta)| = 0.0
    e^{i pi} + 1 = 1.2246467991473532e-16j

    Text(0.5, 1.0, '$e^{i\\theta} = \\cos\\theta + i\\sin\\theta$')



.. GENERATED FROM PYTHON SOURCE LINES 39-41

e^z is periodic with period 2 pi i
----------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 41-45

.. code-block:: Python


    result = domain_coloring(np.exp, (-2, 2), (-2 * np.pi, 2 * np.pi), resolution=300)
    plot_domain_coloring(result, title=r"$e^z$: phase depends only on Im z")
    print("e^{z + 2 pi i} == e^z:", np.allclose(np.exp(result.z + 2j * np.pi), result.w))



.. image-sg:: /api/gallery/complex_analysis/holomorphic/images/sphx_glr_plot_01_euler_formula_002.png
   :alt: $e^z$: phase depends only on Im z
   :srcset: /api/gallery/complex_analysis/holomorphic/images/sphx_glr_plot_01_euler_formula_002.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    e^{z + 2 pi i} == e^z: True





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

   **Total running time of the script:** (0 minutes 0.250 seconds)


.. _sphx_glr_download_api_gallery_complex_analysis_holomorphic_plot_01_euler_formula.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/holomorphic/plot_01_euler_formula.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_euler_formula.ipynb <plot_01_euler_formula.ipynb>`

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

      :download:`Download Python source code: plot_01_euler_formula.py <plot_01_euler_formula.py>`

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

      :download:`Download zipped: plot_01_euler_formula.zip <plot_01_euler_formula.zip>`


.. only:: html

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

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