.. 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_03_wessel_argand_plane.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_03_wessel_argand_plane.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_03_wessel_argand_plane.py:


Wessel and Argand's complex plane: multiplication rotates and scales
====================================================================

Caspar Wessel (1797) and Jean-Robert Argand (1806) drew :math:`a + bi`
as the point :math:`(a, b)`. Addition is then vector addition, and
multiplication by :math:`w = re^{i\varphi}` rotates every point by
:math:`\varphi` and scales it by :math:`r`. Multiplying by :math:`i` is
a quarter turn, so :math:`i^2 = -1` is a half turn.

.. GENERATED FROM PYTHON SOURCE LINES 13-19

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.complex_analysis import map_grid
    from mathematicskit.complex_analysis.visualizers import plot_mapped_grid








.. GENERATED FROM PYTHON SOURCE LINES 20-22

Powers of i: four quarter turns
-------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 22-36

.. code-block:: Python


    fig, ax = plt.subplots(figsize=(5, 5))
    z = 1.0 + 0.0j
    for k in range(4):
        ax.annotate("", xy=(z.real, z.imag), xytext=(0, 0), arrowprops={"arrowstyle": "->", "lw": 2})
        ax.text(1.12 * z.real - 0.05, 1.12 * z.imag - 0.05, f"$i^{k}$")
        z *= 1j
    theta = np.linspace(0, 2 * np.pi, 200)
    ax.plot(np.cos(theta), np.sin(theta), "k:", lw=0.8)
    ax.set_xlim(-1.4, 1.4)
    ax.set_ylim(-1.4, 1.4)
    ax.set_aspect("equal")
    ax.set_title("Multiplying by i is a quarter turn")




.. image-sg:: /api/gallery/complex_analysis/holomorphic/images/sphx_glr_plot_03_wessel_argand_plane_001.png
   :alt: Multiplying by i is a quarter turn
   :srcset: /api/gallery/complex_analysis/holomorphic/images/sphx_glr_plot_03_wessel_argand_plane_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    Text(0.5, 1.0, 'Multiplying by i is a quarter turn')



.. GENERATED FROM PYTHON SOURCE LINES 37-39

Multiplication by w = 1 + i: rotate 45 degrees, scale by sqrt 2
----------------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 39-45

.. code-block:: Python


    w = 1.0 + 1.0j
    print(f"|w| = {abs(w):.6f}, arg w = {np.degrees(np.angle(w)):.1f} degrees")
    grid = map_grid(lambda z: w * z, x_range=(0.0, 1.0), y_range=(0.0, 1.0), n_lines=6)
    plot_mapped_grid(grid)




.. image-sg:: /api/gallery/complex_analysis/holomorphic/images/sphx_glr_plot_03_wessel_argand_plane_002.png
   :alt: plot 03 wessel argand plane
   :srcset: /api/gallery/complex_analysis/holomorphic/images/sphx_glr_plot_03_wessel_argand_plane_002.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    |w| = 1.414214, arg w = 45.0 degrees

    array([<Axes: xlabel='Re z', ylabel='Im z'>,
           <Axes: xlabel='Re w', ylabel='Im w'>], dtype=object)



.. GENERATED FROM PYTHON SOURCE LINES 46-48

Moduli multiply, arguments add
------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 48-54

.. code-block:: Python


    z1, z2 = 2.0 * np.exp(0.3j), 1.5 * np.exp(1.1j)
    print(f"|z1 z2| = {abs(z1 * z2):.6f}  vs  |z1||z2| = {abs(z1) * abs(z2):.6f}")
    print(f"arg(z1 z2) = {np.angle(z1 * z2):.6f}  vs  arg z1 + arg z2 = {np.angle(z1) + np.angle(z2):.6f}")

    plt.show()




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

 .. code-block:: none

    |z1 z2| = 3.000000  vs  |z1||z2| = 3.000000
    arg(z1 z2) = 1.400000  vs  arg z1 + arg z2 = 1.400000





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

   **Total running time of the script:** (0 minutes 0.172 seconds)


.. _sphx_glr_download_api_gallery_complex_analysis_holomorphic_plot_03_wessel_argand_plane.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_03_wessel_argand_plane.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_03_wessel_argand_plane.ipynb <plot_03_wessel_argand_plane.ipynb>`

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

      :download:`Download Python source code: plot_03_wessel_argand_plane.py <plot_03_wessel_argand_plane.py>`

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

      :download:`Download zipped: plot_03_wessel_argand_plane.zip <plot_03_wessel_argand_plane.zip>`


.. only:: html

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

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