.. 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_02_cauchy_riemann_equations.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_02_cauchy_riemann_equations.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_02_cauchy_riemann_equations.py:


The Cauchy-Riemann equations: u_x = v_y, u_y = -v_x
===================================================

A function :math:`f = u + iv` is complex differentiable exactly where
its real and imaginary parts satisfy the Cauchy-Riemann equations. The
residual :math:`\max(|u_x - v_y|, |u_y + v_x|)` vanishes everywhere for
:math:`z^2` and :math:`e^z`, but not for :math:`\bar z`,
:math:`|z|^2`, or :math:`\operatorname{Re} z`. :math:`|z|^2` is the
interesting case: it satisfies the equations only at :math:`z = 0`.

.. 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_riemann, complex_derivative, complex_grid








.. GENERATED FROM PYTHON SOURCE LINES 20-22

Holomorphic vs. non-holomorphic functions
-----------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 22-36

.. code-block:: Python


    functions = {
        "z^2": lambda z: z**2,
        "exp(z)": np.exp,
        "conj(z)": np.conj,
        "|z|^2": lambda z: abs(z) ** 2,
        "Re(z)": lambda z: z.real,
    }
    z0 = 0.7 - 0.4j
    for name, f in functions.items():
        r = cauchy_riemann(f, z0)
        print(f"{name:8s} u_x={r.u_x:+.3f} v_y={r.v_y:+.3f} u_y={r.u_y:+.3f} v_x={r.v_x:+.3f}  residual={r.residual:.1e}")
    print("d/dz z^2 at z0:", complex_derivative(lambda z: z**2, z0), "= 2 z0 =", 2 * z0)





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

 .. code-block:: none

    z^2      u_x=+1.400 v_y=+1.400 u_y=+0.800 v_x=-0.800  residual=5.6e-11
    exp(z)   u_x=+1.855 v_y=+1.855 u_y=+0.784 v_x=-0.784  residual=5.6e-11
    conj(z)  u_x=+1.000 v_y=-1.000 u_y=+0.000 v_x=+0.000  residual=2.0e+00
    |z|^2    u_x=+1.400 v_y=+0.000 u_y=-0.800 v_x=+0.000  residual=1.4e+00
    Re(z)    u_x=+1.000 v_y=+0.000 u_y=+0.000 v_x=+0.000  residual=1.0e+00
    d/dz z^2 at z0: (1.400000000040258-0.8000000000230045j) = 2 z0 = (1.4-0.8j)




.. GENERATED FROM PYTHON SOURCE LINES 37-39

Where does the squared modulus satisfy the equations?
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 39-48

.. code-block:: Python


    z = complex_grid((-1, 1), (-1, 1), 41)
    residual = np.vectorize(lambda p: cauchy_riemann(lambda w: abs(w) ** 2, p).residual)(z)
    fig, ax = plt.subplots()
    image = ax.imshow(residual, origin="lower", extent=(-1, 1, -1, 1), cmap="viridis")
    fig.colorbar(image, ax=ax, label="Cauchy-Riemann residual")
    ax.set_title(r"$|z|^2$ is complex differentiable only at $z = 0$")
    ax.set_xlabel("Re z")
    ax.set_ylabel("Im z")



.. image-sg:: /api/gallery/complex_analysis/holomorphic/images/sphx_glr_plot_02_cauchy_riemann_equations_001.png
   :alt: $|z|^2$ is complex differentiable only at $z = 0$
   :srcset: /api/gallery/complex_analysis/holomorphic/images/sphx_glr_plot_02_cauchy_riemann_equations_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    Text(48.922222222222274, 0.5, 'Im z')




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

   **Total running time of the script:** (0 minutes 0.123 seconds)


.. _sphx_glr_download_api_gallery_complex_analysis_holomorphic_plot_02_cauchy_riemann_equations.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_02_cauchy_riemann_equations.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_cauchy_riemann_equations.ipynb <plot_02_cauchy_riemann_equations.ipynb>`

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

      :download:`Download Python source code: plot_02_cauchy_riemann_equations.py <plot_02_cauchy_riemann_equations.py>`

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

      :download:`Download zipped: plot_02_cauchy_riemann_equations.zip <plot_02_cauchy_riemann_equations.zip>`


.. only:: html

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

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