:orphan:

Examples
========

This gallery walks through every public feature of ``mathematicskit.complex_analysis``:
Euler's formula, the complex plane, and the Cauchy-Riemann equations;
contour integrals, Cauchy's integral theorem and formula, and
Liouville's theorem; residues, Laurent series, the argument principle,
and Rouché's theorem; essential singularities; conformal maps (Möbius
transformations, the Joukowski airfoil, and Schwarz's lemma); and
domain coloring of complex functions.

Each script in this gallery is self-contained and can be run directly with
``python examples/complex_analysis/<section>/<script>.py``.

Sections
--------

- **holomorphic** -- Euler's formula, the Wessel-Argand plane, and the
  Cauchy-Riemann equations.
- **contour_integrals** -- Cauchy's integral theorem and integral
  formula, and Liouville's theorem.
- **residues** -- the residue theorem, the argument principle, the
  fundamental theorem of algebra, Laurent series, and Rouché's theorem.
- **series** -- the Casorati-Weierstrass theorem on essential
  singularities.
- **conformal_maps** -- Riemann's mapping theorem via the Cayley
  transform, the Joukowski airfoil, Möbius transformations, and
  Schwarz's lemma.
- **domain_coloring** -- phase portraits of complex functions.


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Conformal maps
--------------

Möbius transformations, the Riemann mapping theorem, and the Joukowski airfoil.


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    <div class="sphx-glr-thumbcontainer" tooltip="Riemann (1851) asserted that every simply connected proper subdomain of the plane can be mapped conformally onto the unit disk. For the upper half-plane the map is explicit: the Cayley transform w = (z - i)/(z + i), a Möbius transformation. The image of the Cartesian grid is a family of circles that still meet at right angles, because conformal maps preserve angles.">

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  .. image:: /api/gallery/complex_analysis/conformal_maps/images/thumb/sphx_glr_plot_01_riemann_mapping_theorem_thumb.png
    :alt:

  :doc:`/api/gallery/complex_analysis/conformal_maps/plot_01_riemann_mapping_theorem`

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      <div class="sphx-glr-thumbnail-title">Riemann's mapping theorem: the upper half-plane mapped conformally onto the disk</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="J(z) = z + 1/z flattens the unit circle onto [-2, 2]. A circle through z = 1 that encloses -1 maps to an airfoil with a sharp trailing edge, where J&#x27;(1) = 0 and the map stops being conformal. Joukowski used this in 1910 to carry the solvable flow around a cylinder over to a wing profile.">

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  .. image:: /api/gallery/complex_analysis/conformal_maps/images/thumb/sphx_glr_plot_02_joukowski_airfoil_thumb.png
    :alt:

  :doc:`/api/gallery/complex_analysis/conformal_maps/plot_02_joukowski_airfoil`

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      <div class="sphx-glr-thumbnail-title">The Joukowski map: turning circles into airfoils</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="August Ferdinand Möbius&#x27;s 1855 theory of Kreisverwandtschaft (&quot;circle relationship&quot;) studied the maps w = (az + b)/(cz + d), ad - bc \ne 0. Every one of them sends circles and lines to circles and lines, where a line is a circle through \infty. This script maps a family of circles and checks that each image is again a circle by fitting one to it.">

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  .. image:: /api/gallery/complex_analysis/conformal_maps/images/thumb/sphx_glr_plot_03_mobius_circles_thumb.png
    :alt:

  :doc:`/api/gallery/complex_analysis/conformal_maps/plot_03_mobius_circles`

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      <div class="sphx-glr-thumbnail-title">Möbius transformations map circles to circles</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="If f maps the unit disk into itself with f(0) = 0, then |f(z)| \le |z| and |f&#x27;(0)| \le 1, with equality only for rotations 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.">

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  .. image:: /api/gallery/complex_analysis/conformal_maps/images/thumb/sphx_glr_plot_04_schwarz_lemma_thumb.png
    :alt:

  :doc:`/api/gallery/complex_analysis/conformal_maps/plot_04_schwarz_lemma`

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      <div class="sphx-glr-thumbnail-title">Schwarz's lemma: self-maps of the disk fixing 0 cannot expand</div>
    </div>


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    </div>

Contour integrals
-----------------

Cauchy's integral theorem and Cauchy's integral formula.


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    <div class="sphx-glr-thumbcontainer" tooltip="For f holomorphic inside and on a closed contour \gamma, \oint_\gamma f(z)\,dz = 0, whatever the shape of \gamma. 1/z shows what happens when the hypothesis fails: its integral is 2\pi i times the number of times \gamma winds around the singularity at 0.">

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  .. image:: /api/gallery/complex_analysis/contour_integrals/images/thumb/sphx_glr_plot_01_cauchy_integral_theorem_thumb.png
    :alt:

  :doc:`/api/gallery/complex_analysis/contour_integrals/plot_01_cauchy_integral_theorem`

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      <div class="sphx-glr-thumbnail-title">Cauchy's integral theorem: the integral of a holomorphic function around a closed contour is zero</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="For f holomorphic inside a contour \gamma,">

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  .. image:: /api/gallery/complex_analysis/contour_integrals/images/thumb/sphx_glr_plot_02_cauchy_integral_formula_thumb.png
    :alt:

  :doc:`/api/gallery/complex_analysis/contour_integrals/plot_02_cauchy_integral_formula`

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      <div class="sphx-glr-thumbnail-title">Cauchy's integral formula: boundary values determine f and every derivative</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Cauchy&#x27;s integral formula on the circle |z| = R gives the estimate |f&#x27;(0)| \le M(R)/R, where M(R) is the maximum of |f| on the circle. If f is entire and bounded, letting R \to \infty forces f&#x27; \equiv 0. So every nonconstant entire function, such as \sin z, is unbounded. Applied to 1/p(z), the theorem proves the fundamental theorem of algebra.">

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  .. image:: /api/gallery/complex_analysis/contour_integrals/images/thumb/sphx_glr_plot_03_liouville_theorem_thumb.png
    :alt:

  :doc:`/api/gallery/complex_analysis/contour_integrals/plot_03_liouville_theorem`

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      <div class="sphx-glr-thumbnail-title">Liouville's theorem: a bounded entire function is constant</div>
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Domain coloring
---------------

Phase portraits: coloring each point z by the phase and modulus of f(z).


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    <div class="sphx-glr-thumbcontainer" tooltip="Each point z is colored by w = f(z): hue for the phase \arg w, brightness bands for every doubling of |w|. A zero is where all colors meet counter-clockwise, a pole where they meet clockwise, and the order is the number of times each color appears around the point. The technique was named by Frank Farris in the 1990s and developed systematically by Elias Wegert.">

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  .. image:: /api/gallery/complex_analysis/domain_coloring/images/thumb/sphx_glr_plot_01_domain_coloring_thumb.png
    :alt:

  :doc:`/api/gallery/complex_analysis/domain_coloring/plot_01_domain_coloring`

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      <div class="sphx-glr-thumbnail-title">Domain coloring: reading zeros, poles, and branch cuts from a phase portrait</div>
    </div>


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    </div>

Holomorphic functions
---------------------

Euler's formula and the Cauchy-Riemann equations for complex differentiability.


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    <div class="sphx-glr-thumbcontainer" tooltip="Euler&#x27;s 1748 formula wraps the real line around the unit circle: e^{i\theta} is the point at angle \theta, and e^{i\pi} + 1 = 0. The domain coloring of e^z shows the consequence for the whole plane: the hue (phase) depends only on \operatorname{Im} z, so e^z repeats every 2\pi i.">

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  .. image:: /api/gallery/complex_analysis/holomorphic/images/thumb/sphx_glr_plot_01_euler_formula_thumb.png
    :alt:

  :doc:`/api/gallery/complex_analysis/holomorphic/plot_01_euler_formula`

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      <div class="sphx-glr-thumbnail-title">Euler's formula: e^{i theta} = cos theta + i sin theta</div>
    </div>


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

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  .. image:: /api/gallery/complex_analysis/holomorphic/images/thumb/sphx_glr_plot_02_cauchy_riemann_equations_thumb.png
    :alt:

  :doc:`/api/gallery/complex_analysis/holomorphic/plot_02_cauchy_riemann_equations`

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      <div class="sphx-glr-thumbnail-title">The Cauchy-Riemann equations: u_x = v_y, u_y = -v_x</div>
    </div>


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

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  .. image:: /api/gallery/complex_analysis/holomorphic/images/thumb/sphx_glr_plot_03_wessel_argand_plane_thumb.png
    :alt:

  :doc:`/api/gallery/complex_analysis/holomorphic/plot_03_wessel_argand_plane`

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      <div class="sphx-glr-thumbnail-title">Wessel and Argand's complex plane: multiplication rotates and scales</div>
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Residues
--------

The residue theorem and the argument principle.


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    <div class="sphx-glr-thumbcontainer" tooltip="For f meromorphic inside \gamma, \oint_\gamma f = 2\pi i \sum_k n(\gamma, z_k)\operatorname{Res}(f, z_k). Growing a circle past the poles of f(z) = e^{z}/((z - 1)(z + 2)(z - 3i)) adds each residue in turn, and the same theorem evaluates the real integral \int_{-\infty}^\infty dx/(1 + x^4) = \pi/\sqrt 2.">

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  .. image:: /api/gallery/complex_analysis/residues/images/thumb/sphx_glr_plot_01_residue_theorem_thumb.png
    :alt:

  :doc:`/api/gallery/complex_analysis/residues/plot_01_residue_theorem`

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      <div class="sphx-glr-thumbnail-title">Cauchy's residue theorem: the contour integral is 2 pi i times the enclosed residues</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="\frac{1}{2\pi i}\oint_\gamma f&#x27;/f = N - P, the number of zeros minus poles inside \gamma. Equivalently, the image curve f(\gamma) winds N - P times around 0. For a quintic, counting roots inside disks of growing radius needs no root finder at all.">

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  .. image:: /api/gallery/complex_analysis/residues/images/thumb/sphx_glr_plot_02_argument_principle_thumb.png
    :alt:

  :doc:`/api/gallery/complex_analysis/residues/plot_02_argument_principle`

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      <div class="sphx-glr-thumbnail-title">The argument principle: counting zeros minus poles by winding</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Gauss&#x27;s 1799 dissertation gave the first substantial proof that every polynomial of degree n has n complex roots. The modern topological proof counts windings: on a huge circle p(z) \approx z^n winds n times around 0, on a tiny circle about a non-root it winds 0 times, so as the circle grows the image curve must pass through 0 -- and the argument principle says it does so exactly n times.">

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  .. image:: /api/gallery/complex_analysis/residues/images/thumb/sphx_glr_plot_03_fundamental_theorem_of_algebra_thumb.png
    :alt:

  :doc:`/api/gallery/complex_analysis/residues/plot_03_fundamental_theorem_of_algebra`

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      <div class="sphx-glr-thumbnail-title">The fundamental theorem of algebra, by winding numbers</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Pierre Alphonse Laurent (1843, after Weierstrass in 1841) expanded a function holomorphic in an annulus r &lt; |z - a| &lt; R as \sum_{n=-\infty}^{\infty} a_n (z-a)^n, with every coefficient a contour integral a_n = \frac{1}{2\pi i}\oint f(z)(z-a)^{-n-1}dz. For f(z) = 1/((z-1)(z-2)) the expansion depends on the annulus: in 1 &lt; |z| &lt; 2 it is a_n = -2^{-n-1} for n \ge 0 and a_n = -1 for n &lt; 0.">

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  .. image:: /api/gallery/complex_analysis/residues/images/thumb/sphx_glr_plot_04_laurent_series_thumb.png
    :alt:

  :doc:`/api/gallery/complex_analysis/residues/plot_04_laurent_series`

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      <div class="sphx-glr-thumbnail-title">Laurent series: expansions with negative powers in an annulus</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Eugène Rouché (1862): if |g| &lt; |f| on a closed contour, then f and f + g have the same number of zeros inside it. Picture f(\gamma) as a person walking around a lamppost at 0 and g as the leash to a dog at f + g: a leash shorter than the distance to the post means the dog circles it the same number of times. Here z^5 + 3z + 1 has all five zeros in |z| &lt; 2 (dominated by z^5) and exactly one in |z| &lt; 1 (dominated by 3z).">

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  .. image:: /api/gallery/complex_analysis/residues/images/thumb/sphx_glr_plot_05_rouche_theorem_thumb.png
    :alt:

  :doc:`/api/gallery/complex_analysis/residues/plot_05_rouche_theorem`

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      <div class="sphx-glr-thumbnail-title">Rouché's theorem: counting zeros by domination</div>
    </div>


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    </div>

Series and singularities
------------------------

The behaviour of functions near isolated singularities.


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    <div class="sphx-glr-thumbcontainer" tooltip="An isolated singularity is removable, a pole, or essential, depending on whether the Laurent series has no negative powers, finitely many, or infinitely many. Casorati and Weierstrass showed that near an essential singularity the values of f are dense in the plane. For e^{1/z} at 0 we find a point within \delta = 10^{-3} of 0 where f hits an arbitrary target w, and the phase portrait shows every color packed into every neighbourhood of 0.">

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  .. image:: /api/gallery/complex_analysis/series/images/thumb/sphx_glr_plot_01_casorati_weierstrass_thumb.png
    :alt:

  :doc:`/api/gallery/complex_analysis/series/plot_01_casorati_weierstrass`

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      <div class="sphx-glr-thumbnail-title">The Casorati-Weierstrass theorem: near an essential singularity f comes close to every value</div>
    </div>


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    </div>


.. toctree::
   :hidden:
   :includehidden:


   /api/gallery/complex_analysis/conformal_maps/index.rst
   /api/gallery/complex_analysis/contour_integrals/index.rst
   /api/gallery/complex_analysis/domain_coloring/index.rst
   /api/gallery/complex_analysis/holomorphic/index.rst
   /api/gallery/complex_analysis/residues/index.rst
   /api/gallery/complex_analysis/series/index.rst


.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-gallery

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

      :download:`Download all examples in Python source code: complex_analysis_python.zip </api/gallery/complex_analysis/complex_analysis_python.zip>`

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

      :download:`Download all examples in Jupyter notebooks: complex_analysis_jupyter.zip </api/gallery/complex_analysis/complex_analysis_jupyter.zip>`


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

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