Breakthroughs in Complex Analysis
=================================


.. include:: /_generated/nav/complex_analysis.rst

.. epigraph::

   "The shortest path between two truths in the real domain passes
   through the complex domain."
   -- attributed to Jacques Hadamard

Complex numbers entered mathematics as a device for solving cubic
equations. Complex analysis began when mathematicians started to treat
them as a *plane* on which functions could be differentiated and
integrated. Differentiability turns out to be far more restrictive
there than on the real line, and that rigidity is what makes the
theory powerful. This chronology traces the ideas behind
:mod:`mathematicskit.complex_analysis`, from Euler's formula, through
Cauchy's integral calculus, to Riemann's geometric view of functions as
conformal maps.

.. contents:: Timeline
   :local:
   :depth: 1

1748 -- Euler's Formula
-----------------------

In his *Introductio in analysin infinitorum* Leonhard Euler compared
the power series of the exponential, sine, and cosine and found

.. math::

   e^{i\theta} = \cos\theta + i\sin\theta,

which makes :math:`e^{i\pi} + 1 = 0` a special case. The formula
turned trigonometry into algebra with exponentials, and it gave every
complex number a polar form :math:`z = re^{i\theta}`. It also shows
that the complex exponential is periodic, with period :math:`2\pi i`.

*Implementation:* :func:`mathematicskit.complex_analysis.systems.contours.circle_contour`
parametrizes circles as :math:`c + re^{it}`. The example traces
:math:`e^{i\theta}` around the unit circle and uses
:func:`~mathematicskit.complex_analysis.systems.domain_coloring.domain_coloring`
to show the :math:`2\pi i` periodicity of :math:`e^z`.

*References:* L. Euler, *Introductio in analysin infinitorum*, vol. 1
(Lausanne, 1748), Ch. 8, §138.

.. minigallery:: ../../examples/complex_analysis/holomorphic/plot_01_euler_formula.py

1797-1806 -- Wessel and Argand's Complex Plane
----------------------------------------------

For two centuries after Rafael Bombelli used them to solve cubics in
1572, "imaginary" numbers were rules for calculation with no meaning of
their own. The Norwegian-Danish surveyor Caspar Wessel, in a paper read
to the Royal Danish Academy in 1797, and the Paris bookkeeper
Jean-Robert Argand, in a pamphlet of 1806, independently gave them one:
:math:`a + bi` is the point :math:`(a, b)` of a plane. Addition is
vector addition, and multiplication by :math:`re^{i\varphi}` rotates by
:math:`\varphi` and scales by :math:`r`, so

.. math::

   |z_1 z_2| = |z_1|\,|z_2|, \qquad \arg(z_1 z_2) = \arg z_1 + \arg z_2 .

Multiplying by :math:`i` is a quarter turn, which makes
:math:`i^2 = -1` a half turn. Wessel's paper went unnoticed for a
century. Gauss's 1831 advocacy made the plane standard, and it is the
setting of everything else on this page.

*Implementation:* :func:`mathematicskit.complex_analysis.systems.conformal_maps.map_grid`
maps a coordinate grid through :math:`z \mapsto wz`, showing
multiplication as a rotation and scaling, drawn with
:func:`~mathematicskit.complex_analysis.visualizers.plots.plot_mapped_grid`.

*References:* C. Wessel, "Om Directionens analytiske Betegning,"
*Nye Samling af det Kongelige Danske Videnskabernes Selskabs Skrifter*
5 (1799), 469-518 (read 1797); J.-R. Argand, *Essai sur une manière de
représenter les quantités imaginaires dans les constructions
géométriques* (Paris, 1806).

.. minigallery:: ../../examples/complex_analysis/holomorphic/plot_03_wessel_argand_plane.py

1799 -- The Fundamental Theorem of Algebra
------------------------------------------

Every polynomial of degree :math:`n \ge 1` with complex coefficients
has exactly :math:`n` complex roots, counted with multiplicity. Jean le
Rond d'Alembert (1746), Euler, and Lagrange all attempted proofs. Carl
Friedrich Gauss criticized them in his 1799 doctoral dissertation and
gave a largely geometric proof of his own, which still assumed a
topological fact about curves that was only proved in 1920; he
published three further proofs, the last in 1849. The cleanest modern
proof counts windings. On a large circle :math:`|z| = R`,
:math:`p(z) \approx z^n` winds :math:`n` times around 0. On a tiny
circle around a point that is not a root, it winds 0 times. The winding
number can only change when the image curve crosses 0, and by the
argument principle it equals the number of roots inside.

*Implementation:* :func:`mathematicskit.complex_analysis.systems.residues.argument_principle`
counts the winding of :math:`p(\gamma)` about 0 on circles from
:func:`~mathematicskit.complex_analysis.systems.contours.circle_contour`.
The example checks the count against :func:`numpy.roots` as the radius
grows to enclose all :math:`n` roots.

*References:* C. F. Gauss, *Demonstratio nova theorematis omnem
functionem algebraicam rationalem integram unius variabilis in factores
reales primi vel secundi gradus resolvi posse* (Helmstedt, 1799); B.
Fine and G. Rosenberger, *The Fundamental Theorem of Algebra* (Springer,
1997).

.. minigallery:: ../../examples/complex_analysis/residues/plot_03_fundamental_theorem_of_algebra.py

1814 -- The Cauchy-Riemann Equations
------------------------------------

Jean le Rond d'Alembert (1752, in fluid dynamics) and Euler (1777) had
met the equations

.. math::

   \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \qquad
   \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}

for the real and imaginary parts of :math:`f = u + iv`. Augustin-Louis
Cauchy derived them in his 1814 memoir on definite integrals, published
in 1827. Bernhard Riemann's 1851 dissertation made them the
*definition* of a complex function: :math:`f` is complex differentiable
exactly when its parts satisfy the equations. Functions like
:math:`\bar z` or :math:`|z|^2`, which are smooth as maps of the plane,
fail them.

*Implementation:* :func:`mathematicskit.complex_analysis.systems.holomorphic.cauchy_riemann`
estimates :math:`u_x, u_y, v_x, v_y` with
:func:`~mathematicskit.calculus.systems.finite_differences.central_difference`
and reports the residual of the equations.
:func:`~mathematicskit.complex_analysis.systems.holomorphic.complex_derivative`
estimates :math:`f'(z)`. The tests confirm that the residual vanishes
for :math:`z^3`, :math:`e^z`, :math:`\sin z`, and :math:`1/z`, and has
the predicted nonzero values for :math:`\bar z`, :math:`|z|`, and
:math:`\operatorname{Re} z`.

*References:* A.-L. Cauchy, "Mémoire sur les intégrales définies,"
*Mémoires présentés par divers savants à l'Académie royale des
sciences* 1 (1827), 599-799 (read 1814); B. Riemann, *Grundlagen für
eine allgemeine Theorie der Functionen einer veränderlichen complexen
Grösse* (Göttingen, 1851).

.. minigallery:: ../../examples/complex_analysis/holomorphic/plot_02_cauchy_riemann_equations.py

1825 -- Cauchy's Integral Theorem
---------------------------------

In a memoir on definite integrals "taken between imaginary limits,"
Cauchy showed that the integral of a holomorphic function between two
points does not depend on the path, provided no singularity lies
between the paths. Equivalently, around any closed contour
:math:`\gamma` bounding a region where :math:`f` is holomorphic,

.. math::

   \oint_\gamma f(z)\,dz = 0.

Cauchy's proof assumed :math:`f'` continuous. Édouard Goursat removed
that assumption in 1900. The theorem fails as soon as a singularity is
enclosed: :math:`\oint dz/z = 2\pi i` around the origin, which is the
seed of the residue calculus.

*Implementation:* :func:`mathematicskit.complex_analysis.systems.contours.contour_integral`
integrates :math:`f(\gamma(t))\gamma'(t)` with :func:`scipy.integrate.quad`
(``complex_func=True``) on each smooth piece of a
:class:`~mathematicskit.complex_analysis.core.base.Contour` built by
:func:`~mathematicskit.complex_analysis.systems.contours.circle_contour`
or :func:`~mathematicskit.complex_analysis.systems.contours.polygon_contour`.
:func:`~mathematicskit.complex_analysis.systems.contours.winding_number`
computes :math:`\frac{1}{2\pi i}\oint dz/(z - z_0)`. The tests check
:math:`\oint z^n\,dz = 0` for :math:`n \ne -1`, the integral of an
entire function around a triangle, and Green's-theorem area
:math:`\oint \bar z\,dz = 2i\cdot\text{area}`.

*References:* A.-L. Cauchy, *Mémoire sur les intégrales définies,
prises entre des limites imaginaires* (Paris, 1825); E. Goursat, "Sur
la définition générale des fonctions analytiques, d'après Cauchy,"
*Transactions of the American Mathematical Society* 1 (1900), 14-16.

.. minigallery:: ../../examples/complex_analysis/contour_integrals/plot_01_cauchy_integral_theorem.py

1826 -- Cauchy's Calculus of Residues
-------------------------------------

Cauchy introduced the *residue* of a function at a pole, the
coefficient :math:`a_{-1}` of :math:`1/(z - z_0)` in its expansion
there, as a "new kind of calculus analogous to the infinitesimal
calculus." The integral around a closed contour then reduces to a sum
over the enclosed singularities:

.. math::

   \oint_\gamma f(z)\,dz = 2\pi i \sum_k n(\gamma, z_k)\operatorname{Res}(f, z_k).

The residue theorem evaluates many real integrals that resist every
real-variable technique, by closing the real line with a large arc.

*Implementation:* :func:`mathematicskit.complex_analysis.systems.residues.residue`
computes :math:`a_{-1}` on a small circle with the periodic
trapezoidal rule, which converges geometrically for these integrands.
:func:`~mathematicskit.complex_analysis.systems.residues.residue_theorem`
computes both sides of the theorem in a
:class:`~mathematicskit.complex_analysis.core.base.ResidueTheoremResult`.
The tests check residues at simple poles, at a fourth-order pole, and at
the essential singularity of :math:`e^{1/z}`, and they evaluate
:math:`\int_{\mathbb R} dx/(1 + x^2) = \pi` by residues.

*References:* A.-L. Cauchy, "Sur un nouveau genre de calcul analogue au
calcul infinitésimal," *Exercices de mathématiques* 1 (1826), 11-24.

.. minigallery:: ../../examples/complex_analysis/residues/plot_01_residue_theorem.py

1831 -- Cauchy's Integral Formula
---------------------------------

In exile in Turin, Cauchy found that a holomorphic function inside a
contour is determined entirely by its values on the contour:

.. math::

   f^{(n)}(z_0) = \frac{n!}{2\pi i}\oint_\gamma \frac{f(z)}{(z - z_0)^{n+1}}\,dz.

It follows that a function differentiable once in the complex sense is
differentiable infinitely often and equals its Taylor series in any
disk where it is holomorphic. Cauchy used these results for his
"calculus of limits," which bounded the errors of power-series
solutions in celestial mechanics.

*Implementation:* :func:`mathematicskit.complex_analysis.systems.contours.cauchy_integral_formula`
evaluates the formula for any :math:`n` by
:func:`~mathematicskit.complex_analysis.systems.contours.contour_integral`.
The tests recover :math:`e^{z_0}` and its first three derivatives from
values on a circle, and :math:`\cos z_0` from :math:`\sin z` on a
square.

*References:* A.-L. Cauchy, "Sur la mécanique céleste et sur un nouveau
calcul appelé calcul des limites" (Turin, 1831); F. Smithies, *Cauchy
and the Creation of Complex Function Theory* (Cambridge University
Press, 1997), Ch. 6.

.. minigallery:: ../../examples/complex_analysis/contour_integrals/plot_02_cauchy_integral_formula.py

1843 -- Laurent Series
----------------------

Near an isolated singularity a function has no Taylor series, but it
still has an expansion if negative powers are allowed. Pierre Alphonse
Laurent submitted the theorem to the Paris Academy in 1843; Karl
Weierstrass had found it in 1841 in a paper not published until 1894.
A function holomorphic in an annulus :math:`r < |z - a| < R` equals

.. math::

   f(z) = \sum_{n=-\infty}^{\infty} a_n (z-a)^n, \qquad
   a_n = \frac{1}{2\pi i}\oint_{|z-a|=\rho} \frac{f(z)}{(z-a)^{n+1}}\,dz,

for any :math:`r < \rho < R`. The coefficient :math:`a_{-1}` is the
residue, and the negative-power part classifies the singularity:
removable, a pole, or essential. The same function has different
expansions in different annuli, separated by its singularities.

*Implementation:* :func:`mathematicskit.complex_analysis.systems.contours.contour_integral`
computes each coefficient directly from the integral formula. The
example recovers the closed-form coefficients of
:math:`1/((z-1)(z-2))` in the three annuli around 0 and shows the
partial sums converging inside :math:`1 < |z| < 2`.
:func:`~mathematicskit.complex_analysis.systems.laurent.laurent_coefficients`
computes every coefficient at once: the trapezoidal rule on a circle
turns the integrals into one :func:`numpy.fft.fft`. The tests check it
against the Taylor series of :math:`e^z`, the two expansions of
:math:`1/(z(1-z))`, and the residue.

*References:* P. A. Laurent, "Extension du théorème de M. Cauchy
relatif à la convergence du développement d'une fonction suivant les
puissances ascendantes de la variable," *Comptes Rendus* 17 (1843),
348-349 (report by Cauchy); K. Weierstrass, "Darstellung einer
analytischen Function einer complexen Veränderlichen, deren absoluter
Betrag zwischen zwei gegebenen Grenzen liegt" (1841), in
*Mathematische Werke*, vol. 1 (Berlin, 1894), 51-66.

.. minigallery:: ../../examples/complex_analysis/residues/plot_04_laurent_series.py

1844 -- Liouville's Theorem
---------------------------

A function holomorphic on the whole plane (an *entire* function) that
is bounded must be constant. Joseph Liouville stated the result in 1844
for doubly periodic functions, and Cauchy published a proof of the
general statement the same year. It follows from Cauchy's integral
formula on a circle of radius :math:`R` around :math:`a`, which gives
the *Cauchy estimates*

.. math::

   |f^{(n)}(a)| \le \frac{n!\,M(R)}{R^n}, \qquad M(R) = \max_{|z-a|=R}|f(z)| .

If :math:`|f| \le M` everywhere, letting :math:`R \to \infty` forces
:math:`f'(a) = 0` at every point. So :math:`\sin z` and :math:`e^z`,
bounded on the real line, must grow without bound off it. The theorem
also gives a two-line proof of the fundamental theorem of algebra: if
:math:`p` had no root, :math:`1/p` would be bounded and entire.

*Implementation:* :func:`mathematicskit.complex_analysis.systems.contours.cauchy_integral_formula`
computes derivatives from contour integrals. The example evaluates the
Cauchy bounds for a polynomial and for :math:`\sin z` on growing
circles.

*References:* J. Liouville, lectures of 1847 published by C. W.
Borchardt, "Leçons sur les fonctions doublement périodiques," *Journal
für die reine und angewandte Mathematik* 88 (1880), 277-310; A.-L.
Cauchy, "Mémoires sur les fonctions complémentaires," *Comptes Rendus*
19 (1844), 1377-1384.

.. minigallery:: ../../examples/complex_analysis/contour_integrals/plot_03_liouville_theorem.py

1851 -- Riemann's Mapping Theorem
---------------------------------

Riemann's dissertation treated complex functions geometrically, as
maps of one region of the plane onto another. Where :math:`f' \ne 0`,
such a map rotates and scales every small neighbourhood by
:math:`f'(z)`, so it preserves angles: it is *conformal*. Riemann
asserted that every simply connected region other than the whole plane
can be mapped conformally and one-to-one onto the unit disk. His
argument relied on the Dirichlet principle, which Karl Weierstrass
criticized. William Fogg Osgood gave the first rigorous proof in 1900.
For the upper half-plane the map is explicit: the Cayley transform
:math:`w = (z - i)/(z + i)`.

*Implementation:* :func:`mathematicskit.complex_analysis.systems.conformal_maps.mobius_transform`
evaluates Möbius transformations :math:`(az + b)/(cz + d)`, including
the Cayley transform, and
:func:`~mathematicskit.complex_analysis.systems.conformal_maps.map_grid`
maps a coordinate grid through any function. The tests check that the
Cayley transform sends the real axis to the unit circle and the upper
half-plane into the disk, that composition corresponds to the matrix
product, that the cross-ratio is invariant, and that a conformal map
preserves right angles.

*References:* B. Riemann, *Grundlagen für eine allgemeine Theorie der
Functionen einer veränderlichen complexen Grösse* (Göttingen, 1851);
W. F. Osgood, "On the existence of the Green's function for the most
general simply connected plane region," *Transactions of the American
Mathematical Society* 1 (1900), 310-314.

.. minigallery:: ../../examples/complex_analysis/conformal_maps/plot_01_riemann_mapping_theorem.py

1855 -- Möbius Transformations
------------------------------

August Ferdinand Möbius's 1855 memoir on *Kreisverwandtschaft*
("circle relationship") studied the maps of the extended plane

.. math::

   w = \frac{az + b}{cz + d}, \qquad ad - bc \ne 0 .

They are the only one-to-one conformal maps of the Riemann sphere onto
itself, and they send circles and lines to circles and lines, where a
line is a circle through :math:`\infty`. A circle through the pole
:math:`z = -d/c` becomes a line. Composition corresponds to multiplying
the matrices :math:`\begin{pmatrix} a & b\\ c & d\end{pmatrix}`, and
the cross-ratio of four points is invariant. Felix Klein's 1872
Erlangen program later recognized them as the symmetries of inversive
geometry, and Henri Poincaré used them as the isometries of hyperbolic
geometry.

*Implementation:* :func:`mathematicskit.complex_analysis.systems.conformal_maps.mobius_transform`
evaluates the map on circles from
:func:`~mathematicskit.complex_analysis.systems.contours.circle_contour`.
The example fits a circle to each image to a residual of about
:math:`10^{-15}`, and shows that a circle through the pole maps to a
line.
:func:`~mathematicskit.complex_analysis.systems.conformal_maps.classify_mobius`
sorts a transformation into elliptic, parabolic, hyperbolic, or
loxodromic by :math:`(a+d)^2/(ad-bc)` and returns its fixed points.

*References:* A. F. Möbius, "Die Theorie der Kreisverwandtschaft in
rein geometrischer Darstellung" (1855), in *Gesammelte Werke*, vol. 2
(Leipzig: Hirzel, 1886); T. Needham, *Visual Complex Analysis* (Oxford,
1997), Ch. 3.

.. minigallery:: ../../examples/complex_analysis/conformal_maps/plot_03_mobius_circles.py

1855 -- The Argument Principle
------------------------------

In Turin in 1831 Cauchy counted the roots of an equation inside a
contour by following how the argument of the function changes around
it. In 1855 he published the general form, counting zeros and poles
together:

.. math::

   \frac{1}{2\pi i}\oint_\gamma \frac{f'(z)}{f(z)}\,dz = N - P.

The left side is the winding number of the image curve
:math:`f(\gamma)` about 0. Eugène Rouché's 1862 theorem and the Nyquist
stability criterion of control theory both follow from it.

*Implementation:* :func:`mathematicskit.complex_analysis.systems.residues.argument_principle`
counts :math:`N - P` by integrating :math:`f'/f` when :math:`f'` is
given. Otherwise it unwraps the phase of :math:`f` along the contour
(:func:`numpy.unwrap`). The tests count zeros and poles with
multiplicity by both methods, and match root counts in the unit disk
against :func:`numpy.roots`.

*References:* F. Smithies, *Cauchy and the Creation of Complex Function
Theory* (Cambridge University Press, 1997), 177; L. V. Ahlfors,
*Complex Analysis*, 3rd ed. (McGraw-Hill, 1979), Ch. 4, Sec. 5.2.

.. minigallery:: ../../examples/complex_analysis/residues/plot_02_argument_principle.py

1862 -- Rouché's Theorem
------------------------

Eugène Rouché, in an 1862 memoir on Lagrange's series, proved that if
:math:`|g(z)| < |f(z)|` everywhere on a closed contour, then :math:`f`
and :math:`f + g` have the same number of zeros inside it. The
argument principle explains why. The image curves :math:`f(\gamma)` and
:math:`(f+g)(\gamma)` are like a person walking around a lamppost at 0
and a dog on a leash of length :math:`|g|`. If the leash is always
shorter than the distance to the post, the dog circles the post exactly
as many times as the person does. Choosing which term dominates on
which circle locates roots without computing them: :math:`z^5 + 3z + 1`
has all five roots in :math:`|z| < 2`, where :math:`z^5` dominates, and
exactly one in :math:`|z| < 1`, where :math:`3z` does.

*Implementation:* :func:`mathematicskit.complex_analysis.systems.residues.argument_principle`
counts zeros of the dominant term and of the full function on each
circle. The example checks the domination inequality and compares the
counts with :func:`numpy.roots`.
:func:`~mathematicskit.complex_analysis.systems.residues.rouche_condition`
checks the domination inequality on sampled contour points.

*References:* E. Rouché, "Mémoire sur la série de Lagrange," *Journal
de l'École Polytechnique* 22 (1862), 193-224; L. V. Ahlfors, *Complex
Analysis*, 3rd ed. (McGraw-Hill, 1979), Ch. 4, Sec. 5.2.

.. minigallery:: ../../examples/complex_analysis/residues/plot_05_rouche_theorem.py

1902-1910 -- The Kutta-Joukowski Airfoil
----------------------------------------

Potential flow around a circular cylinder is easy to write down with
complex functions, but a cylinder is not a wing. Wilhelm Kutta (1902)
and Nikolai Joukowski (1906) showed that the lift on a body equals
:math:`\rho V \Gamma`, where :math:`\Gamma` is the circulation, fixed
by requiring smooth flow off a sharp trailing edge. In 1910 Joukowski
used the conformal map

.. math::

   J(z) = z + \frac{c^2}{z}

to carry the cylinder flow over to a wing section. It flattens the
circle :math:`|z| = c` onto the segment :math:`[-2c, 2c]`, and a
slightly shifted circle through :math:`z = c` maps to an airfoil whose
sharp trailing edge sits where :math:`J'(c) = 0` and the map stops
being conformal. Because conformal maps carry solutions of Laplace's
equation to solutions, the flow around the airfoil is known exactly.
These profiles were the first wing sections designed by theory.

*Implementation:* :func:`mathematicskit.complex_analysis.systems.conformal_maps.joukowski_map`
maps circles from
:func:`~mathematicskit.complex_analysis.systems.contours.circle_contour`
to airfoils, and
:func:`~mathematicskit.complex_analysis.systems.holomorphic.complex_derivative`
confirms that :math:`J'` vanishes at the trailing edge.

*References:* W. M. Kutta, "Auftriebskräfte in strömenden
Flüssigkeiten," *Illustrierte Aeronautische Mitteilungen* 6 (1902),
133-135; N. E. Joukowski, "Über die Konturen der Tragflächen der
Drachenflieger," *Zeitschrift für Flugtechnik und
Motorluftschiffahrt* 1 (1910), 281-284.

.. minigallery:: ../../examples/complex_analysis/conformal_maps/plot_02_joukowski_airfoil.py

1998-2012 -- Domain Coloring and Phase Portraits
------------------------------------------------

The graph of a complex function lives in four real dimensions, so it
cannot be drawn directly. Computer graphics made a substitute
practical: color each point :math:`z` of the domain by the value
:math:`f(z)`, with hue giving the argument and brightness the modulus.
Frank Farris named the technique *domain coloring* in 1998, and Elias
Wegert's *Visual Complex Functions* (2012) developed these *phase
portraits* into a systematic tool. Much of the theory on this page can
be read off such a picture. Around a zero of order :math:`k` every hue
appears :math:`k` times counter-clockwise, around a pole clockwise,
branch cuts show as color discontinuities, and essential singularities
show every color infinitely often, as the Casorati-Weierstrass theorem
predicts.

*Implementation:* :func:`mathematicskit.complex_analysis.systems.domain_coloring.domain_coloring`
samples :math:`f` on a grid from
:func:`~mathematicskit.complex_analysis.utils.grids.complex_grid` and
builds the HSV image, drawn with
:func:`~mathematicskit.complex_analysis.visualizers.plots.plot_domain_coloring`.

*References:* F. A. Farris, review of T. Needham, *Visual Complex
Analysis*, *American Mathematical Monthly* 105 (1998), 570-576; E.
Wegert, *Visual Complex Functions: An Introduction with Phase
Portraits* (Basel: Birkhäuser, 2012).

.. minigallery:: ../../examples/complex_analysis/domain_coloring/plot_01_domain_coloring.py