{{Short description|Branch of mathematics studying functions of a complex variable}}
{{distinguish|Complexity analysis }}
{{More footnotes|date=March 2021}}
{{Complex analysis sidebar}}

'''Complex analysis''', traditionally known as the '''theory of functions of a complex variable''', is the branch of [[mathematical analysis]] that studies complex-valued functions of one or more complex variables. It is used in many branches of mathematics, including [[functional analysis]], [[algebraic geometry]], [[number theory]], [[analytic combinatorics]], and [[applied mathematics]], as well as in [[physics]], including the branches of [[hydrodynamics]], [[thermodynamics]], [[quantum mechanics]], and [[twistor theory]]. It also has applications in engineering fields, such as [[nuclear engineering|nuclear]], [[aerospace engineering|aerospace]], [[mechanical engineering|mechanical]], and [[electrical engineering]].<ref>{{Cite web|url=https://gateway.newton.ac.uk/event/ofbw51|title=Industrial Applications of Complex Analysis|date=October 30, 2019|access-date=November 20, 2023|website=Newton Gateway to Mathematics}}</ref>

Complex analysis primarily studies [[holomorphic functions]], which are [[differentiable function]]s of a complex variable. In contrast with a differentiable real function, a holomorphic function is always [[infinitely differentiable]] and equal to the sum of its [[Taylor series]] in some [[neighborhood (mathematics)|neighborhood]] of each point of its [[domain of a function|domain]]. This makes methods and results of complex analysis significantly different from those of real analysis. Complex functions also behave very differently under [[contour integration]]: the integral of a holomorphic function over a contour in the complex plane does not depend on the details of the contour, only on how it winds around the singularities of the function. Being able to move a given contour to a more suitable one often leads to major simplifications in proofs.

The theory of [[several complex variables]] generalizes one-variable complex function theory to more than one complex dimension. While many of the techniques of a single complex variable are used and generalized in this setting, the theory of several complex variables makes use of additional techniques such as [[Banach algebra]]s and [[sheaf theory]]. It is often concerned with questions of interest in [[algebraic geometry]] and [[symmetric spaces]]. 

== History ==
[[File:Augustin-Louis Cauchy 1901.jpg|thumb|170px|[[Augustin-Louis Cauchy]], one of the founders of complex analysis]]
Complex analysis is one of the classical branches in mathematics, with roots in the 18th century and just prior. Important mathematicians associated with complex numbers include [[Euler]], [[Carl Friedrich Gauss|Gauss]], [[Bernhard Riemann|Riemann]], [[Cauchy]], [[Weierstrass]], and many more in the 20th century. Complex analysis, in particular the theory of [[conformal mapping]]s, has many physical applications and is also used throughout [[analytic number theory]]. In modern times, it has become very popular through a new boost from [[complex dynamics]] and the pictures of [[fractal]]s produced by iterating [[holomorphic functions]].  Another important application of complex analysis is in [[string theory]] which examines conformal invariants in [[quantum field theory]].

== Complex functions ==
[[Image:Exponentials_of_complex_number_within_unit_circle-2.svg|thumb|right|320px|An [[exponentiation|exponential]] function {{math|''A''<sup>''n''</sup>}} of a discrete ([[integer]]) variable {{mvar|n}}, similar to [[geometric progression]]]]<!-- I detest clueless Fr.wikipedia with their plain-ISO-8859-text-only notation: can somebody find an image of the same, but with a good symbolic notation? -->
A [[complex function]] is a [[function (mathematics)|function]] from [[complex number]]s to complex numbers. In other words, it is a function that has a subset of the complex numbers as a [[Domain of a function|domain]] and the complex numbers as a [[codomain]]. Complex functions are generally assumed to have a domain that contains a nonempty [[open subset]] of the [[complex plane]].

For any complex function, the values <math>z</math> from the domain and their images <math>f(z)</math> in the range may be separated into [[Real number|real]] and [[Imaginary number|imaginary]] parts:

: <math>z=x+iy \quad \text{ and } \quad f(z) = f(x+iy)=u(x,y)+iv(x,y),</math>

where <math>x,y,u(x,y),v(x,y)</math> are all real-valued.

In other words, a complex function <math>f:\mathbb{C}\to\mathbb{C}</math> may be decomposed into two real-valued functions (<math>u</math>, <math>v</math>) of two real variables (<math>x</math>, <math>y</math>):
: <math>u:\mathbb{R}^2\to\mathbb{R} \quad</math> and <math>\quad v:\mathbb{R}^2\to\mathbb{R}.</math>

A complex [[Function (mathematics)|function]] is [[continuous function|continuous]] if and only if its associated [[vector-valued function]] of two variables is also continuous. However, this identification does not extend to [[differentiability]]. The definition of the [[derivative]] of a complex function is very similar to that of a real function, but the differentiability of the associated real function of two variables does not imply that the derivative of the complex function exists. In particular, if a complex function has a derivative, it has derivatives of every order and equals the sum of its [[Taylor series]] in a [[neighborhood (mathematics)|neighborhood]] of every point of its domain. 

It follows that two differentiable functions that are equal in a [[neighborhood (mathematics)|neighborhood]] of a point are equal on the intersection of their domain if the domains are [[connected space|connected]]. The latter property is the basis of the principle of [[analytic continuation]] which allows extending every real or complex [[analytic function]] in a unique way for getting a complex analytic function whose domain is the whole complex plane with a finite number of [[arc (geometry)|curve arc]]s removed. Many basic and [[special functions|special]] complex functions are defined in this way, including the [[exponential function#Complex plane|complex exponential function]], [[complex logarithm|complex logarithm functions]], and [[trigonometric functions#In the complex plane|trigonometric functions]].

== Holomorphic functions ==
{{main|Holomorphic function}}
Complex functions that are [[differentiable]] at every point of an [[open set|open subset]] <math>\Omega</math> of the complex plane are said to be ''holomorphic on'' {{nowrap|<math>\Omega</math>.}} In the context of complex analysis, the derivative of <math>f</math> at <math>z_0</math> is defined to be<ref>{{cite book |last1=Rudin |first1=Walter |title=Real and Complex Analysis |date=1987 |publisher=McGraw-Hill Education |isbn=978-0-07-054234-1 |page=197 |url=https://59clc.files.wordpress.com/2011/01/real-and-complex-analysis.pdf#page=212 |language=en}}</ref>

: <math>f'(z_0) = \lim_{z \to z_0} \frac{f(z)-f(z_0)}{z-z_0}.</math>

Superficially, this definition is formally analogous to that of the derivative of a real function.  However, complex derivatives and differentiable functions behave in significantly different ways compared to their real counterparts.  In particular, for this limit to exist, the value of the difference quotient must approach the same complex number, regardless of the manner in which we approach <math>z_0</math> in the complex plane.  Consequently, complex differentiability has much stronger implications than real differentiability. For instance, holomorphic functions are [[infinitely differentiable]], whereas the existence of the ''n''th derivative need not imply the existence of the (''n'' + 1)th derivative for real functions.  Furthermore, all holomorphic functions satisfy the stronger condition of [[analytic function|analyticity]], meaning that the function is, at every point in its domain, locally given by a convergent power series. In essence, this means that functions holomorphic on <math>\Omega</math> can be approximated arbitrarily well by polynomials in some neighborhood of every point in <math>\Omega</math>. This stands in sharp contrast to differentiable real functions; there are infinitely differentiable real functions that are ''nowhere'' analytic; see {{slink|Non-analytic smooth function|A smooth function which is nowhere real analytic}}.

Most elementary functions, including the [[exponential function]], the [[trigonometric function]]s, and all [[polynomial|polynomial functions]], extended appropriately to complex arguments as functions {{nowrap|<math>\mathbb{C}\to\mathbb{C}</math>,}} are holomorphic over the entire complex plane, making them ''[[entire functions]]'', while rational functions <math>p/q</math>, where ''p'' and ''q'' are polynomials, are holomorphic on domains that exclude points where ''q'' is zero.  Such functions that are holomorphic everywhere except a set of isolated points are known as ''meromorphic functions''.  On the other hand, the functions {{nowrap|<math>z\mapsto \Re(z)</math>,}} {{nowrap|<math>z\mapsto |z|</math>,}} and <math>z\mapsto \bar{z}</math> are not holomorphic anywhere on the complex plane, as can be shown by their failure to satisfy the Cauchy–Riemann conditions (see below).

An important property of holomorphic functions is the relationship between the partial derivatives of their real and imaginary components, known as the [[Cauchy–Riemann conditions]].  If <math>f:\mathbb{C}\to\mathbb{C}</math>, defined by {{nowrap|<math>f(z) = f(x + iy) = u(x, y) + iv(x, y)</math>,}} where {{nowrap|<math>x, y, u(x, y),v(x, y) \in \R</math>,}} is holomorphic on a [[Region (mathematics)|region]] {{nowrap|<math>\Omega</math>,}} then for all <math>z_0\in \Omega</math>,
:<math>\frac{\partial f}{\partial\bar{z}}(z_0) = 0,\ \text{where } \frac\partial{\partial\bar{z}} \mathrel{:=} \frac12\left(\frac\partial{\partial x} + i\frac\partial{\partial y}\right).</math>
In terms of the real and imaginary parts of the function, ''u'' and ''v'', this is equivalent to the pair of equations <math>u_x = v_y</math> and <math>u_y=-v_x</math>, where the subscripts indicate partial differentiation.  However, the Cauchy–Riemann conditions do not characterize holomorphic functions, without additional continuity conditions (see [[Looman–Menchoff theorem]]).

Holomorphic functions exhibit some remarkable features.  For instance, [[Picard theorem|Picard's theorem]] asserts that the range of an entire function can take only three possible forms: {{nowrap|<math>\mathbb{C}</math>,}} {{nowrap|<math>\mathbb{C}\setminus\{z_0\}</math>,}} or <math>\{z_0\}</math> for some {{nowrap|<math>z_0\in\mathbb{C}</math>.}} In other words, if two distinct complex numbers <math>z</math> and <math>w</math> are not in the range of an entire function {{nowrap|<math>f</math>,}} then <math>f</math> is a constant function.  Moreover, a holomorphic function on a connected open set is determined by its restriction to any nonempty open subset.

==Conformal map==
{{excerpt|Conformal map}}

== Major results ==
[[Image:Complex-plot.png|right|thumb|262px|[[Domain coloring|Color wheel graph]] of the function
{{math|''f''(''x'') {{=}} {{sfrac|(''x''<sup>2</sup> − 1)(''x'' − 2 − ''i'')<sup>2</sup>|''x''<sup>2</sup> + 2 + 2''i''}}}}.<br />
[[Hue]] represents the [[Argument (complex analysis)|argument]], [[brightness]] the [[Absolute value#Complex numbers|magnitude.]]]]
One of the central tools in complex analysis is the [[line integral]]. The line integral around a closed path of a function that is holomorphic everywhere inside the area bounded by the closed path is always zero, as is stated by the [[Cauchy integral theorem]]. The values of such a holomorphic function inside a disk can be computed by a path integral on the disk's boundary (as shown in [[Cauchy's integral formula]]). Path integrals in the complex plane are often used to determine complicated real integrals, and here the theory of [[residue (complex analysis)|residue]]s among others is applicable (see [[methods of contour integration]]). A "pole" (or [[isolated singularity]]) of a function is a point where the function's value becomes unbounded, or "blows up". If a function has such a pole, then one can compute the function's residue there, which can be used to compute path integrals involving the function; this is the content of the powerful [[residue theorem]]. The remarkable behavior of holomorphic functions near essential singularities is described by [[Picard theorem#Big Picard|Picard's theorem]]. Functions that have only poles but no [[Essential singularity|essential singularities]] are called [[meromorphic]]. [[Laurent series]] are the complex-valued equivalent to [[Taylor series]], but can be used to study the behavior of functions near singularities through infinite sums of more well understood functions, such as polynomials.

A [[bounded function]] that is holomorphic in the entire complex plane must be constant; this is [[Liouville's theorem (complex analysis)|Liouville's theorem]]. It can be used to provide a natural and short proof for the [[Fundamental Theorem of Algebra|fundamental theorem of algebra]] which states that the [[field (mathematics)|field]] of complex numbers is [[algebraically closed field|algebraically closed]].

If a function is holomorphic throughout a [[Connected space|connected]] domain then its values are fully determined by its values on any smaller subdomain.  The function on the larger domain is said to be [[analytic continuation|analytically continued]] from its values on the smaller domain.  This allows the extension of the definition of functions, such as the [[Riemann zeta function]], which are initially defined in terms of infinite sums that converge only on limited domains to almost the entire complex plane.  Sometimes, as in the case of the [[natural logarithm]], it is impossible to analytically continue a holomorphic function to a non-simply connected domain in the complex plane but it is possible to extend it to a holomorphic function on a closely related surface known as a [[Riemann surface]].

All this refers to complex analysis in one variable. There is also a very rich theory of [[Function of several complex variables|complex analysis in more than one complex dimension]] in which the analytic properties such as [[power series]] expansion carry over whereas most of the geometric properties of holomorphic functions in one complex dimension (such as [[conformality]]) do not carry over. The [[Riemann mapping theorem]] about the conformal relationship of certain domains in the complex plane, which may be the most important result in the one-dimensional theory, fails dramatically in higher dimensions.

A major application of certain [[Complex Hilbert space|complex space]]s is in [[quantum mechanics]] as [[wave function]]s.

== See also ==
* [[Complex geometry]]
* [[Hypercomplex analysis]]
* [[List of complex analysis topics]]
* [[Monodromy theorem]]
* [[Riemann–Roch theorem]]
* [[Runge's theorem]]
* [[Vector calculus]]

== References ==
{{reflist}}

== Sources ==
* [[Mark J. Ablowitz|Ablowitz, M. J.]] & [[Athanassios S. Fokas|A. S. Fokas]], ''Complex Variables: Introduction and Applications'' (Cambridge, 2003).
* [[Lars Ahlfors|Ahlfors, L.]], ''Complex Analysis'' (McGraw-Hill, 1953).
* [[Henri Cartan|Cartan, H.]], ''Théorie élémentaire des fonctions analytiques d'une ou plusieurs variables complexes.'' (Hermann, 1961). English translation, ''Elementary Theory of Analytic Functions of One or Several Complex Variables.'' (Addison-Wesley, 1963).
* [[Constantin Carathéodory|Carathéodory, C.]], ''Funktionentheorie.'' (Birkhäuser, 1950). English translation, ''Theory of Functions of a Complex Variable'' (Chelsea, 1954). [2 volumes.]
* [[George F. Carrier|Carrier, G. F.]], [[Max Krook|M. Krook]], & C. E. Pearson, [https://archive.org/details/functionsofcompl00carr/ ''Functions of a Complex Variable: Theory and Technique.''] (McGraw-Hill, 1966).
* [[John B. Conway|Conway, J. B.]], ''Functions of One Complex Variable.'' (Springer, 1973).
* Fisher, S., ''Complex Variables.'' (Wadsworth & Brooks/Cole, 1990).
* [[Andrew Forsyth|Forsyth, A.]], [https://archive.org/details/theoryoffunction00fors/ ''Theory of Functions of a Complex Variable''] (Cambridge, 1893).
* [[Eberhard Freitag|Freitag, E.]] & R. Busam, ''Funktionentheorie''. (Springer, 1995). English translation, ''Complex Analysis''. (Springer, 2005).
* [[Édouard Goursat|Goursat, E.]], [https://archive.org/details/courseinmathemat02gouruoft/ ''Cours d'analyse mathématique, tome 2'']. (Gauthier-Villars, 1905). English translation, [https://archive.org/details/coursemathema0102gourrich/ ''A course of mathematical analysis, vol. 2, part 1: Functions of a complex variable'']. (Ginn, 1916).
* [[Peter Henrici (mathematician)|Henrici, P.]], ''Applied and Computational Complex Analysis'' (Wiley).  [Three volumes: 1974, 1977, 1986.]
* [[Erwin Kreyszig|Kreyszig, E.]], ''Advanced Engineering Mathematics.'' (Wiley, 1962).
* [[Mikhail Lavrentyev|Lavrentyev, M.]] & B. Shabat, ''Методы теории функций комплексного переменного.'' (''Methods of the Theory of Functions of a Complex Variable''). (1951, in Russian).
* [[Aleksei Ivanovich Markushevich|Markushevich, A. I.]], ''Theory of Functions of a Complex Variable'', (Prentice-Hall, 1965). [Three volumes.]
* [[Jerrold E. Marsden|Marsden]] & Hoffman, ''Basic Complex Analysis.'' (Freeman, 1973).
* [[Tristan Needham|Needham, T.]], ''Visual Complex Analysis.'' (Oxford, 1997). http://usf.usfca.edu/vca/
* [[Reinhold Remmert|Remmert, R.]], ''Theory of Complex Functions''. (Springer, 1990).
* [[Walter Rudin|Rudin, W.]], ''Real and Complex Analysis.'' (McGraw-Hill, 1966).
* Shaw, W. T., ''Complex Analysis with Mathematica'' (Cambridge, 2006).
* [[Elias M. Stein|Stein, E.]] & R. Shakarchi, ''Complex Analysis.'' (Princeton, 2003).
* [[Aleksei Sveshnikov|Sveshnikov, A. G.]] & [[Andrey Nikolayevich Tikhonov|A. N. Tikhonov]], ''Теория функций комплексной переменной.'' (Nauka, 1967). English translation, [https://archive.org/details/SveshnikovTikhonovTheTheoryOfFunctionsOfAComplexVariable ''The Theory Of Functions Of A Complex Variable''] (MIR, 1978).
* [[Edward Charles Titchmarsh|Titchmarsh, E. C.]], [https://archive.org/details/in.ernet.dli.2015.2588/ ''The Theory of Functions.''] (Oxford, 1932).
* Wegert, E., ''Visual Complex Functions''. (Birkhäuser, 2012).
* [[Edmund T. Whittaker|Whittaker, E. T.]] & [[George N. Watson|G. N. Watson]], ''[[A Course of Modern Analysis]].'' (Cambridge, 1902). [https://archive.org/details/courseofmodernan00whit/ 3rd ed. (1920)]

== External links ==
{{Sister project links| wikt=complex analysis | commons=Category:Complex analysis | b=no | n=no | q=Complex analysis | s=no | v=no | voy=no | species=no | d=no}}
* [http://mathworld.wolfram.com/ComplexAnalysis.html Wolfram Research's MathWorld Complex Analysis Page]
* [https://www.jirka.org/ca/ Guide to Cultivating Complex Analysis: Working the Complex Field] by Jiri Lebl ([[Creative Commons|Creative Commons BY-NC-SA]])

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