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Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied ...
that investigates functions of
complex numbers In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
. It is helpful in many branches of mathematics, including
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
,
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...
,
analytic combinatorics In combinatorics, the symbolic method is a technique for counting combinatorial objects. It uses the internal structure of the objects to derive formulas for their generating functions. The method is mostly associated with Philippe Flajolet an ...
,
applied mathematics Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathemati ...
; as well as in
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
, including the branches of
hydrodynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) a ...
,
thermodynamics Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws ...
, and particularly
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
. By extension, use of complex analysis also has applications in engineering fields such as
nuclear Nuclear may refer to: Physics Relating to the nucleus of the atom: *Nuclear engineering *Nuclear physics *Nuclear power *Nuclear reactor *Nuclear weapon *Nuclear medicine *Radiation therapy *Nuclear warfare Mathematics *Nuclear space *Nuclear ...
,
aerospace Aerospace is a term used to collectively refer to the atmosphere and outer space. Aerospace activity is very diverse, with a multitude of commercial, industrial and military applications. Aerospace engineering consists of aeronautics and ast ...
,
mechanical Mechanical may refer to: Machine * Machine (mechanical), a system of mechanisms that shape the actuator input to achieve a specific application of output forces and movement * Mechanical calculator, a device used to perform the basic operations ...
and
electrical engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the l ...
. As a
differentiable function In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
of a complex variable is equal to its
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
(that is, it is
analytic Generally speaking, analytic (from el, ἀναλυτικός, ''analytikos'') refers to the "having the ability to analyze" or "division into elements or principles". Analytic or analytical can also have the following meanings: Chemistry * ...
), complex analysis is particularly concerned with
analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
s of a complex variable (that is,
holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex deriv ...
s).


History

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 Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ...
,
Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
, Riemann,
Cauchy Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He w ...
,
Weierstrass Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics ...
, and many more in the 20th century. Complex analysis, in particular the theory of
conformal mapping In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths. More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-preserving) at a point u_0\in ...
s, has many physical applications and is also used throughout
analytic number theory In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Diri ...
. In modern times, it has become very popular through a new boost from
complex dynamics Complex dynamics is the study of dynamical systems defined by iteration of functions on complex number spaces. Complex analytic dynamics is the study of the dynamics of specifically analytic functions. Techniques *General **Montel's theorem ** P ...
and the pictures of
fractal In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as ill ...
s produced by iterating
holomorphic functions In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex deriv ...
. Another important application of complex analysis is in
string theory In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and intera ...
which examines conformal invariants in
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
.


Complex functions

A complex function is a function from
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s to complex numbers. In other words, it is a function that has a subset of the complex numbers as a
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function **Domain of holomorphy of a function * ...
and the complex numbers as a
codomain In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either th ...
. Complex functions are generally supposed to have a domain that contains a nonempty
open subset In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suff ...
of the
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
. For any complex function, the values z from the domain and their images f(z) in the range may be separated into
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
and imaginary parts: : z=x+iy \quad \text \quad f(z) = f(x+iy)=u(x,y)+iv(x,y), where x,y,u(x,y),v(x,y) are all real-valued. In other words, a complex function f:\mathbb\to\mathbb may be decomposed into : u:\mathbb^2\to\mathbb \quad and \quad v:\mathbb^2\to\mathbb, i.e., into two real-valued functions (u, v) of two real variables (x, y). Similarly, any complex-valued function on an arbitrary
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
can be considered as an
ordered pair In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In con ...
of two
real-valued function In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain. Real-valued functions of a real variable (commonly called ''real f ...
s: or, alternatively, as a vector-valued function from into \mathbb R^2. Some properties of complex-valued functions (such as continuity) are nothing more than the corresponding properties of vector valued functions of two real variables. Other concepts of complex analysis, such as
differentiability In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
, are direct generalizations of the similar concepts for real functions, but may have very different properties. In particular, every differentiable complex function is
analytic Generally speaking, analytic (from el, ἀναλυτικός, ''analytikos'') refers to the "having the ability to analyze" or "division into elements or principles". Analytic or analytical can also have the following meanings: Chemistry * ...
(see next section), and two differentiable functions that are equal in a
neighborhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
of a point are equal on the intersection of their domain (if the domains are
connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
). The latter property is the basis of the principle of
analytic continuation In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a ...
which allows extending every real
analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
in a unique way for getting a complex analytic function whose domain is the whole complex plane with a finite number of curve arcs removed. Many basic and special complex functions are defined in this way, including the
complex exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, al ...
, complex logarithm functions, and
trigonometric functions In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...
.


Holomorphic functions

Complex functions that are differentiable at every point of an
open subset In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suff ...
\Omega of the complex plane are said to be ''holomorphic on'' In the context of complex analysis, the derivative of f at z_0 is defined to be : f'(z_0) = \lim_ \frac. 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 z_0 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 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 \Omega can be approximated arbitrarily well by polynomials in some neighborhood of every point in \Omega. This stands in sharp contrast to differentiable real functions; there are infinitely differentiable real functions that are ''nowhere'' analytic; see . Most elementary functions, including the
exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
, the
trigonometric function In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...
s, and all polynomial functions, extended appropriately to complex arguments as functions are holomorphic over the entire complex plane, making them ''entire'' ''functions'', while rational functions p/q, 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 and z\mapsto \bar 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 f:\mathbb\to\mathbb, defined by where is holomorphic on a
region In geography, regions, otherwise referred to as zones, lands or territories, are areas that are broadly divided by physical characteristics ( physical geography), human impact characteristics ( human geography), and the interaction of humanity an ...
then for all z_0\in \Omega, :\frac(z_0) = 0,\ \text \frac\partial \mathrel \frac12\left(\frac\partial + i\frac\partial\right). In terms of the real and imaginary parts of the function, ''u'' and ''v'', this is equivalent to the pair of equations u_x = v_y and u_y=-v_x, where the subscripts indicate partial differentiation. However, the Cauchy–Riemann conditions do not characterize holomorphic functions, without additional continuity conditions (see
Looman–Menchoff theorem In the mathematical field of complex analysis, the Looman–Menchoff theorem states that a continuous complex-valued function defined in an open set of the complex plane is holomorphic if and only if it satisfies the Cauchy–Riemann equations. It ...
). Holomorphic functions exhibit some remarkable features. For instance,
Picard's theorem In complex analysis, Picard's great theorem and Picard's little theorem are related theorems about the range of an analytic function. They are named after Émile Picard. The theorems Little Picard Theorem: If a function f: \mathbb \to\mathbb i ...
asserts that the range of an entire function can take only three possible forms: or \ for some In other words, if two distinct complex numbers z and w are not in the range of an entire function then f 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

Conformal mapping are locally invertible complex analytic function in two dimensions for orientation preservation.


Application of Conformal mapping

* In aerospace engineering * In Biomedical sciences * In Brain mapping * Genetic mapping * Geodesics * In Geometry * In Geophysics * In Google * In Literature * in Engineering * In Electronics * In Protein synthesis * In Geography, in Cartography.


Major results

One of the central tools in complex analysis is the
line integral In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; '' contour integral'' is used as well, ...
. 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 residues among others is applicable (see
methods of contour integration In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane. Contour integration is closely related to the calculus of residues, a method of complex analysis. ...
). 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 In complex analysis, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals and infinite series as wel ...
. The remarkable behavior of holomorphic functions near essential singularities is described by
Picard's theorem In complex analysis, Picard's great theorem and Picard's little theorem are related theorems about the range of an analytic function. They are named after Émile Picard. The theorems Little Picard Theorem: If a function f: \mathbb \to\mathbb i ...
. Functions that have only poles but no
essential singularities In complex analysis, an essential singularity of a function is a "severe" singularity near which the function exhibits odd behavior. The category ''essential singularity'' is a "left-over" or default group of isolated singularities that a ...
are called meromorphic.
Laurent series In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion c ...
are the complex-valued equivalent to
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
, 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 In mathematics, a function ''f'' defined on some set ''X'' with real or complex values is called bounded if the set of its values is bounded. In other words, there exists a real number ''M'' such that :, f(x), \le M for all ''x'' in ''X''. ...
that is holomorphic in the entire complex plane must be constant; this is Liouville's theorem. It can be used to provide a natural and short proof for the
fundamental theorem of algebra The fundamental theorem of algebra, also known as d'Alembert's theorem, or the d'Alembert–Gauss theorem, states that every non- constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomia ...
which states that the field of complex numbers is
algebraically closed In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, because ...
. If a function is holomorphic throughout a
connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
domain then its values are fully determined by its values on any smaller subdomain. The function on the larger domain is said to be
analytically continued In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a ne ...
from its values on the smaller domain. This allows the extension of the definition of functions, such as the
Riemann zeta function The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
, 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 The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
, 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 In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...
. All this refers to complex analysis in one variable. There is also a very rich theory of complex analysis in more than one complex dimension in which the analytic properties such as
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...
expansion carry over whereas most of the geometric properties of holomorphic functions in one complex dimension (such as
conformality In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths. More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-preserving) at a point u_0\in ...
) do not carry over. The
Riemann mapping theorem In complex analysis, the Riemann mapping theorem states that if ''U'' is a non-empty simply connected open subset of the complex number plane C which is not all of C, then there exists a biholomorphic mapping ''f'' (i.e. a bijective holomorphi ...
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 spaces is in
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
as
wave function A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...
s.


See also

* Hypercomplex analysis *
Vector calculus Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subjec ...
*
Complex dynamics Complex dynamics is the study of dynamical systems defined by iteration of functions on complex number spaces. Complex analytic dynamics is the study of the dynamics of specifically analytic functions. Techniques *General **Montel's theorem ** P ...
*
List of complex analysis topics Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied ...
* Monodromy theorem *
Real analysis In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include conv ...
*
Riemann–Roch theorem The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. It rel ...
* Runge's theorem


References

* Ablowitz, M. J. & A. S. Fokas, ''Complex Variables: Introduction and Applications'' (Cambridge, 2003). * Ahlfors, L., ''Complex Analysis'' (McGraw-Hill, 1953). * 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). * Carathéodory, C., ''Funktionentheorie.'' (Birkhäuser, 1950). English translation, ''Theory of Functions of a Complex Variable'' (Chelsea, 1954). volumes.* Carrier, G. F., M. Krook, & C. E. Pearson
''Functions of a Complex Variable: Theory and Technique.''
(McGraw-Hill, 1966). * Conway, J. B., ''Functions of One Complex Variable.'' (Springer, 1973). * Fisher, S., ''Complex Variables.'' (Wadsworth & Brooks/Cole, 1990). * Forsyth, A.
''Theory of Functions of a Complex Variable''
(Cambridge, 1893). * Freitag, E. & R. Busam, ''Funktionentheorie''. (Springer, 1995). English translation, ''Complex Analysis''. (Springer, 2005). * Goursat, E.
''Cours d'analyse mathématique, tome 2''
(Gauthier-Villars, 1905). English translation
''A course of mathematical analysis, vol. 2, part 1: Functions of a complex variable''
(Ginn, 1916). * Henrici, P., ''Applied and Computational Complex Analysis'' (Wiley). hree volumes: 1974, 1977, 1986.* Kreyszig, E., ''Advanced Engineering Mathematics.'' (Wiley, 1962). * Lavrentyev, M. & B. Shabat, ''Методы теории функций комплексного переменного.'' (''Methods of the Theory of Functions of a Complex Variable''). (1951, in Russian). * Markushevich, A. I., ''Theory of Functions of a Complex Variable'', (Prentice-Hall, 1965). hree volumes.* Marsden & Hoffman, ''Basic Complex Analysis.'' (Freeman, 1973). * Needham, T., ''Visual Complex Analysis.'' (Oxford, 1997). http://usf.usfca.edu/vca/ * Remmert, R., ''Theory of Complex Functions''. (Springer, 1990). * Rudin, W., ''Real and Complex Analysis.'' (McGraw-Hill, 1966). * Shaw, W. T., ''Complex Analysis with Mathematica'' (Cambridge, 2006). * Stein, E. & R. Shakarchi, ''Complex Analysis.'' (Princeton, 2003). * Sveshnikov, A. G. & A. N. Tikhonov, ''Теория функций комплексной переменной.'' (Nauka, 1967). English translation
''The Theory Of Functions Of A Complex Variable''
(MIR, 1978). * Titchmarsh, E. C.
''The Theory of Functions.''
(Oxford, 1932). * Wegert, E., ''Visual Complex Functions''. (Birkhäuser, 2012). * Whittaker, E. T. &
G. N. Watson George Neville Watson (31 January 1886 – 2 February 1965) was an English mathematician, who applied complex analysis to the theory of special functions. His collaboration on the 1915 second edition of E. T. Whittaker's ''A Course of Mode ...
, ''
A Course of Modern Analysis ''A Course of Modern Analysis: an introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions'' (colloquially known as Whittaker and Watson) is a landmark textb ...
.'' (Cambridge, 1902)
3rd ed. (1920)


External links



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