Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of
mathematical analysis that investigates
functions of
complex numbers. 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,
analytic combinatorics,
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 mathemat ...
; as well as in
physics, 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) and ...
,
thermodynamics, and particularly
quantum mechanics. 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,
mechanical 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 of a complex variable is equal to its
Taylor series (that is, it is
analytic), complex analysis is particularly concerned with
analytic functions 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 derivativ ...
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 ma ...
,
Gauss,
Riemann,
Cauchy,
Weierstrass, and many more in the 20th century. Complex analysis, in particular the theory of
conformal mappings, 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 Diric ...
. In modern times, it has become very popular through a new boost from
complex dynamics 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 illu ...
s produced by iterating
holomorphic functions. 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 interac ...
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 numbers to complex numbers. In other words, it is a function that has a subset of the complex numbers as a
domain and the complex numbers as a
codomain. 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
from the domain and their images
in the range may be separated into
real and
imaginary parts:
:
where
are all real-valued.
In other words, a complex function
may be decomposed into
:
and
i.e., into two real-valued functions (
,
) of two real variables (
,
).
Similarly, any complex-valued function on an arbitrary
set 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 functions: or, alternatively, as a
vector-valued function from into
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, are direct generalizations of the similar concepts for real functions, but may have very different properties. In particular, every
differentiable complex function is
analytic (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). The latter property is the basis of the principle of
analytic continuation which allows extending every real
analytic function 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.
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 ...
of the complex plane are said to be ''holomorphic on'' In the context of complex analysis, the derivative of
at
is defined to be
:
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
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
can be approximated arbitrarily well by polynomials in some neighborhood of every point in
. 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
trigonometric functions, 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
, 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
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
, defined by where is holomorphic on a
region then for all
,
:
In terms of the real and imaginary parts of the function, ''u'' and ''v'', this is equivalent to the pair of equations
and
, 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's theorem 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
and
are not in the range of an entire function then
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. 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
In mathematics, the Cauchy integral theorem (also known as the Cauchy–Goursat theorem) in complex analysis, named after Augustin-Louis Cauchy (and Édouard Goursat), is an important statement about line integrals for holomorphic functions in t ...
. 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). 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's theorem. Functions that have only poles but no
essential singularities 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, 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''. A fun ...
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 which states that the
field of complex numbers is
algebraically closed.
If a function is holomorphic throughout a
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
analytically continued 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.
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 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 spaces is in
quantum mechanics as
wave functions.
See also
*
Hypercomplex analysis In mathematics, hypercomplex analysis is the basic extension of real analysis and complex analysis to the study of functions where the argument is a hypercomplex number. The first instance is functions of a quaternion variable, where the argument i ...
*
Vector calculus
*
Complex dynamics
*
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
In complex analysis, the monodromy theorem is an important result about analytic continuation of a complex-analytic function to a larger set. The idea is that one can extend a complex-analytic function (from here on called simply ''analytic fun ...
*
Real analysis
*
Riemann–Roch theorem
*
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 Modern ...
, ''
A Course of Modern Analysis.'' (Cambridge, 1902)
3rd ed. (1920)
External links
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