Complex analysis is one of the classical branches in mathematics with its roots in the 19th century and some even before. Important names are
Euler,
Gauss,
Riemann,
Cauchy,
Weierstrass,and many more in the 20th century. Traditionally, complex analysis, in particular the theory of
conformal mappings, has many physical applications and is also used throughout analytical
number theory. In modern times, it became very popular through a new boost of
complex dynamics and the pictures of
fractals produced by iterating holomorphic functions, the most popular being the
Mandelbrot set. Another important application of complex analysis today is in
string theorywhich is a conformally invariant
quantum field theory.
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of
mathematicsinvestigating
functionsof
complex numbers. It is useful in many branches of mathematics, including
number theory and
applied mathematics, and in
physics.
Complex analysis is particularly concerned with the
analytic functionsof complex variables, which are commonly divided into two main classes: the
holomorphic functionsand the meromorphic functions. Because the separable
realand
imaginaryparts of any analytic function must satisfy
Laplace's equation, complex analysis is widely applicable to two-dimensional problems in
physics.
A complex function is a function in which the
independent variable and the
dependent variableare both complex numbers. More precisely, a complex function is a function whose
domain Ω is a
subset of the
complex plane and whose
range is also a subset of the complex plane. For any complex function, both the independent variable and the dependent variable may be separated into
realand
imaginary parts:
z=x+iy and w=f(z)=u(z)+iv(z)
Where
x and
y are real numbers and
u(z),
v(z) are real-valued functions. In other words, the components of the function
f(z), u=u(x,y) and
v=v(x,y) can be interpreted as real valued functions of the two real variables,
x and
y.
The basic concepts of complex analysis are often introduced by extending the elementary real functions (e.g., exponentials, logarithms, and trigonometric functions) into the complex domain.
Holomorphic functions are the central object of study of
complex analysis; they are
functions defined on an
open subsetof
the complex number planeC with values in C that are complex-differentiable at every point. This is a much stronger condition than
real differentiability and implies that the function is
infinitely often differentiable and can be described by its
Taylor series. The term
analytic functionis often used interchangeably with "holomorphic function", although the former term is also used in the broader sense of a function (real, complex, or of more general type) that is equal to its Taylor series in a neighborhood of each point in its domain. The fact that the class of analytic functions coincides with the class of holomorphic functions is a major theorem in complex analysis. Holomorphic functions are sometimes said to be
regular functions. A function that is holomorphic on the whole complex plane is called an
entire function. The phrase "holomorphic at a point a" means not just differentiable at a, but differentiable everywhere within some open disk centered at a in the complex plane.
One central tool in complex analysis is the
line integral. The integral around a closed path of a function which is holomorphic everywhere inside the area bounded by the closed path is always zero; this is the
Cauchy integral theorem. The values of a holomorphic function inside a disk can be computed by a certain path integral on the disk's boundary (
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 useful (see
methods of contour integration). If a function has a pole or singularity at some point, that is, at that point its values "blow up" and have no finite value, then one can compute the function's residue at that pole, and these residues 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 the
Weierstrass-Casorati theorem. Functions which have only poles but no essential singularities are called
meromorphic.
Laurent series are similar to
Taylor series but can be used to study the behavior of functions near singularities.
An important property of holomorphic functions is that if a function is holomorphic throughout a
simply connecteddomain then its values are fully determined by its values on any smaller subdomain. The function on the larger domain is said to be
analytically continuedfrom its values on the smaller domain. This allows the extension of the definition of functions such as the
Riemann zeta functionwhich 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
complex analysis in more than one complex dimension where the analytic properties such as power series expansion still remain true whereas most of the geometric properties of holomorphic functions in one complex dimension (such as
conformality) are no longer true. 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. It is also applied in many subjects throughout engineering, particularly in
power engineering.
Ever since the groundbreaking work of J. J. Kohn in the early 1960s, there has been a significant interaction between the theory of partial differential equations and the function theory of several complex variables. Partial Differential Equations and Complex Analysis explores the background and plumbs the depths of this symbiosis.
