Everything about Complex Analysis totally explained
Complex analysis, traditionally known as the
theory of functions of a complex variable, is the branch of
mathematics investigating
functions of
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 functions of complex variables, which are commonly divided into two main classes: the
holomorphic functions and the
meromorphic functions. Because the separable
real and
imaginary parts of any analytic function must satisfy
Laplace's equation, complex analysis is widely applicable to two-dimensional problems in
physics.
History
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 theory which is a conformally invariant
quantum field theory.
Complex functions
A complex function is a function in which the
independent variable and the
dependent variable are 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
real and
imaginary parts:
» and
» where
In other words, the real and imaginary parts of a differentiable function of a complex variable are
harmonic functions because they satisfy
Laplace's equation.
Holomorphic functions
Holomorphic functions are complex functions defined on an
open subset of the complex plane which are
differentiable. Complex differentiability has much stronger consequences than usual (real) differentiability. For instance, holomorphic functions are infinitely differentiable, a fact that's far from true for real differentiable functions. Most elementary functions, including the
exponential function, the
trigonometric functions, and all
polynomial functions, are holomorphic.
See also:
analytic function,
holomorphic sheaf and
vector bundles.
Major results
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.
A bounded function which 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.
An important property of holomorphic functions is that if a function is holomorphic throughout a
simply 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 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's impossible to analytically continue a holomorphic function to a non-simply connected domain in the complex plane but it's 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.
Further Information
Get more info on 'Complex Analysis'.
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