complex analysis

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.

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.

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 and are real-valued functions.

In other words, the components of the function f(z),

and

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.

Derivatives and the Cauchy-Riemann equations

Just as in real analysis, a "smooth" complex function w = f(z) may have a derivative at a particular point in its domain Ω. In fact, the definition of the derivative

is analogous to the real case, with one very important difference. In real analysis, the limit can only be approached by moving along the one-dimensional number line. In complex analysis, the limit can be approached from any direction in the two-dimensional complex plane.

If this limit, the derivative, exists for every point z in Ω, then f(z) is said to be differentiable on Ω. It can be shown that any differentiable f(z) is analytic. This is a much more powerful result than the analogous theorem that can be proved for real-valued functions of real numbers. In the calculus of real numbers, we can construct a function f(x) that has a first derivative everywhere, but for which the second derivative does not exist at one or more points in the function's domain. But in the complex plane, if a function f(z) is differentiable in a neighborhood it must also be infinitely differentiable in that neighborhood. (See "Holomorphic functions are analytic" for a proof.)

By applying the methods of vector calculus to compute the partial derivatives of the two real functions u(x, y) and v(x, y) into which f(z) can be decomposed, and by considering two paths leading to a point z in Ω, it can be shown that the derivative exists if and only if

Equating the real and imaginary parts of these two expressions we obtain the traditional formulation of the Cauchy-Riemann Equations:

or, in another common notation,

By differentiating this system of two partial differential equations first with respect to x, and then with respect to y, we can easily show that

or, in another common notation,

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

Main article: Holomorphic function
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 is far from true for real differentiable functions. Most elementary functions, including the exponential function, the trigonometric functions, and all polynomial functions, are holomorphic.

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 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, 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.

History

The Mandelbrot set, the most common example of a fractal.
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 applications in engineering, but it 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.

References

• Needham T., Visual Complex Analysis (Oxford, 1997).
• Henrici P., Applied and Computational Complex Analysis (Wiley). [Three volumes: 1974, 1977, 1986.]
• Kreyszig, E, Advanced Engineering Mathematics, 9 ed., Ch.13-18 (Wiley, 2006).
• Scheidemann, V., Introduction to complex analysis in several variables (Birkhauser, 2005)
• Shaw, W.T., Complex Analysis with Mathematica (Cambridge, 2006).
• Marsden & Hoffman, Basic complex analysis (Freeman, 1999).

Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".
function expresses dependence between two quantities, one of which is given (the independent variable, argument of the function, or its "input") and the other produced (the dependent variable, value of the function, or "output").
In mathematics, a complex number is a number of the form

where a and b are real numbers, and i is the imaginary unit, with the property i ² = −1.
Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study.
Applied mathematics is a branch of mathematics that concerns itself with the mathematical techniques typically used in the application of mathematical knowledge to other domains.
In mathematics, an analytic function is a function that is locally given by a convergent power series. Analytic functions can be thought of as a bridge between polynomials and general functions.
Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point.
In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function.
In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2.4871773339…. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as π and
imaginary number (or purely imaginary number) is a complex number whose square is a negative real number. Imaginary numbers were defined in 1572 by Rafael Bombelli. At the time, such numbers were thought not to exist, much as zero and the negative numbers were regarded by
In mathematics, Laplace's equation is a partial differential equation named after its discoverer, Pierre-Simon Laplace. The solutions of Laplace's equation are important in many fields of science, notably the fields of electromagnetism, astronomy, and fluid dynamics, because they
Physics is the science of matter[1] and its motion[2][3], as well as space and time[4][5] —the science that deals with concepts such as force, energy, mass, and charge.
In mathematics, an independent variable is any of the arguments, i.e. "inputs", to a function. These are contrasted with the dependent variable, which is the value, i.e. the "output", of the function.
In mathematics, an independent variable is any of the arguments, i.e. "inputs", to a function. These are contrasted with the dependent variable, which is the value, i.e. the "output", of the function.
subset of a set B if A is "contained" inside B. Notice that A and B may coincide. The relationship of one set being a subset of another is called inclusion or containment.
complex plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis. It can be thought of as a modified Cartesian plane, with the real part of a complex number represented by a displacement along the
In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2.4871773339…. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as π and
imaginary number (or purely imaginary number) is a complex number whose square is a negative real number. Imaginary numbers were defined in 1572 by Rafael Bombelli. At the time, such numbers were thought not to exist, much as zero and the negative numbers were regarded by
derivative is a measurement of how a function changes when the values of its inputs change. Loosely speaking, a derivative can be thought of as how much a quantity is changing at some given point.
In mathematics, an analytic function is a function that is locally given by a convergent power series. Analytic functions can be thought of as a bridge between polynomials and general functions.
neighbourhood (or neighborhood) is one of the basic concepts in a topological space. Intuitively speaking, a neighbourhood of a point is a set containing the point where you can wiggle the point a bit without leaving the set.
In complex analysis, a field of mathematics, a complex-valued function f of a complex variable
• is holomorphic at a point a iff it is differentiable at every point within some open disk centered at a, and
• is analytic at a

Vector calculus (also called vector analysis) is a field of mathematics concerned with multivariate real analysis of vectors in a metric space with two or more dimensions (some results can only be applied to three dimensions[1]).
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary).
If and only if, in logic and fields that rely on it such as mathematics and philosophy, is a logical connective between statements which means that the truth of either one of the statements
In mathematics, the Cauchy-Riemann differential equations in complex analysis, named after Augustin Cauchy and Bernhard Riemann, are two partial differential equations which provide a necessary but not sufficient condition for a function to be holomorphic.