Complex analysis

Plot of the function f(x)=(x2-1)(x-2-i)2/(x2+2+2i). The hue represents the function argument, while the saturation represents the magnitude.

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 (or, more generally, 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.

Contents

History

The Mandelbrot set, 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 physical applications and is also used throughout analytic 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:

z = x + iy\, and
w = f(z) = u(z) + iv(z)\,
where x,y \in \mathbb{R}\, 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.

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

f^\prime(z) = \frac{dw}{dz} = \lim_{h \to 0}\frac{f(z+h) - f(z)}{h}\,

is analogous to the real case but 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, and for the derivative to exist, the limiting value must be the same no matter what the direction of approach of h to 0.

(The claim that "in real analysis, the limit can only be approached by moving along the one dimensional number line" should not be confused with directional derivatives. It may be explained that in directional derivatives, one still moves along the one dimensional x line but it can be in "discrete" units; that is, if one follows y = x2 curve, that does not mean that one is moving on the plane (instead of the one dimensional x line) but means that one is approaching in steps of discrete units.)

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 existence of derivative implies

f^\prime(z) = \frac{\partial u}{\partial x} + i\frac{\partial v}{\partial x} = \frac{\partial v}{\partial y} - i\frac{\partial u}{\partial y}.\,

Equating the real and imaginary parts of these two expressions we obtain the traditional formulation of the Cauchy–Riemann equations:[1]

\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \qquad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}\, or, in another common notation, u_x=v_y \qquad u_y=-v_x.\,

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

\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 \qquad \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} = 0\, or, in another common notation, u_{xx} + u_{yy} = v_{xx} + v_{yy} = 0.\,

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.

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 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.

See also

Notes

  1. ↑ Saff, E. B.; A. D. Snider (2003). Fundamentals of Complex Analysis. Pearson Education Inc.. pp. 73. ISBN 0-13-907874. 

References

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