Complex analysis

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Complex analysis is the branch of mathematics investigating functions of complex numbers, and is of enormous practical use in many branches of mathematics, including applied mathematics.

Complex analysis is particularly concerned with analytic functions of complex variables, known as holomorphic functions.

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 defined on a subset of the complex plane with complex values.

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,u(z),v(z)\in {\mathbb  {R}}.

It follows that the components of the function,

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 extension of real functions (exponentials, logarithms, trigonometric functions) to the complex domain is frequently used as an introduction to complex analysis.

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 path 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, maybe 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

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.

See also

References

  • Needham T., Visual Complex Analysis (Oxford, 1997).
  • Henrici P., Applied and Computational Complex Analysis (Wiley). [Three volumes: 1974, 1977, 1986.]
  • Shaw, W.T., Complex Analysis with Mathematica (Cambridge, 2006).

External links