Complex analysis

Complex analysis is the branch of mathematics investigating the theory of holomorphic or analytic functions, i.e. functions defined in some region in the plane of complex numbers taking complex values which are differentiable as complex functions. Surprisingly, complex differentiability has much stronger consequences than usual (real) differentiability. E.g., another characterisation of holomorphic functions is that they are representable as power series in every open disc in their domain of definition. In particular, holomorphic functions are infinitely differentiable, a fact that is far from true for real differentiable functions. Most elementary functions such as all polynomials, the exponential and the trigonometric functions, are holomorphic.

Another very important property of holomorphic functions is that if a function is holomorphic throughout a simply connected domain then its value is fully determined by its value on a 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 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 be a holomorphic domain on a closely related surface known as a Riemann surface.

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 and e.g. the Riemann Mapping Theorem, maybe the most important result in the one-dimensional theory, fails dramatically.

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, many more in the 20th century. Traditionally complex analysis, in particular conformal mapping, had lots of applications in engineering. 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.