Complex analysis is the branch of mathematics investigating the holomorphic or analytic functions, i.e. functions defined in some region of the complex plane taking complex values which are differentiable as complex functions. Complex differentiability has much stronger consequences than usual (real) differentiability. For instance, every holomorphic function is representable as power series in every open disc in its 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.
One central tool in complex analysis is the path integral. The values of a holomorphic function inside a disk can be computed by a certain path integral on the circle surrounding the disk (Cauchys formula). Path integrals in the complex plane are often used to determine complicated real integrals, and here the theory of residues is useful. If a function has a pole or singularity at some point, meaning that its values "explode" and it does not have a finite value there, then one can define 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 residue theorem. The remarkable behavior of holomorphic functions near essential poles is described by the Weierstrass-Casorati theorem.
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 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. 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.
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 the theory of conformal mappings, has lots of applications in engineering, but 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.