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# Complex analysis

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Complex analysis is the branch of mathematics investigating the theory of holomorphic 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.

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 not 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.

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Last edited July 17, 2001 9:50 am (diff)
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