Difference between revisions of "Complex analysis"

Revision as of 01:06, April 25, 2008

Complex analysis is the study of functions supported on the [[complex number]s.

Much of complex analysis is devoted to studying holomorphic functions that are infinitely differentiable. These functions take complex values in the complex plane and are differentiable as complex functions.

Complex analysis relies heavily on contour integration, which enables computation of difficult integrals by examining singularities of the function in regions of the complex plane near the limits of integration.

The central result in complex analysis is the Cauchy integral theorem, and a powerful claim of complex analysis is Picard's great theorem.

The Cauchy-Riemann equations provide conditions a function must satisfy in order for a complex generalization of the derivative (the "complex derivative"). When the complex derivative can be defined "everywhere," the function is called "analytic".

Additional concepts in complex analysis include the following:

@@ Line 1: / Line 1: @@
-Complex analysis is the study of [[complex number]]s of the form:
+Complex analysis is the study of functions supported on the [[complex number]s.
-: <math>i = \sqrt{-1}</math>
-From this definition a "complex plane" is constructed, consisting of z = x + iy, where x and y are real numbers:
-: <math>z = x + iy\,</math>, and
-: <math>w = f(z) = u(z) + iv(z)\,</math>
-: where <math>x,y \in \mathbb{R}\,</math> and <math>u(z), v(z)\,</math> are real-valued functions.
-so that real numbers are on the x-axis and imaginary numbers are on the y-axis.
-Traditionally, mathematical objects are derived from [[physics]] or [[geometry]]. The complex numbers on the other hand were constructed out of the blue as an ad-hoc solution to certain problems that appeared to be unsolvable with real numbers. Initially unanimously rejected by the leading mathematicians, they were eventually accepted by a growing number of mathematicians due to a series of spectacular results. Not all of those results can be proved with [[elementary techniques]] however, and are thus considered questionable by more rigorous mathematicians.
 Much of complex analysis is devoted to studying [[holomorphic functions]] that are infinitely differentiable.  These functions take complex values in the complex plane and are differentiable as complex functions.

Difference between revisions of "Complex analysis"

Revision as of 01:06, April 25, 2008

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