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Complex analysis (view source)

Revision as of 01:11, December 29, 2006

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Complex analysis is the study of mathematics based on an imaginary square root for negative one, called "i" by [[Leonhard Euler]]:

: <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.

so that real numbers are on the x-axis and imaginary numbers are on the y-axis.

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:

Analytic Continuation
Argument Principle
Branch Cut and Branch Point
Cauchy Principal Value
Complex Residue
Conformal Mapping
Contour Integration
de Moivre's Identity
Euler Formula
Inside-Outside Theorem
Jordan's Lemma
Laurent Series
Liouville's Conformality Theorem
Monogenic Function
Morera's Theorem
Permanence of Algebraic Form
Pole
Polygenic Function
Aschlafly
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