Difference between revisions of "Complex analysis"

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Complex analysis is the study of [[complex number]]s of the form:
  
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: <math>i = \sqrt{-1}</math>
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From this definition a "complex plane" is constructed, consisting of z = x + iy, where x and y are real numbers:
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: <math>z = x + iy\,</math>, and
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: <math>w = f(z) = u(z) + iv(z)\,</math>
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: where <math>x,y \in \mathbb{R}\,</math> and <math>u(z), v(z)\,</math> are real-valued functions.
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so that real numbers are on the x-axis and imaginary numbers are on the y-axis.
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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.
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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.
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The central result in complex analysis is the [[Cauchy integral theorem]], and a powerful claim of complex analysis is Picard's great theorem.
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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]]".
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Additional concepts in complex analysis include the following:
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*[[Analytic Continuation]]
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*[[Argument Principle]]
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*[[Branch Cut]] and [[Branch Point]]
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*[[Residue theory]] and [[Cauchy's residue theorem]]
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*[[Conformal transformation]]
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*[[Contour Integration]]
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*[[Euler's formula]]
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*[[Laurent Series]]
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*[[Morera's Theorem]]
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*[[Polygenic Function]]
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*[[Elliptic function]]s
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[[category:mathematics]]
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[[category:complex analysis]]

Revision as of 20:17, 11 November 2007

Complex analysis is the study of complex numbers of the form:

From this definition a "complex plane" is constructed, consisting of z = x + iy, where x and y are real numbers:

, and
where and are real-valued functions.

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: