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−  Complex analysis is the study of [[complex number]]s of the form:
 
   
−  : <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
 
−  : <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 realvalued functions.
 
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−  so that real numbers are on the xaxis and imaginary numbers are on the yaxis.
 
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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 [[CauchyRiemann 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]]
 
−  *[[Argument Principle]]
 
−  *[[Branch Cut]] and [[Branch Point]]
 
−  *[[Residue theory]] and [[Cauchy's residue theorem]]
 
−  *[[Conformal transformation]]
 
−  *[[Contour Integration]]
 
−  *[[Euler's formula]]
 
−  *[[Laurent Series]]
 
−  *[[Morera's Theorem]]
 
−  *[[Polygenic Function]]
 
−  *[[Elliptic function]]s
 
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−  [[category:mathematics]]
 
−  [[category:complex analysis]]
 