~~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 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]]~~

~~*[[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]]~~