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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]]
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