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Line 1: − Complex analysis is the study of [[complex number]]s of the form: − : <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: − − : <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. − − 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]] − *[[Residue theory]] and [[Cauchy's residue theorem]] − *[[Conformal transformation]] − *[[Contour Integration]] − *[[Euler's formula]] − *[[Laurent Series]] − *[[Morera's Theorem]] − *[[Polygenic Function]] − *[[Elliptic function]]s − − [[category:mathematics]] − [[category:complex analysis]]
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