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Line 1: − + − − : <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. Line 17:
Line 7: − + − + − + − + − + − Complex Residue+ − + − + − de Moivre's Identity+ − + − Inside-Outside Theorem+ − Jordan's Lemma+ − − Liouville's Conformality Theorem − Monogenic Function − Morera's Theorem − Permanence of Algebraic Form − Pole − Polygenic Function − + +
→top: re-categorisation
Complex analysis is the study of [[complex number]]s of the form:
'''Complex analysis''' is the study of functions supported on the [[complex number]]s.
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.
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.
The central result in complex analysis is the [[Cauchy integral theorem]], and a powerful claim of complex analysis is Picard's great theorem.
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]]".
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:
Additional concepts in complex analysis include the following:
Analytic Continuation
*[[Analytic Continuation]]
Argument Principle
*[[Argument Principle]]
Branch Cut and Branch Point
*[[Branch Cut]] and [[Branch Point]]
Cauchy Principal Value
*[[Residue theory]] and [[Cauchy's residue theorem]]
*[[Conformal transformation]]
Conformal Mapping
*[[Contour Integration]]
Contour Integration
*[[Euler's formula]]
*[[Laurent Series]]
Euler Formula
*[[Morera's Theorem]]
*[[Polygenic Function]]
*[[Elliptic function]]s
Laurent Series
[[category:mathematics]]
[[Category:Mathematics]]
[[Category:Complex Analysis]]