Difference between revisions of "Complex analysis"

@@ Line 1: / 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]]

Difference between revisions of "Complex analysis"

Revision as of 02:06, November 12, 2007

Navigation menu

Search