Complex analysis - Revision history

DavidB4-bot: /* top */re-categorisation

2018-08-14T01:02:42Z

top: re-categorisation

DavidB4-bot: /* top */clean up & uniformity

2016-07-13T06:38:48Z

top: clean up & uniformity

JoelR at 06:22, January 28, 2009

2009-01-28T06:22:39Z

DaBoss3 at 01:06, April 25, 2008

2008-04-25T01:06:44Z

DaBoss3: Removed wrong and irrelevent material

2008-04-25T01:06:37Z

Removed wrong and irrelevent material

Mettmach at 12:22, January 28, 2008

2008-01-28T12:22:02Z

TheGuy: Undo revision 332029 by Special:Contributions/Jameson (User talk:Jameson)

2007-11-12T02:17:36Z

Undo revision 332029 by Special:Contributions/Jameson (User talk:Jameson)

Jameson: Removing all content from page

2007-11-12T02:06:13Z

Removing all content from page

Jaques at 02:18, April 14, 2007

2007-04-14T02:18:34Z

Jaques at 21:08, April 7, 2007

2007-04-07T21:08:35Z

← Older revision		Revision as of 01:02, August 14, 2018
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	[[Category:Mathematics]]		[[Category:Mathematics]]
−	[[Category:Complex ~~analysis~~]]	+	[[Category:Complex Analysis]]

@@ Line 23: / Line 23: @@
 *[[Elliptic function]]s
-[[category:mathematics]]
+[[Category:Mathematics]]
-[[category:complex analysis]]
+[[Category:Complex analysis]]

← Older revision		Revision as of 06:22, January 28, 2009
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−	Complex analysis is the study of functions supported on the [[complex number]]s.	+	'''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.

← Older revision		Revision as of 01:06, April 25, 2008
Line 1:		Line 1:
−	Complex analysis is the study of functions supported on the [[complex number]s.	+	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.

← Older revision		Revision as of 01:06, April 25, 2008
Line 1:		Line 1:
−	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.
−
−	~~: <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.~~
−
−	~~Traditionally, mathematical objects are derived from~~ [[~~physics]] or [[geometry]]. The~~ complex numbers on the other hand were constructed out of the blue as an ad-hoc solution to certain problems that appeared to be unsolvable with real numbers. Initially unanimously rejected by the leading mathematicians, they were eventually accepted by a growing number ~~of mathematicians due to a series of spectacular results. Not all of those results can be proved with [[elementary techniques~~]~~] however, and are thus considered questionable by more rigorous mathematicians~~.

	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.

← Older revision		Revision as of 12:22, January 28, 2008
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	so that real numbers are on the x-axis and imaginary numbers are on the y-axis.		so that real numbers are on the x-axis and imaginary numbers are on the y-axis.
		+
		+	Traditionally, mathematical objects are derived from [[physics]] or [[geometry]]. The complex numbers on the other hand were constructed out of the blue as an ad-hoc solution to certain problems that appeared to be unsolvable with real numbers. Initially unanimously rejected by the leading mathematicians, they were eventually accepted by a growing number of mathematicians due to a series of spectacular results. Not all of those results can be proved with [[elementary techniques]] however, and are thus considered questionable by more rigorous mathematicians.

	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.

← Older revision	Revision as of 02:17, November 12, 2007
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	+	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]]

← Older revision		Revision as of 02:06, November 12, 2007
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]]~~

← Older revision		Revision as of 02:18, April 14, 2007
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	[[category:mathematics]]		[[category:mathematics]]
		+	[[category:complex analysis]]

@@ Line 27: / Line 27: @@
 *[[Conformal transformation]]
 *[[Contour Integration]]
-*[[Euler's identity]]
+*[[Euler's formula]]
 *[[Laurent Series]]
 *[[Morera's Theorem]]