Spectral centroid - Revision history

en>Magioladitis: Replace unicode entity nbsp for character [NBSP] (or space) per WP:NBSP + other fixes, replaced: → (3) using AWB (10331)

2014-07-29T08:01:40Z

Replace unicode entity nbsp for character [NBSP] (or space) per WP:NBSP + other fixes, replaced: → (3) using AWB (10331)

← Older revision		Revision as of 10:01, 29 July 2014
Line 1:		Line 1:
	~~In [[complex analysis]], a branch of mathematics, the~~ '~~''Schwarz integral formula''', named after [[Hermann Schwarz]], allows one to recover a [[holomorphic function]], [[up to]]~~ an ~~imaginary constant, from the boundary values of its real part.~~		The author's title is Andera and she believes it seems quite good. I am an invoicing officer and I'll be promoted soon. The favorite pastime for him and his children is fashion and he'll be beginning something else along with it. For a while I've been in Alaska but I will have to move in a yr or two.<br><br>Here is my website; best [http://203.250.78.160/zbxe/?document_srl=1792908 cheap psychic readings] [http://www.publicpledge.com/blogs/post/7034 tarot readings] ([http://myfusionprofits.com/groups/find-out-about-self-improvement-tips-here/ myfusionprofits.com])

	~~==Unit disc==~~
	~~Let ''ƒ'' = ''u'' + ''iv'~~' be ~~a function which is holomorphic on the closed unit disc {''z'' ∈ '''C''' \| \|''z''\| ≤ 1}~~. ~~Then~~

	~~: <math> f(z) = \frac{1}{2\pi i} \oint_{\|\zeta\| = 1} \frac{\zeta + z}{\zeta - z} \text{Re}(f(\zeta)) \, \frac{d\zeta}{\zeta}~~
	~~+ i\text{Im}(f(0))</math>~~

	for ~~all \|~~'~~'z''\| < 1~~.

	~~==Upper half-plane==~~
	~~Let~~ '~~'ƒ'' = ''u'' + ''iv'' be~~ a ~~function that is holomorphic on the closed [[upper half-plane]] {''z'' ∈ '''C''' \| Im(''z'') ≥ 0} such that, for some ''α'' > 0, \|''z''~~<~~sup~~>~~''α''~~<~~/sup~~>~~ ''ƒ''(''z'')\|~~ is ~~bounded on the closed upper half-plane. Then~~

	~~: <math>~~
	~~f(z)~~
	=
	~~\frac{1}{\pi i} \int_{-\infty}^\infty \frac{u(\zeta,0)}{\zeta - z} \, d\zeta~~
	=
	~~\frac{1}{\pi i} \int_{-\infty}^\infty \frac{Re(f)(\zeta+0i)}{\zeta - z} \, d\zeta~~
	~~</math>~~

	~~for all Im(''z'') > 0.~~

	~~Note that, as compared to the version on the unit disc, this formula does not have an arbitrary constant added to the integral~~; ~~this is because the additional decay condition makes the conditions for this formula more stringent.~~

	~~== Corollary of Poisson integral formula ==~~

	~~The formula follows from~~ [~~[Poisson integral formula]] applied to ''u'':<ref>~~
	~~{{cite web~~
	~~\|url=~~http://~~books~~.~~google~~.~~com~~/~~books~~?id=~~NVrgftOGG1sC&pg=PA9&ots=FTpLISInOP&dq=Schwarz+formula&sig=tYdkW2Mq4IJg-gTIDWVCEI4HKCE~~
	~~\|title=Lectures on Entire Functions - Google Book Search~~
	~~\|publisher=books.google.com~~
	~~\|accessdate=2008-06-26~~
	~~\|last=~~
	~~\|first=~~
	}}
	~~</ref><ref>The derivation without an appeal to the Poisson formula can be found at:~~ http://~~planetmath~~.~~org~~/~~encyclopedia~~/~~PoissonFormula.html<~~/~~ref>~~

	: ~~<math>u(z) = \frac{1}{2\pi}\int_0^{2\pi} u(e^{i\psi}) \operatorname{Re} {e^{i\psi} + z \over e^{i\psi} - z} \, d\psi\text{ for }\|z\| < 1~~.</~~math>~~

	~~By means of conformal maps, the formula can be generalized to any simply connected open set.~~

	~~== Notes and references ==~~
	~~<references~~ />

	* [[Lars Ahlfors\|Ahlfors, Lars V.]] (1979), ''Complex Analysis'', Third Edition, McGraw-~~Hill, ISBN 0~~-07-~~085008~~-9
	* Remmert, Reinhold (1990), ''Theory of Complex Functions'', Second Edition, Springer, ISBN 0-~~387~~-~~97195-5~~
	* Saff, E. ~~B., and A. D. Snider (1993~~)~~, ''Fundamentals of Complex Analysis for Mathematics, Science, and Engineering'', Second Edition, Prentice Hall, ISBN 0-13-327461-6~~

	~~[[Category:Complex analysis]]~~

103.21.125.79 at 06:20, 4 August 2013

2013-08-04T06:20:11Z

← Older revision		Revision as of 08:20, 4 August 2013
Line 1:		Line 1:
	~~Hi there~~. My ~~[http~~://~~kpupf~~.~~com~~/~~xe/talk/735373 online psychic chat] title~~ is ~~Sophia Meagher although it is not~~ the ~~title on my birth certification~~. ~~Some time in~~ [http://~~www~~.~~sirudang~~.com/~~siroo_Notice/2110 phone psychic] the past she selected to live in Alaska and her mothers and fathers live nearby~~. ~~I am an invoicing officer and I'll be promoted soon. It's not a common factor but what she likes doing is to perform domino but she doesn't have the time recently~~.<br><br>~~My weblog ... free tarot readings ([~~http://~~www~~.~~zavodpm~~.ru/~~blogs~~/~~glennmusserrvji~~/~~14565~~-~~great~~-~~hobby~~-~~advice~~-~~assist~~-~~allow~~-~~you~~-~~get~~-~~going please click the up coming article~~])		In [[complex analysis]], a branch of mathematics, the '''Schwarz integral formula''', named after [[Hermann Schwarz]], allows one to recover a [[holomorphic function]], [[up to]] an imaginary constant, from the boundary values of its real part.

			==Unit disc==
			Let ''ƒ'' = ''u'' + ''iv'' be a function which is holomorphic on the closed unit disc {''z'' ∈ '''C''' \| \|''z''\| ≤ 1}. Then

			: <math> f(z) = \frac{1}{2\pi i} \oint_{\|\zeta\| = 1} \frac{\zeta + z}{\zeta - z} \text{Re}(f(\zeta)) \, \frac{d\zeta}{\zeta}
			+ i\text{Im}(f(0))</math>

			for all \|''z''\| < 1.

			==Upper half-plane==
			Let ''ƒ'' = ''u'' + ''iv'' be a function that is holomorphic on the closed [[upper half-plane]] {''z'' ∈ '''C''' \| Im(''z'') ≥ 0} such that, for some ''α'' > 0, \|''z''<sup>''α''</sup> ''ƒ''(''z'')\| is bounded on the closed upper half-plane. Then

			: <math>
			f(z)
			=
			\frac{1}{\pi i} \int_{-\infty}^\infty \frac{u(\zeta,0)}{\zeta - z} \, d\zeta
			=
			\frac{1}{\pi i} \int_{-\infty}^\infty \frac{Re(f)(\zeta+0i)}{\zeta - z} \, d\zeta
			</math>

			for all Im(''z'') > 0.

			Note that, as compared to the version on the unit disc, this formula does not have an arbitrary constant added to the integral; this is because the additional decay condition makes the conditions for this formula more stringent.

			== Corollary of Poisson integral formula ==

			The formula follows from [[Poisson integral formula]] applied to ''u'':<ref>
			{{cite web
			\|url=http://books.google.com/books?id=NVrgftOGG1sC&pg=PA9&ots=FTpLISInOP&dq=Schwarz+formula&sig=tYdkW2Mq4IJg-gTIDWVCEI4HKCE
			\|title=Lectures on Entire Functions - Google Book Search
			\|publisher=books.google.com
			\|accessdate=2008-06-26
			\|last=
			\|first=
			}}
			</ref><ref>The derivation without an appeal to the Poisson formula can be found at: http://planetmath.org/encyclopedia/PoissonFormula.html</ref>

			: <math>u(z) = \frac{1}{2\pi}\int_0^{2\pi} u(e^{i\psi}) \operatorname{Re} {e^{i\psi} + z \over e^{i\psi} - z} \, d\psi\text{ for }\|z\| < 1.</math>

			By means of conformal maps, the formula can be generalized to any simply connected open set.

			== Notes and references ==
			<references />

			* [[Lars Ahlfors\|Ahlfors, Lars V.]] (1979), ''Complex Analysis'', Third Edition, McGraw-Hill, ISBN 0-07-085008-9
			* Remmert, Reinhold (1990), ''Theory of Complex Functions'', Second Edition, Springer, ISBN 0-387-97195-5
			* Saff, E. B., and A. D. Snider (1993), ''Fundamentals of Complex Analysis for Mathematics, Science, and Engineering'', Second Edition, Prentice Hall, ISBN 0-13-327461-6

			[[Category:Complex analysis]]

en>Mcld: /* Alternative usage */ fix error

2011-10-12T10:34:57Z

Alternative usage: fix error

New page

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