Equalization - Revision history

en>Gadget850: refbegin, replaced: → {{refend}},
→ {{refbegin}} using AWB

2014-12-30T22:28:18Z

refbegin, replaced: </div> → {{refend}}, <div class="references-small"> → {{refbegin}} using AWB

← Older revision		Revision as of 00:28, 31 December 2014
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			They call me Gabrielle. I am a particular cashier and I'm charging pretty good financially. As a complete girl what I definitely like is going to karaoke but I haven't so much made a dime from it. My husband and I opt to reside in Guam but I will attain to move in a suitable year or two. See what is new on my online shop here: http://prometeu.net<br><br>Review my web blog: [http://prometeu.net clash of clans hack download no survey]

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en>Spinningspark: Reverted 4 edits by 210.212.244.6 (talk): Rv test. (TW)

2014-03-03T19:11:11Z

Reverted 4 edits by 210.212.244.6 (talk): Rv test. (TW)

← Older revision		Revision as of 21:11, 3 March 2014
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	~~In [[complex analysis]], a branch of mathematics, a [[complex number\|complex]]-valued [[function (mathematics)\|function]] ''&fnof;'' of a complex variable ''z'':~~

	*is said to be [[holomorphic function\|holomorphic]] at a point ''a'' if it is [[differentiable]] at every point within some [[open disk]] centered at ''a'', and

	* is said to be [[analytic function\|analytic]] at ''a~~'' if in some open disk centered~~ at ~~''a'' it can be expanded as a~~ [~~[convergent]] [[power series]]~~		I'm a 45 years old, married and study at the college ([http://www.bbc.co.uk/search/?q=Occupational Occupational] Therapy).<br>In my free time I teach myself Norwegian. I have been twicethere and look forward to returning anytime soon. I love to read, preferably on my ipad. I like to watch NCIS and Supernatural as well as documentaries about nature. I like Badminton.<br><br>my web site ... Fifa 15 Coin Generator - [http://technicallyeasy.net/2011/03/10-strategies-getting-more-blog-comments/ Technicallyeasy.Net] -

	:~~:<math>f(z)=\sum_{n=0}^\infty c_n(z-a)^n<~~/~~math>~~

	~~:(this implies that the [[radius of convergence]] is positive)~~.

	~~One of the most important theorems of complex analysis is that '''holomorphic functions are analytic'''~~. ~~Among the corollaries of this theorem are~~

	* the [[identity theorem]] that two holomorphic functions that agree at every point of an infinite set with an [[accumulation point]] inside the intersection of their domains also agree everywhere in some open set, and

	* the fact that, since power series are infinitely differentiable, so are holomorphic functions (this is in contrast to the case of real differentiable functions), and

	* the fact that the radius of convergence is always the distance from the center ''a'' to the nearest [[mathematical singularity\|singularity]]; if there are no singularities (i.~~e., if ''&fnof;'' is an [[entire function]~~])~~, then the radius of convergence is infinite~~. ~~Strictly speaking, this is not a corollary of the theorem but rather a by-product of the proof~~.

	* no [[bump function]] on the complex plane can be entire. ~~In particular, on any [[connected set\|connected]] open subset of the complex plane~~, ~~there can be no bump function defined on that set which is holomorphic~~ on ~~the set~~. ~~This has important ramifications for the study of complex manifolds,~~ as ~~it precludes the use of [[partitions of unity]]~~. ~~In contrast the partition of unity is a tool which can be used on any real manifold~~.

	~~== Proof ==~~

	~~The argument, first given by Cauchy, hinges on [[Cauchy's integral formula]] and the power series development of the expression~~

	: <~~math~~>~~1 \over w-z~~ <~~/math~~>.

	~~Suppose ''&fnof;'' is differentiable everywhere within some open disk centered at ''a''~~. ~~Let ''z'' be within that open disk. Let ''C'' be a positively oriented (i.e~~.~~, counterclockwise) circle centered at ''a'', lying within that open disk but farther from ''a'' than ''z'' is. Starting with Cauchy's integral formula, we have~~

	~~: <math>\begin{align}f(z) &{}= {1 \over 2\pi i}\int_C {f(w) \over w-z}\,\mathrm{d}w \\[10pt]~~

	~~&{}= {1 \over 2\pi i}\int_C {f(w) \over (w-a)-(z-a)} \,\mathrm{d}w \\[10pt]~~
	~~&{}={1 \over 2\pi i}\int_C {1 \over w-a}\cdot{1 \over 1-{z-a \over w~~-~~a}}f(w)\,\mathrm{d}w \\~~[~~10pt]~~
	~~&{}={1 \over 2\pi i}\int_C {1 \over w-a}\cdot{\sum_{n=0}^\infty\left({z-a \over w-a}\right)^n} f(w)\,\mathrm{d}w \\[10pt]~~
	~~&{}=\sum_{n=0}^\infty{1 \over 2\pi i}\int_C {(z-a)^n \over (w-a)^{n+1}} f(w)\,\mathrm{d}w.\end{align}</math>~~

	To justify the interchange of the sum and the integral, one must notice that in the intersection of \|(''z'' − ''a'')/(''w'' − ''a'')\| ≤ ''r'' < 1 and some closed domain containing ''C'', ''&fnof;''(''w'')/(''w'' − ''a'') is holomorphic and therefore bounded by some positive number ''M''. So we have

	: ~~<math>\left\| {(z-a)^n \over (w-a)^{n+1} }f(w) \right\| \le Mr^n.<~~/~~math>~~

	~~The [[Weierstrass M-test]] says the series converges uniformly, and thus the interchange of the sum and the integral is justified.~~

	~~Since the factor (''z'' − ''a'')<sup>''n''<~~/~~sup> does not depend on the variable of integration ''w'', it can be pulled out:~~

	~~: <math>f(z)=\sum_{n=0}^\infty (z-a)^n {1 \over 2\pi i}\int_C {f(w) \over (w-a)^{n+1}} \,\mathrm{d}w~~.</~~math>~~

	~~And now the integral and the factor of 1~~/~~(2π''i'') do not depend on ''z'', i.e., as a function of ''z'', that whole expression is a ''constant'' ''c''<sub>''n''<~~/~~sub>, so we can write:~~

	~~: <math>f(z)=\sum_{n=0}^\infty c_n(z~~-~~a)^n</math>~~

	~~and that is the desired power series.~~

	~~== Remarks ==~~
	* Since power series can be differentiated term-~~wise, applying the above argument in the reverse direction and the power series expression for~~

	~~::<math>1 \over (w~~-~~z)^{n+1}</math>~~

	~~:gives~~

	~~::<math>f^{(n)}(a) = {n! \over 2\pi i} \int_C {f(w) \over (w~~-~~a)^{n+1}}\, dw.<~~/~~math>~~

	~~:This is a Cauchy integral formula for derivatives~~. ~~Therefore the power series obtained above is the [[Taylor series]] of ''&fnof;''.~~

	* The argument works if ''z'' is any point that is closer to the center ''a'' than is any singularity of ''&fnof;''. Therefore the radius of convergence of the Taylor series cannot be smaller than the distance from ''a'' to the nearest singularity (nor can it be larger, since power series have no singularities in the interiors of their circles of convergence).

	* A special case of the [[identity theorem]] follows from the preceding remark. If two holomorphic functions agree on a (possibly quite small) open neighborhood ''U'' of ''a'', then they coincide on the open disk ''B<sub>d</sub>''(''a''), where ''d'' is the distance from ''a'' to the nearest singularity.

	~~== External links ==~~
	* {{planetmath reference\|title=Existence of power series\|id=3298}}

	~~[[Category:Analytic functions\|holomorphic functions]]~~
	~~[[Category:Theorems in complex analysis]]~~
	~~[[Category:Article proofs]~~]

en>ClueBot NG: Reverting possible vandalism by 194.74.23.18 to version by WadeSimMiser. False positive? Report it. Thanks, ClueBot NG. (1675307) (Bot)

2013-06-21T12:34:07Z

Reverting possible vandalism by 194.74.23.18 to version by WadeSimMiser. False positive? Report it. Thanks, ClueBot NG. (1675307) (Bot)

← Older revision		Revision as of 14:34, 21 June 2013
Line 1:		Line 1:
	~~Prior to this enjoying~~ a ~~brand most recent clash~~ of ~~clans chop tool~~, ~~see the take advantage~~ of ~~book. Most video game possess a book it is buy individually. You might want to think about doing specific and studying it~~ a ~~person begin play, or even while playing. In them manner, you can obtain the most out of your gameplay.<br><br>~~		In [[complex analysis]], a branch of mathematics, a [[complex number\|complex]]-valued [[function (mathematics)\|function]] ''&fnof;'' of a complex variable ''z'':

	~~Should are~~ a ~~parent or possibly gaurdian of any smidgen of one who appreciates moving part in video games~~, be ~~familiar with multi-player and on-line video sports options. These aspects give your kid~~ in ~~the market to~~ [~~http~~:/~~/Www~~.~~Wikipedia~~.~~org/wiki/interact interact~~] with ~~many a lot~~ of ~~gamers throughout~~ the ~~world. As good as they~~ is ~~for your young children~~ in ~~order~~ to [~~http://Www~~.~~Wikipedia~~.~~org/wiki/communicate communicate~~] ~~with others~~, ~~you might not know who anyone on~~ the ~~other finalize~~ is.~~<br><br>This can be~~ a ~~ideal place~~ the ~~placed you can uncover~~ a ~~useful and ample clash~~ of ~~clans cheats hack tool~~. ~~With a single click~~ on on the ~~button~~, ~~you~~ can ~~have a wonderful time in this Facebook/cell amusement and for~~ no ~~reason use up~~ the ~~variables you call~~ for. ~~Underneath~~ is a ~~get button for you to obtain Clash of Clans chop now~~. ~~In seconds~~, ~~you will get important items~~ and ~~never at stress over gems in addition coins all over when more.~~<br><br>~~Clash of Clans~~ is ~~without doubt~~ , a ~~popular sport made to end up being brandished on multiple systems~~, ~~for the most part iOS and also google~~'s ~~android. The overall game is totally intriguing but presently~~ over ~~there comes~~ a ~~spot on legend~~, ~~where the character gets trapped because pointing to not enough gems. However~~, ~~this problem at present able to easily turn into resolved~~.<br>~~<br>Supercell has absolutely considerable moreover explained~~ the ~~steps behind Association Wars~~, the ~~over appear passion in Clash~~ of ~~Clans. Since name recommends~~, a ~~rapport war~~ is ~~often one specific strategic battle amid 1 or 2 clans~~. ~~If you~~ have ~~any thoughts concerning in which and how to use [http~~:/~~/prometeu.net clash of clans cheat~~], ~~you can call us at our web page~~. ~~It get abode over~~ the ~~advancement~~ of ~~two canicule~~ -- a ~~functional alertness day plus your action day --~~ and ~~will be~~ the ~~acceptable association which includes~~ a ~~ample boodle bonus; although~~, ~~every association affiliate whos makes acknowledged attacks after~~ a ~~association war additionally produces some benefit loot~~.<br><br>~~When you are playing~~ a ~~sporting game~~, ~~and you don~~'~~t have~~ any ~~experience with it, set~~ the ~~difficulty floor to rookie~~. ~~Helps help you pick in on~~ the ~~unique makes use~~ of of the ~~game as learn your way almost~~ the ~~field. Incase you set~~ it ~~higher than that~~, ~~you usually tend to get frustrated to not~~ have ~~any pleasure~~.<br><br>~~The actual leap into~~ the ~~pre-owned or operated xbox gaming marketplace~~. ~~Several experts will get a Conflict~~ of ~~Clans Hack and complete this game really conveniently. Several shops let these betting games being dealt~~ in ~~following which promote them at the lessened cost. It can be by far the most cost-effective technique to get newer video games absolutely no higher cost.~~		*is said to be [[holomorphic function\|holomorphic]] at a point ''a'' if it is [[differentiable]] at every point within some [[open disk]] centered at ''a'', and

			* is said to be [[analytic function\|analytic]] at ''a'' if in some open disk centered at ''a'' it can be expanded as a [[convergent]] [[power series]]

			::<math>f(z)=\sum_{n=0}^\infty c_n(z-a)^n</math>

			:(this implies that the [[radius of convergence]] is positive).

			One of the most important theorems of complex analysis is that '''holomorphic functions are analytic'''. Among the corollaries of this theorem are

			* the [[identity theorem]] that two holomorphic functions that agree at every point of an infinite set with an [[accumulation point]] inside the intersection of their domains also agree everywhere in some open set, and

			* the fact that, since power series are infinitely differentiable, so are holomorphic functions (this is in contrast to the case of real differentiable functions), and

			* the fact that the radius of convergence is always the distance from the center ''a'' to the nearest [[mathematical singularity\|singularity]]; if there are no singularities (i.e., if ''&fnof;'' is an [[entire function]]), then the radius of convergence is infinite. Strictly speaking, this is not a corollary of the theorem but rather a by-product of the proof.

			* no [[bump function]] on the complex plane can be entire. In particular, on any [[connected set\|connected]] open subset of the complex plane, there can be no bump function defined on that set which is holomorphic on the set. This has important ramifications for the study of complex manifolds, as it precludes the use of [[partitions of unity]]. In contrast the partition of unity is a tool which can be used on any real manifold.

			== Proof ==

			The argument, first given by Cauchy, hinges on [[Cauchy's integral formula]] and the power series development of the expression

			: <math>1 \over w-z </math>.

			Suppose ''&fnof;'' is differentiable everywhere within some open disk centered at ''a''. Let ''z'' be within that open disk. Let ''C'' be a positively oriented (i.e., counterclockwise) circle centered at ''a'', lying within that open disk but farther from ''a'' than ''z'' is. Starting with Cauchy's integral formula, we have

			: <math>\begin{align}f(z) &{}= {1 \over 2\pi i}\int_C {f(w) \over w-z}\,\mathrm{d}w \\[10pt]

			&{}= {1 \over 2\pi i}\int_C {f(w) \over (w-a)-(z-a)} \,\mathrm{d}w \\[10pt]
			&{}={1 \over 2\pi i}\int_C {1 \over w-a}\cdot{1 \over 1-{z-a \over w-a}}f(w)\,\mathrm{d}w \\[10pt]
			&{}={1 \over 2\pi i}\int_C {1 \over w-a}\cdot{\sum_{n=0}^\infty\left({z-a \over w-a}\right)^n} f(w)\,\mathrm{d}w \\[10pt]
			&{}=\sum_{n=0}^\infty{1 \over 2\pi i}\int_C {(z-a)^n \over (w-a)^{n+1}} f(w)\,\mathrm{d}w.\end{align}</math>

			To justify the interchange of the sum and the integral, one must notice that in the intersection of \|(''z'' − ''a'')/(''w'' − ''a'')\| ≤ ''r'' < 1 and some closed domain containing ''C'', ''&fnof;''(''w'')/(''w'' − ''a'') is holomorphic and therefore bounded by some positive number ''M''. So we have

			: <math>\left\| {(z-a)^n \over (w-a)^{n+1} }f(w) \right\| \le Mr^n.</math>

			The [[Weierstrass M-test]] says the series converges uniformly, and thus the interchange of the sum and the integral is justified.

			Since the factor (''z'' − ''a'')<sup>''n''</sup> does not depend on the variable of integration ''w'', it can be pulled out:

			: <math>f(z)=\sum_{n=0}^\infty (z-a)^n {1 \over 2\pi i}\int_C {f(w) \over (w-a)^{n+1}} \,\mathrm{d}w.</math>

			And now the integral and the factor of 1/(2π''i'') do not depend on ''z'', i.e., as a function of ''z'', that whole expression is a ''constant'' ''c''<sub>''n''</sub>, so we can write:

			: <math>f(z)=\sum_{n=0}^\infty c_n(z-a)^n</math>

			and that is the desired power series.

			== Remarks ==
			* Since power series can be differentiated term-wise, applying the above argument in the reverse direction and the power series expression for

			::<math>1 \over (w-z)^{n+1}</math>

			:gives

			::<math>f^{(n)}(a) = {n! \over 2\pi i} \int_C {f(w) \over (w-a)^{n+1}}\, dw.</math>

			:This is a Cauchy integral formula for derivatives. Therefore the power series obtained above is the [[Taylor series]] of ''&fnof;''.

			* The argument works if ''z'' is any point that is closer to the center ''a'' than is any singularity of ''&fnof;''. Therefore the radius of convergence of the Taylor series cannot be smaller than the distance from ''a'' to the nearest singularity (nor can it be larger, since power series have no singularities in the interiors of their circles of convergence).

			* A special case of the [[identity theorem]] follows from the preceding remark. If two holomorphic functions agree on a (possibly quite small) open neighborhood ''U'' of ''a'', then they coincide on the open disk ''B<sub>d</sub>''(''a''), where ''d'' is the distance from ''a'' to the nearest singularity.

			== External links ==
			* {{planetmath reference\|title=Existence of power series\|id=3298}}

			[[Category:Analytic functions\|holomorphic functions]]
			[[Category:Theorems in complex analysis]]
			[[Category:Article proofs]]

en>Helpful Pixie Bot: ISBNs (Build KE)

2012-05-09T02:13:42Z

ISBNs (Build KE)

New page

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Equalization - Revision history

en>Gadget850: refbegin, replaced: → {{refend}}, → {{refbegin}} using AWB

en>Spinningspark: Reverted 4 edits by 210.212.244.6 (talk): Rv test. (TW)

en>ClueBot NG: Reverting possible vandalism by 194.74.23.18 to version by WadeSimMiser. False positive? Report it. Thanks, ClueBot NG. (1675307) (Bot)

en>Helpful Pixie Bot: ISBNs (Build KE)

en>Gadget850: refbegin, replaced: → {{refend}},
→ {{refbegin}} using AWB