Pulse-width modulation - Revision history

175.142.17.150: /* Types */

2015-01-11T08:30:31Z

Types

← Older revision		Revision as of 10:30, 11 January 2015
Line 1:		Line 1:
	~~My name~~: ~~Harold Luong~~<br>~~My age~~: ~~32 years old~~<br>~~Country~~: ~~Netherlands~~<br>~~Town~~: ~~Tholen~~ <br>~~Post code~~: ~~4691 Jr~~<br>~~Address~~: ~~Kotterstraat 142~~<br><br>~~Feel free to surf to my website~~ [http://~~asphalt8hackerz~~.~~wordpress~~.com ~~asphalt 8 hack~~]		== ヤンの三種類に達するの「 '3'色 'それは ==

			それはすぐに、赤色の「色」のタッチは静か分より弱多く、急速な拡大の最後の激怒の目を表面化、ナマの赤」の色が [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-13.html 時計カシオ] '他の二つである、再びローリング雷雲ですヤンかなり「色」。<br><br>色」！ヤンの三種類に達するの「 [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-4.html カシオ腕時計ソーラー] '3'色 [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-5.html カシオ時計] 'それは！'<br><br>が完全に見えた三万円、「カラー」、少し静かな広場を浮上し、突然沸騰し、再び、多くの人が感動を流し直面し、時間のこの短い期間に一日は、彼らがある3 [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-0.html カシオ時計価格] 3「カラー」Danlei、この壮大な他のを見て、本当にまれな世紀と見なされる！<br><br>このその後、Danleiの3ヤン「色」の存在を見て、高齢者のムーの骨が3つだけ「カラー」Danleiであれば、高いプラットフォームは、いくつかのDanta長老の心は、非常に緊張したアップとなっている回償還が奇数シャオヤンと清華は、競争する強さを持って存在する
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			== 北境界の深さに王と ==

			言葉は、それが本当にやや面倒であるので、彼はダーク外にあぐらをかい鉱山エリアを座っていた、この境界線の中で状況を観察し始めた [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-3.html カシオ gps 時計]。<br><br>この観察は、移動を気にし、1時間後、表示されないかを判断するための事故の後、シャオヤン安堵のほんの少しのため息を持っていたほぼ一時間ということで、王は、措置をとることを躊躇北の背後にある周囲散歩の深さに向かって一歩一歩 [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-4.html カシオ腕時計ソーラー]。<br><br>'ブーム [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-9.html カシオ腕時計チタン]！'<br>北境界の深さに王と<br>は、偶然ではない、すぐに10〜10フィート太い黒「色」サンダー回りのモーメントは、一般的にはPythonのような遅い、ダークレイのいくつかの注目を喚起ノースキングヘッドにコイル状のスロー蠕動本体は、ララの雷を笑う、連続体はダークサンダーから来ました [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-1.html カシオ時計メンズ]。<br>シャオヤンマインドコントロールで<br>、北王は停止し、それを見上げた黒 [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-12.html 電波腕時計カシオ] '色'サンダーを、ゆっくりとストレッチ
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			== 丁寧に磨き、熱を ==

			「薬」は、物質が、それは真剣に、彼は「運動」地心Qinglian制御する場合であっても、シャオヤンのために、絶対に耐えられないことですが精錬の故障率を上げるため、より少ないでしょう過ちので、能力で発射するが、それでも「薬」素材ウェイ、保険の各2株を改良する代わりに選ぶ、「医学」木材フルキャストを下に突進することはできません [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-0.html casio 腕時計]。<br>丁寧に磨き、熱を<br>とシャオヤンはこのアイデアを持っており、現時点では顔に小さな王女、劉Lingさん、さらにはそれがされている傲慢傲慢ヤン李、を含む多くの人が、「色」威厳のある、慎重に制御する各菌株「医学」材料ということ、オールクリア、この時点で、損失」の薬」の材料は、人々が気を悪くする方法があるでしょう [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-0.html カシオ電波ソーラー腕時計]。<br><br>巨大な正方形の上に、すべての参加者が黙っている、唯一のバック平方フォワード」スイング」では、「医学」材料そっと放出さパチパチ炎を洗練 [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-14.html casio 腕時計 phys]。<br>このようなセキュリティにおけるhuanhangrn
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			== 霧で満たさ ==

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en>Ninney: Reverted 1 edit by 14.139.110.178 (talk) to last revision by Willardmcg. (TW)

2014-02-21T12:03:20Z

Reverted 1 edit by 14.139.110.178 (talk) to last revision by Willardmcg. (TW)

← Older revision		Revision as of 14:03, 21 February 2014
Line 1:		Line 1:
	~~[[Image~~:Gamma abs 3D.png\|right\|thumb\|The absolute value of the [[Gamma function]]. This shows that a function becomes infinite at the poles (left). On the right, the Gamma function does not have poles, it just increases quickly.]]		My name: Harold Luong<br>My age: 32 years old<br>Country: Netherlands<br>Town: Tholen <br>Post code: 4691 Jr<br>Address: Kotterstraat 142<br><br>Feel free to surf to my website [http://asphalt8hackerz.wordpress.com asphalt 8 hack]
	In the mathematical field of [[complex analysis]], a '''pole''' of a [[meromorphic function]] is a certain type of [[mathematical singularity\|singularity]] that behaves like the singularity of <math> \scriptstyle \frac{1}{z^n} </math> at ''z'' = 0. For a pole of the function ''f''(''z'') at point ''a'' the function [[Infinity#Complex_analysis\|approaches infinity]] <~~!--uniformly--~~> ~~as ''z'' approaches ''a''.~~

	~~== Definition ==~~
	~~Formally, suppose ''U'' is an [[open subset]] of the [[complex plane]] '''C''', ''a'' is an element of ''U'' and ''f''~~ : ''U'' \ {''a''} → '''C''' is a function which is [[holomorphic]] over its domain. If there exists a holomorphic function ''g'' : ''U'' → '''C''' and a positive integer ''n'', such that for all ''z'' in ''U'' \ {''a''}
	~~:<math> f(z) = \frac{g(z)}{(z-a)^n}~~ <~~/math~~>
	~~holds, then ''a'' is called a '''pole of ''f'''''. The smallest such ''n'' is called the '''order of the pole'''. A pole of order 1 is called a '''simple pole'''.~~

	A few authors allow the order of a pole to be zero, in which case a pole of order zero is either a regular point or a [[removable singularity]]. However, it is more usual to require the order of a pole to be positive.

	~~From above several equivalent characterizations can be deduced:~~

	~~If ''n'' is the order of pole ''a'', then necessarily ''g''(''a'') ≠ 0 for the function ''g'' in the above expression. So we can put~~

	:<~~math~~>~~f(z) = \frac{1}{h(z)}</math>~~

	for some ''h'' that is holomorphic in an open neighborhood of ''a'' and has a zero of order ''n'' at ''a''. So informally one might say that poles occur as reciprocals of zeros of holomorphic functions.

	~~Also, by the holomorphy of ''g'', ''f'' can be expressed as:~~

	:<~~math~~>~~f(z) = \frac{a_{-n}}{ (z - a)^n } + \cdots + \frac{a_{-1}}{ (z - a) } + \sum_{k\, \geq \,0} a_k (z - a)^k.</math>~~

	This is a [[Laurent series]] with finite ''principal part''. The holomorphic function <math>\scriptstyle \sum_{k\,\ge\,0} a_k(z\, - \,a)^k</math> (on ''U'') is called the ''regular part'' of ''f''. So the point ''a'' is a pole of order ''n'' of ''f'' if and only if all the terms in the Laurent series expansion of ''f'' around ''a'' below degree −''n'' vanish and the term in degree −''n'' is not zero.

	~~==Pole at infinity==~~
	A complex function can be defined as having a pole at the [[point at infinity]]. In this case ''U'' has to be a neighborhood of infinity, such as the exterior of any closed ball. To use the previous definition, a meaning for ''g'' being holomorphic at ∞ is needed. Alternately, a definition can be given starting from the definition at a finite point by suitably mapping the point at infinity to a finite point. The map <math>\scriptstyle z \mapsto \frac{1}{z}</math> does that. Then, by definition, a function ''f'' holomorphic in a neighborhood of infinity has a pole at infinity if the function <math>\scriptstyle f(\frac{1}{z})</math> (which will be holomorphic in a neighborhood of <math>\scriptstyle z = 0</math>), has a pole at <math>\scriptstyle z = 0</math>, the order of which will be regarded as the order of the pole of ''f'' at infinity.

	~~==Pole of a function on a complex manifold==~~
	~~In general, having a function <math>\scriptstyle f~~:~~\; M\, \rightarrow \,\mathbb{C}~~<~~/math~~> that is holomorphic in a neighborhood, <math>\scriptstyle U</math>, of the point <math>\scriptstyle a</math>, in the [[complex manifold]] ''M'', it is said that ''f'' has a pole at ''a'' of order ''n'' if, having a [[Atlas (topology)\|chart]] <math>\scriptstyle \phi:~~\; U\, \rightarrow \,\mathbb{C}~~<~~/math~~>~~, the function~~ <~~math~~>\scriptstyle f\, \circ \,\phi^{-1}:\; \mathbb{C}\, \rightarrow \,\mathbb{C}</math> has a pole of order ''n'' at <math>\scriptstyle \phi(a)</math> (which can be taken as being zero if a convenient choice of the chart is made).
	]
	~~The pole at infinity is the simplest nontrivial example of this definition in which ''M'' is taken~~ to ~~be the [[Riemann sphere]] and the chart is taken~~ to ~~be <math>\scriptstyle \phi(z)\, = \,\frac{1}{z}</math>.~~

	~~== Examples ==~~
	* The function

	~~::<math>f(z) = \frac{3}{z}</math>~~

	~~: has a pole of order 1 or simple pole at <math>\scriptstyle z\, = \,0</math>.~~

	* The function

	~~:: <math>f(z) = \frac{z+2}{(z-5)^2(z+7)^3}</math>~~

	~~: has a pole of order 2 at <math>\scriptstyle z\, = \,5</math> and a pole of order 3 at <math>\scriptstyle z\, = \,-7</math>.~~

	* The function

	~~:: <math>f(z) = \frac{z-4}{e^z-1}</math>~~

	: has poles of order 1 at <math>\scriptstyle z\, = \,2\pi ni\text{ for } n\, = \,\dots,\, -1,\, 0,\, 1,\, \dots.</math> To see that, write <math>\scriptstyle e^z</math> in [[Taylor series]] around the origin.

	* The function

	~~::<math>f(z) = z</math>~~

	~~: has a single pole at infinity of order 1.~~

	~~== Terminology and generalizations ==~~
	~~If the first derivative of a function ''f'' has a simple pole at ''a'', then ''a'' is a [[branch point]] of ''f''. (The converse need not be true).~~

	~~A non-removable singularity that is not a pole or a [[branch point]] is called an [[essential singularity]].~~

	~~A complex function which is holomorphic except for some isolated singularities and whose only singularities are poles is called [[meromorphic]].~~

	~~==See also==~~
	* [[Control theory#Stability]]
	* [[Filter design]]
	* [[Filter (signal processing)]]
	* [[Nyquist stability criterion]]
	* [[Pole–zero plot]]
	* [[Residue (complex analysis)]]
	* [[Zero (complex analysis)]]

	~~== External links ==~~
	* {{MathWorld \| urlname= Pole \| title= Pole}}
	* [http://~~math.fullerton.edu/mathews/c2003/SingularityZeroPoleMod~~.~~html Module for Zeros and Poles by John H~~. ~~Mathews]~~

	~~[[Category:Meromorphic functions]~~]

en>Denisarona: Reverted edits by 220.225.196.142 (talk) to last version by Monkbot

2014-02-03T07:56:15Z

Reverted edits by 220.225.196.142 (talk) to last version by Monkbot

← Older revision		Revision as of 09:56, 3 February 2014
Line 1:		Line 1:
	~~Hello~~, I'~~m Fabian~~, a ~~24 year old from La Roche~~-~~Sur~~-~~Yon~~, ~~France~~.<br>~~My hobbies include~~ (~~but are~~ not ~~limited~~ to) ~~Machining~~, ~~Target Shooting and watching Arrested Development~~.<br><br>~~Here~~ is ~~my web blog~~; [~~http~~://~~Www~~.~~Uzaleznienia~~.~~Edu.pl~~/~~index~~.~~php~~/~~aktualnosci~~/~~item~~/1-~~ciernie-gwiazd~~/ ~~Fifa 15 Coin Generator~~]		[[Image:Gamma abs 3D.png\|right\|thumb\|The absolute value of the [[Gamma function]]. This shows that a function becomes infinite at the poles (left). On the right, the Gamma function does not have poles, it just increases quickly.]]
			In the mathematical field of [[complex analysis]], a '''pole''' of a [[meromorphic function]] is a certain type of [[mathematical singularity\|singularity]] that behaves like the singularity of <math> \scriptstyle \frac{1}{z^n} </math> at ''z'' = 0. For a pole of the function ''f''(''z'') at point ''a'' the function [[Infinity#Complex_analysis\|approaches infinity]] <!--uniformly--> as ''z'' approaches ''a''.

			== Definition ==
			Formally, suppose ''U'' is an [[open subset]] of the [[complex plane]] '''C''', ''a'' is an element of ''U'' and ''f'' : ''U'' \ {''a''} → '''C''' is a function which is [[holomorphic]] over its domain. If there exists a holomorphic function ''g'' : ''U'' → '''C''' and a positive integer ''n'', such that for all ''z'' in ''U'' \ {''a''}
			:<math> f(z) = \frac{g(z)}{(z-a)^n} </math>
			holds, then ''a'' is called a '''pole of ''f'''''. The smallest such ''n'' is called the '''order of the pole'''. A pole of order 1 is called a '''simple pole'''.

			A few authors allow the order of a pole to be zero, in which case a pole of order zero is either a regular point or a [[removable singularity]]. However, it is more usual to require the order of a pole to be positive.

			From above several equivalent characterizations can be deduced:

			If ''n'' is the order of pole ''a'', then necessarily ''g''(''a'') ≠ 0 for the function ''g'' in the above expression. So we can put

			:<math>f(z) = \frac{1}{h(z)}</math>

			for some ''h'' that is holomorphic in an open neighborhood of ''a'' and has a zero of order ''n'' at ''a''. So informally one might say that poles occur as reciprocals of zeros of holomorphic functions.

			Also, by the holomorphy of ''g'', ''f'' can be expressed as:

			:<math>f(z) = \frac{a_{-n}}{ (z - a)^n } + \cdots + \frac{a_{-1}}{ (z - a) } + \sum_{k\, \geq \,0} a_k (z - a)^k.</math>

			This is a [[Laurent series]] with finite ''principal part''. The holomorphic function <math>\scriptstyle \sum_{k\,\ge\,0} a_k(z\, - \,a)^k</math> (on ''U'') is called the ''regular part'' of ''f''. So the point ''a'' is a pole of order ''n'' of ''f'' if and only if all the terms in the Laurent series expansion of ''f'' around ''a'' below degree −''n'' vanish and the term in degree −''n'' is not zero.

			==Pole at infinity==
			A complex function can be defined as having a pole at the [[point at infinity]]. In this case ''U'' has to be a neighborhood of infinity, such as the exterior of any closed ball. To use the previous definition, a meaning for ''g'' being holomorphic at ∞ is needed. Alternately, a definition can be given starting from the definition at a finite point by suitably mapping the point at infinity to a finite point. The map <math>\scriptstyle z \mapsto \frac{1}{z}</math> does that. Then, by definition, a function ''f'' holomorphic in a neighborhood of infinity has a pole at infinity if the function <math>\scriptstyle f(\frac{1}{z})</math> (which will be holomorphic in a neighborhood of <math>\scriptstyle z = 0</math>), has a pole at <math>\scriptstyle z = 0</math>, the order of which will be regarded as the order of the pole of ''f'' at infinity.

			==Pole of a function on a complex manifold==
			In general, having a function <math>\scriptstyle f:\; M\, \rightarrow \,\mathbb{C}</math> that is holomorphic in a neighborhood, <math>\scriptstyle U</math>, of the point <math>\scriptstyle a</math>, in the [[complex manifold]] ''M'', it is said that ''f'' has a pole at ''a'' of order ''n'' if, having a [[Atlas (topology)\|chart]] <math>\scriptstyle \phi:\; U\, \rightarrow \,\mathbb{C}</math>, the function <math>\scriptstyle f\, \circ \,\phi^{-1}:\; \mathbb{C}\, \rightarrow \,\mathbb{C}</math> has a pole of order ''n'' at <math>\scriptstyle \phi(a)</math> (which can be taken as being zero if a convenient choice of the chart is made).
			]
			The pole at infinity is the simplest nontrivial example of this definition in which ''M'' is taken to be the [[Riemann sphere]] and the chart is taken to be <math>\scriptstyle \phi(z)\, = \,\frac{1}{z}</math>.

			== Examples ==
			* The function

			::<math>f(z) = \frac{3}{z}</math>

			: has a pole of order 1 or simple pole at <math>\scriptstyle z\, = \,0</math>.

			* The function

			:: <math>f(z) = \frac{z+2}{(z-5)^2(z+7)^3}</math>

			: has a pole of order 2 at <math>\scriptstyle z\, = \,5</math> and a pole of order 3 at <math>\scriptstyle z\, = \,-7</math>.

			* The function

			:: <math>f(z) = \frac{z-4}{e^z-1}</math>

			: has poles of order 1 at <math>\scriptstyle z\, = \,2\pi ni\text{ for } n\, = \,\dots,\, -1,\, 0,\, 1,\, \dots.</math> To see that, write <math>\scriptstyle e^z</math> in [[Taylor series]] around the origin.

			* The function

			::<math>f(z) = z</math>

			: has a single pole at infinity of order 1.

			== Terminology and generalizations ==
			If the first derivative of a function ''f'' has a simple pole at ''a'', then ''a'' is a [[branch point]] of ''f''. (The converse need not be true).

			A non-removable singularity that is not a pole or a [[branch point]] is called an [[essential singularity]].

			A complex function which is holomorphic except for some isolated singularities and whose only singularities are poles is called [[meromorphic]].

			==See also==
			* [[Control theory#Stability]]
			* [[Filter design]]
			* [[Filter (signal processing)]]
			* [[Nyquist stability criterion]]
			* [[Pole–zero plot]]
			* [[Residue (complex analysis)]]
			* [[Zero (complex analysis)]]

			== External links ==
			* {{MathWorld \| urlname= Pole \| title= Pole}}
			* [http://math.fullerton.edu/mathews/c2003/SingularityZeroPoleMod.html Module for Zeros and Poles by John H. Mathews]

			[[Category:Meromorphic functions]]

en>GoShow: Reverted edit(s) by 49.204.4.52 identified as test/vandalism using STiki edits.

2012-08-29T13:43:47Z

Reverted edit(s) by 49.204.4.52 identified as test/vandalism using STiki edits.

New page

Hello, I'm Fabian, a 24 year old from La Roche-Sur-Yon, France.<br>My hobbies include (but are not limited to) Machining, Target Shooting and watching Arrested Development.<br><br>Here is my web blog; [http://Www.Uzaleznienia.Edu.pl/index.php/aktualnosci/item/1-ciernie-gwiazd/ Fifa 15 Coin Generator]