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| [[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''.
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| == Definition ==
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| 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''}
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| :<math> f(z) = \frac{g(z)}{(z-a)^n} </math>
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| 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'''.
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| 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.
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| From above several equivalent characterizations can be deduced:
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| 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
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| :<math>f(z) = \frac{1}{h(z)}</math> | |
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| 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.
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| Also, by the holomorphy of ''g'', ''f'' can be expressed as:
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| :<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> | |
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| 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.
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| ==Pole at infinity==
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| 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.
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| ==Pole of a function on a complex manifold==
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| 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).
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| ]
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| 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>.
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| == Examples ==
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| * The function
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| ::<math>f(z) = \frac{3}{z}</math>
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| : has a pole of order 1 or simple pole at <math>\scriptstyle z\, = \,0</math>.
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| * The function
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| :: <math>f(z) = \frac{z+2}{(z-5)^2(z+7)^3}</math>
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| : has a pole of order 2 at <math>\scriptstyle z\, = \,5</math> and a pole of order 3 at <math>\scriptstyle z\, = \,-7</math>.
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| * The function
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| :: <math>f(z) = \frac{z-4}{e^z-1}</math>
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| : 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.
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| * The function
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| ::<math>f(z) = z</math>
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| : has a single pole at infinity of order 1.
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| == Terminology and generalizations ==
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| 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).
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| A non-removable singularity that is not a pole or a [[branch point]] is called an [[essential singularity]].
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| A complex function which is holomorphic except for some isolated singularities and whose only singularities are poles is called [[meromorphic]].
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| ==See also==
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| * [[Control theory#Stability]]
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| * [[Filter design]]
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| * [[Filter (signal processing)]]
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| * [[Nyquist stability criterion]]
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| * [[Pole–zero plot]]
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| * [[Residue (complex analysis)]]
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| * [[Zero (complex analysis)]]
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| == External links ==
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| * {{MathWorld | urlname= Pole | title= Pole}}
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| * [http://math.fullerton.edu/mathews/c2003/SingularityZeroPoleMod.html Module for Zeros and Poles by John H. Mathews]
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| [[Category:Meromorphic functions]]
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My name: Harold Luong
My age: 32 years old
Country: Netherlands
Town: Tholen
Post code: 4691 Jr
Address: Kotterstraat 142
Feel free to surf to my website asphalt 8 hack