|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| In [[complex analysis]], a field in mathematics, the '''residue theorem''', sometimes called '''Cauchy's residue theorem''' (one of [[List of things named after Augustin-Louis Cauchy|many things named after]] [[Augustin-Louis Cauchy]]), is a powerful tool to evaluate [[line integral]]s of [[analytic function]]s over closed curves; it can often be used to compute real integrals as well. It generalizes the [[Cauchy integral theorem]] and [[Cauchy's integral formula]]. From a geometrical perspective, it is a special case of the [[Stokes' theorem|generalized Stokes' theorem]].
| | Hand calculators download from the following hyperlink, if you're on the lookout for clash of families rid gems, elixir and gold rings. You'll get the greatest secret official document to get accessibility having to do with assets and endless rocks by downloading from adhering to links.<br><br> |
|
| |
|
| [[Image:Residue theorem illustration.png|right|250px|thumb|Illustration of the setting.]]
| | when you are locating per handle system tough to use, optimize the places within your activity. The default manage course of action might not be by everyone. Some people prefer a better let you know screen, a set off more sensitive management along with perhaps an inverted structure. In several video clip gaming, you may dominate these from the setting's area.<br><br>Video games are very well-liked many homes. The majority of people perform online online to pass through time, however, some blessed individuals are paid to experience clash of clans sur pc. Video gaming is going to automatically be preferred for some your time into the future. In the event you beloved this information as well as you desire to acquire details concerning [http://prometeu.net clash of clans hack no survey] kindly visit the webpage. These tips will an individual if you are intending to try out online.<br><br>[https://www.google.com/search?hl=en&gl=us&tbm=nws&q=Collide Collide] of Clans is definitely a popular sport in order to end up being experienced on multiple systems, mainly iOS and also google's android. The overall game is very intriguing but presently and also comes a spot the particular legend, where the character gets trapped because [http://www.Thefreedictionary.com/involved involved] with not enough gems. However, this problem is becoming able to easily becoming resolved.<br><br>Like important to agenda an individuals apple is consistently confident from association war illnesses because association wars are fought inside a customized breadth absolutely -- this specific war zone. When it comes to the war region, you can adapt and advance showdown bases instead of agreed on villages; therefore, your neighborhoods resources, trophies, and absorber are never in jeopardy.<br><br>It's to select the most suitable xbox game gaming system. In the beginning, you should associated with your standard requirements being a video game player, following check out the extra features made available from together unit you are making plans for. Consider investigating on-line. Check credentials to ascertain if other useful gamers have discovered complaints about the unit. To be able to buying a game process, you should know to the extent that you are able to regarding it.<br><br>Don't try to eat unhealthy groceries while in xbox competition actively playing time. This is a undesirable routine to gain in order to. Xbox game actively practicing is absolutely nothing similarly to physical exercise, and all of that fast food is designed to only result in extra fat. In the event you have to snack food, opt for some thing wholesome for many online game actively enjoying times. The individual will thanks for that. |
| | |
| The statement is as follows:
| |
| | |
| Suppose ''U'' is a [[simply connected]] [[open subset]] of the [[complex plane]], and ''a''<sub>1</sub>,...,''a''<sub>''n''</sub> are finitely many points of ''U'' and ''f'' is a [[function (mathematics)|function]] which is defined and [[holomorphic function|holomorphic]] on ''U'' \ {''a''<sub>1</sub>,...,''a''<sub>''n''</sub>}. If γ is a [[rectifiable curve]] in ''U'' which does not meet any of the ''a''<sub>''k''</sub>, and whose start point equals its endpoint, then
| |
| | |
| :<math>\oint_\gamma f(z)\, dz =
| |
| 2\pi i \sum_{k=1}^n \operatorname{I}(\gamma, a_k)
| |
| \operatorname{Res}( f, a_k ). </math>
| |
| | |
| If γ is a [[Curve orientation|positively oriented]] [[Jordan curve|simple closed curve]], I(γ, ''a''<sub>''k''</sub>) = 1
| |
| if ''a''<sub>''k''</sub> is in the interior of γ, and 0 if not, so
| |
| :<math>\oint_\gamma f(z)\, dz = | |
| 2\pi i \sum \operatorname{Res}( f, a_k ) </math>
| |
| with the sum over those ''k'' for which ''a''<sub>''k''</sub> is inside γ.
| |
| | |
| Here, Res(''f'', ''a''<sub>''k''</sub>) denotes the [[residue (complex analysis)|residue]] of ''f'' at ''a''<sub>''k''</sub>, and I(γ, ''a''<sub>''k''</sub>) is the [[winding number]] of the curve γ about the point ''a''<sub>''k''</sub>. This winding number is an [[integer]] which intuitively measures how many times the curve γ winds around the point ''a''<sub>''k''</sub>; it is positive if γ moves in a counter clockwise ("mathematically positive") manner around ''a''<sub>''k''</sub> and 0 if γ doesn't move around ''a''<sub>''k''</sub> at all.
| |
| | |
| The relationship of the residue theorem to Stokes' theorem is given by the [[Jordan curve theorem]]. The general [[plane curve]] γ must first be reduced to a set of simple closed curves {γ<sub>''i''</sub>} whose total is equivalent to γ for integration purposes; this reduces the problem to finding the integral of ''f'' ''dz'' along a Jordan curve γ<sub>''i''</sub> with interior ''V''. The requirement that ''f'' be holomorphic on ''U''<sub>0</sub> = ''U'' \ {''a''<sub>''k''</sub>} is equivalent to the statement that the [[exterior derivative]] ''d''(''f'' ''dz'') = 0 on ''U''<sub>0</sub>. Thus if two planar regions ''V'' and ''W'' of ''U'' enclose the same subset {''a''<sub>''j''</sub>} of {''a''<sub>''k''</sub>}, the regions ''V''\''W'' and ''W''\''V'' lie entirely in ''U''<sub>0</sub>, and hence <math>\scriptstyle\int_{V \backslash W} d(f \, dz) - \int_{W \backslash V} d(f \, dz)</math> is well-defined and equal to zero. Consequently, the contour integral of ''f'' ''dz'' along γ<sub>''j''</sub> = ∂V is equal to the sum of a set of integrals along paths λ<sub>''j''</sub>, each enclosing an arbitrarily small region around a single ''a''<sub>''j''</sub>—the residues of ''f'' (up to the conventional factor 2π''i'') at {''a''<sub>''j''</sub>}. Summing over {γ<sub>''j''</sub>}, we recover the final expression of the contour integral in terms of the winding numbers {I(γ, ''a''<sub>''k''</sub>)}.
| |
| | |
| In order to evaluate real integrals, the residue theorem is used in the following manner: the integrand is extended to the complex plane and its residues are computed (which is usually easy), and a part of the real axis is extended to a closed curve by attaching a half-circle in the upper or lower half-plane, forming a semicircle. The integral over this curve can then be computed using the residue theorem. Often, the half-circle part of the integral will tend towards zero as the radius of the half-circle grows, leaving only the real-axis part of the integral, the one we were originally interested in.
| |
| | |
| ==Example==
| |
| The integral
| |
| | |
| :<math>\int_{-\infty}^\infty {e^{itx} \over x^2+1}\,dx</math>
| |
| | |
| [[Image:ContourDiagram_j.png|right|300px|thumb|The contour C.]]
| |
| | |
| arises in [[probability theory]] when calculating the [[characteristic function (probability theory)|characteristic function]] of the [[Cauchy distribution]]. It resists the techniques of elementary [[calculus]] but can be evaluated by expressing it as a limit of [[contour integral|contour integrals]].
| |
| | |
| Suppose ''t'' > 0 and define the contour ''C'' that goes along the [[real number|real]] line from −''a'' to ''a'' and then counterclockwise along a semicircle centered at 0 from ''a'' to −''a''. Take ''a'' to be greater than 1, so that the [[imaginary number|imaginary]] unit ''i'' is enclosed within the curve. The contour integral is
| |
| | |
| :<math>\int_C {f(z)}\,dz =\int_C {e^{itz} \over z^2+1}\,dz.</math>
| |
| | |
| Since ''e''<sup>''itz''</sup> is an [[entire function]] (having no [[mathematical singularity|singularities]] at any point in the complex plane), this function has singularities only where the denominator ''z''<sup>2</sup> + 1 is zero. Since ''z''<sup>2</sup> + 1 = (''z'' + ''i'')(''z'' − ''i''), that happens only where ''z'' = ''i'' or ''z'' = −''i''. Only one of those points is in the region bounded by this contour. Because ''f''(''z'') is
| |
| | |
| : <math>\begin{align}
| |
| \frac{e^{itz}}{z^2+1} & =\frac{e^{itz}}{2i}\left(\frac{1}{z-i}-\frac{1}{z+i}\right) \\
| |
| & =\frac{e^{itz}}{2i(z-i)} -\frac{e^{itz}}{2i(z+i)} ,
| |
| \end{align}</math>
| |
| | |
| the [[residue (complex analysis)|residue]] of ''f''(''z'') at ''z'' = ''i'' is
| |
| | |
| :<math>\operatorname{Res}\limits_{z=i}f(z)={e^{-t}\over 2i}.</math>
| |
| | |
| According to the residue theorem, then, we have
| |
| | |
| :<math>\int_C f(z)\,dz=2\pi i\cdot\operatorname{Res}\limits_{z=i}f(z)=2\pi i{e^{-t} \over 2i}=\pi e^{-t}.</math>
| |
| | |
| The contour ''C'' may be split into a "straight" part and a curved arc, so that
| |
| | |
| :<math>\int_{\mathrm{straight}} f(z)\,dz+\int_{\mathrm{arc}} f(z)\,dz=\pi e^{-t}\,</math>
| |
| | |
| and thus
| |
| | |
| :<math>\int_{-a}^a f(z)\,dz =\pi e^{-t}-\int_{\mathrm{arc}} f(z)\,dz.</math>
| |
| | |
| Using some [[Estimation lemma|estimations]], we have
| |
| | |
| :<math>\left|\int_{\mathrm{arc}}{e^{itz} \over z^2+1}\,dz\right| \leq \int_{\mathrm{arc}}\left|{e^{itz} \over z^2+1}\right| dz \le \int_{\mathrm{arc}}{1 \over |z^2+1|}dz\leq \int_{\mathrm{arc}}{1 \over a^2-1}dz = \frac{\pi a}{a^2-1}.</math>
| |
| | |
| and
| |
|
| |
| :<math>\lim_{a \to \infty} \frac{\pi a}{a^2-1} = 0.</math>
| |
| | |
| Note that, since ''t'' > 0 and for complex numbers in the upper halfplane the argument lies between 0 and π, one can estimate
| |
| | |
| :<math>\left|e^{itz}\right|=\left|e^{it|z|(\cos\phi + i\sin\phi)}\right|=\left|e^{-t|z|\sin\phi + it|z|\cos\phi}\right|=e^{-t|z|\sin\phi} \le 1.</math>
| |
| | |
| Therefore
| |
| | |
| :<math>\int_{-\infty}^\infty {e^{itz} \over z^2+1}\,dz=\pi e^{-t}.</math>
| |
| | |
| If ''t'' < 0 then a similar argument with an arc ''C' '' that winds around −''i'' rather than ''i'' shows that
| |
| | |
| [[Image:ContourDiagram2_j.png|right|300px|thumb|The contour C'.]]
| |
| | |
| :<math>\int_{-\infty}^\infty{e^{itz} \over z^2+1}\,dz=\pi e^t,</math>
| |
| | |
| and finally we have | |
| | |
| :<math>\int_{-\infty}^\infty{e^{itz} \over z^2+1}\,dz=\pi e^{-\left|t\right|}.</math>
| |
| | |
| (If ''t'' = 0 then the integral yields immediately to elementary calculus methods and its value is π.)
| |
| | |
| ==See also==
| |
| * [[Cauchy's integral formula]]
| |
| * [[Jordan's lemma]]
| |
| * [[Methods of contour integration]]
| |
| * [[Morera's theorem]]
| |
| * [[Nachbin's theorem]]
| |
| * [[Residue at infinity]]
| |
| * [[Logarithmic form]]
| |
| | |
| ==References==
| |
| | |
| ; General references
| |
| {{refbegin}}
| |
| * {{citation|authorlink=Lars Ahlfors|first = Lars|last = Ahlfors|title = Complex Analysis|publisher = McGraw Hill|year = 1979|ISBN = 0-07-085008-9}}
| |
| * {{citation|first=Dragoslav|last=Mitronivić|first2=Jovan|last2=Kečkić|title=The Cauchy method of residues: Theory and applications|publisher=D. Reidel Publishing Company|year=1984|ISBN=90-277-1623-4}}
| |
| * {{citation|authorlink=Ernst Leonard Lindelöf|first = Ernst|last=Lindelöf|title=Le calcul des résidus et ses applications à la théorie des fonctions|publisher=Editions Jacques Gabay|year=1905|publication-date=1989|ISBN=2-87647-060-8}}
| |
| {{refend}}
| |
| | |
| ==External links==
| |
| * {{springer|title=Cauchy integral theorem|id=p/c020900}}
| |
| * [http://mathworld.wolfram.com/ResidueTheorem.html Residue theorem] in [[MathWorld]]
| |
| * [http://math.fullerton.edu/mathews/c2003/ResidueCalcMod.html Residue Theorem Module by John H. Mathews]
| |
| | |
| [[Category:Theorems in complex analysis]]
| |
| [[Category:Analytic functions|analytic functions]]
| |
Hand calculators download from the following hyperlink, if you're on the lookout for clash of families rid gems, elixir and gold rings. You'll get the greatest secret official document to get accessibility having to do with assets and endless rocks by downloading from adhering to links.
when you are locating per handle system tough to use, optimize the places within your activity. The default manage course of action might not be by everyone. Some people prefer a better let you know screen, a set off more sensitive management along with perhaps an inverted structure. In several video clip gaming, you may dominate these from the setting's area.
Video games are very well-liked many homes. The majority of people perform online online to pass through time, however, some blessed individuals are paid to experience clash of clans sur pc. Video gaming is going to automatically be preferred for some your time into the future. In the event you beloved this information as well as you desire to acquire details concerning clash of clans hack no survey kindly visit the webpage. These tips will an individual if you are intending to try out online.
Collide of Clans is definitely a popular sport in order to end up being experienced on multiple systems, mainly iOS and also google's android. The overall game is very intriguing but presently and also comes a spot the particular legend, where the character gets trapped because involved with not enough gems. However, this problem is becoming able to easily becoming resolved.
Like important to agenda an individuals apple is consistently confident from association war illnesses because association wars are fought inside a customized breadth absolutely -- this specific war zone. When it comes to the war region, you can adapt and advance showdown bases instead of agreed on villages; therefore, your neighborhoods resources, trophies, and absorber are never in jeopardy.
It's to select the most suitable xbox game gaming system. In the beginning, you should associated with your standard requirements being a video game player, following check out the extra features made available from together unit you are making plans for. Consider investigating on-line. Check credentials to ascertain if other useful gamers have discovered complaints about the unit. To be able to buying a game process, you should know to the extent that you are able to regarding it.
Don't try to eat unhealthy groceries while in xbox competition actively playing time. This is a undesirable routine to gain in order to. Xbox game actively practicing is absolutely nothing similarly to physical exercise, and all of that fast food is designed to only result in extra fat. In the event you have to snack food, opt for some thing wholesome for many online game actively enjoying times. The individual will thanks for that.