Artificial gravity - Revision history

en>Xezbeth: Disambiguated: David Baker → David A. Baker

2015-01-12T11:35:11Z

Disambiguated: David Baker → David A. Baker

← Older revision		Revision as of 13:35, 12 January 2015
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en>Martarius: /* Linear acceleration */ -fs

2014-02-21T22:56:58Z

Linear acceleration: -fs

← Older revision		Revision as of 00:56, 22 February 2014
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	[[Image:argument principle1.svg\|frame\|right\|The simple contour ''C'' (black), the zeros of ''f'' (blue) and the poles of ''f'' (red). Here we have <math>\oint_{C} {f'(z) \over f(z)}\, dz=2\pi i (4-5)</math>.]]		Nice to meet you, my title is Ling and I completely dig that name. Interviewing is how I make a residing and it's something I truly appreciate. Bottle tops gathering is the only hobby his wife doesn't approve of. He presently life in Idaho and his parents reside nearby.<br><br>Here is my weblog: extended auto warranty ([http://dressupgap.com/profile/ev2052 click homepage])

	In [[complex analysis]], the '''argument principle''' (or '''Cauchy's argument principle''') relates the difference between the number of [[Zero (complex analysis)\|zeros]] and [[Pole (complex analysis)\|poles]] of a [[meromorphic function]] to ~~a [[contour integral]] of the function's [[logarithmic derivative]].~~

	~~Specifically~~, ~~if ''f''(''z'')~~ is ~~a meromorphic function inside and on some closed contour ''C'', and ''f'' has no [[Zero (complex analysis)\|zeros]] or [[Pole (complex analysis)\|poles]] on ''C'', then~~
	~~: <math>\oint_{C} {f'(z) \over f(z)}\, dz=2\pi i (N-P)</math>~~
	~~where ''N''~~ and ''P'' denote respectively the number of zeros and poles of ''f''(''z'') inside the contour ''C'', with each zero and pole counted as many times as its [[Multiplicity (mathematics)\|multiplicity]] and [[Pole (complex analysis)\|order]], respectively, indicate. This statement of the theorem assumes that ~~the contour ''C'' is simple, that is, without self-intersections, and that it is oriented counter-clockwise~~.

	~~More generally, suppose that ''f''(''z'') is a meromorphic function on an [[open set]] Ω in the [[complex plane]] and that ''C''~~ is a ~~closed curve in Ω which avoids all zeros~~ and ~~poles of ''f'' and is [[contractible space\|contractible]] to a point inside Ω. For each point ''z'' ∈ Ω, let ''n''(''C'',''z'') be the [[winding number]] of ''C'' around ''z'~~'. ~~Then~~
	~~:<math>\oint_{C} \frac{f'(z)}{f(z)}\, dz = 2\pi i \left(\sum_a n(C,a) - \sum_b n(C,b)\right)</math>~~
	~~where the first summation is over all zeros ''a'' of ''f'' counted with their multiplicities, and the second summation~~ is ~~over~~ the ~~poles ''b'~~' of ~~''f'' counted with their orders~~.

	~~==Interpretation of the contour integral==~~
	~~The contour integral <math>\oint_{C} \frac{f'(z)}{f(z)}\, dz</math> can be interpreted in two ways:~~
	* as the total change in ~~the [[argument (complex analysis)\|argument]] of ''f''(''z'') as ''z'' travels around ''C'', explaining the name of the theorem; this follows from~~
	~~:<math>\frac{d}{dz}\log(f(z))=\frac{f'(z)}{f(z)}</math>~~
	and ~~the relation between arguments and logarithms~~.
	* as 2π''i'' times the winding number of the path ''f''(''C'') around the origin, using the substitution ''w'' = ''f''(''z''):
	:<~~math~~>~~\oint_{C} \frac{f'(z)}{f(z)}\, dz = \oint_{f(C)} \frac{1}{w}\, dw~~<~~/math~~>

	~~==Proof of the argument principle==~~
	~~Let ''z''<sub>''N''</sub> be a zero of ''f''. We can write ''f''(''z'') = (''z'' − ''z''<sub>''N''</sub>)<sup>''k''</sup>''g''(''z'') where ''k''~~ is ~~the multiplicity of the zero, and thus ''g''(''z''<sub>''N''</sub>) ≠ 0. We get~~

	: ~~<math>f'~~(~~z)=k(z-z_N)^{k-1}g(z)+(z-z_N)^kg'(z)\,\!</math>~~

	~~and~~

	~~: <math>{f'(z)\over f(z)}={k \over z-z_N}+{g'(z)\over g(z)}.</math>~~

	Since ''g''(''z''<sub>''N''</sub>) ≠ 0, it follows that ''g' ''(''z'')/''g''(''z'') has no singularities at ''z''<sub>''N''</sub>, and thus is analytic at ''z''<sub>N</sub>, which implies that the [[~~Residue (complex analysis)\|residue]] of ''f''′(''z'')/''f''(''z'') at ''z''<sub>''N''</sub> is ''k''.~~

	~~Let ''z''<sub>P</sub> be a pole of ''f''. We can write ''f''(''z'') = (''z'' − ''z''<sub>P</sub>)<sup>−''m''</sup>''h''(''z'') where ''m'' is the order of the pole, and thus~~
	~~''h''(''z''<sub>P</sub>) ≠ 0. Then,~~

	: ~~<math>f'(z)=-m(z-z_P)^{-m-1}h(z)+(z-z_P)^{-m}h'(z)\,\!.<~~/~~math>~~

	~~and~~

	~~: <math>{f'(z)\over f(z)}={-m \over z-z_P}+{h'(z)\over h(z)}<~~/~~math>~~

	~~similarly as above~~. ~~It follows that ''h''′(''z'')~~/~~''h''(''z'') has no singularities at ''z''<sub>P</sub> since ''h''(''z''<sub>P</sub>) ≠ 0 and thus it is analytic at ''z''<sub>P</sub>. We find that the residue of~~
	~~''f''′(''z'')~~/~~''f''(''z'') at ''z''<sub>P</sub> is −''m''.~~

	~~Putting these together, each zero ''z''<sub>''N''</sub> of multiplicity ''k'' of ''f'' creates a simple pole for~~
	~~''f''′(''z'')/''f''(''z'') with the residue being ''k'', and each pole ''z''<sub>P</sub> of order ''m'' of~~
	~~''f'' creates a simple pole for ''f''′(''z'')/''f''(''z'') with the residue being −''m''. (Here, by a simple pole we~~
	~~mean a pole of order one.) In addition, it can be shown that ''f''′(''z'')/''f''(''z'') has no other poles,~~
	~~and so no other residues.~~

	~~By the [[residue theorem]~~] we have that the integral about ''C'' is the product of 2''πi'' and the sum of the residues. Together, the sum of the ''k'' 's for each zero ''z''<sub>''N''</sub> is the number of zeros counting multiplicities of the zeros, and likewise for the poles, and so we have our result.

	~~==Applications and consequences==~~
	~~The argument principle can be used to efficiently locate zeros or poles of meromorphic functions on a computer. Even with rounding errors, the expression <math>{1\over 2\pi i}\oint_{C} {f'(z~~) \over f(z)}\, dz</math> will yield results close to an integer; by determining these integers for different contours ''C'' one can obtain information about the location of the zeros and poles. Numerical tests of the [[Riemann hypothesis]] use this technique to get an upper bound for the number of zeros of [[Riemann Xi function\|Riemann's <math>\xi(s)</math> function]] inside a rectangle intersecting the critical line.

	~~The proof of [[Rouché's theorem]] uses the argument principle.~~

	~~Modern books on feedback control theory quite frequently use the argument principle to serve as the theoretical basis of the [[Nyquist stability criterion]].~~

	~~A consequence of the more general formulation of the argument principle is that, under the same hypothesis, if ''g'' is an analytic function in Ω, then~~

	~~:<math> \frac{1}{2\pi i} \oint_C g(z)\frac{f'(z)}{f(z)}\, dz = \sum_a n(C,a)g(a) - \sum_b n(C,b)g(b).</math>~~

	~~For example, if ''f'' is a [[polynomial]] having zeros ''z''<sub>1</sub>, ..., ''z''<sub>p</sub> inside a simple contour ''C'', and ''g''(''z'') = ''z''<sup>k</sup>, then~~
	~~:<math> \frac{1}{2\pi i} \oint_C z^k\frac{f'(z)}{f(z)}\, dz = z_1^k+z_2^k+\dots+z_p^k,</math>~~
	~~is [[power sum symmetric polynomial]] of the roots of ''f''.~~

	~~Another consequence is if we compute the complex integral:~~

	~~: <math>\oint_C f(z){g'(z) \over g(z)}\, dz</math>~~

	~~for an appropriate choice of ''g'' and ''f'' we have the [[Abel–Plana formula]]:~~

	~~: <math> \sum_{n=0}^{\infty}f(n)-\int_{0}^{\infty}f(x)\,dx= f(0)/2+i\int_{0}^{\infty}\frac{f(it)-f(-it)}{e^{2\pi t}-1}\, dt </math>~~

	~~which expresses the relationship between a discrete sum and its integral.~~

	~~==Generalized argument principle==~~
	~~There is an immediate generalization of the argument principle. The integral~~
	~~: <math>\oint_{C} {f'(z) \over f(z)} g(z) \, dz</math>~~
	~~is equal to g evaluated at the zeroes, minues g evaluated at the poles.~~

	~~==History==~~
	According to the book by [[Frank Smithies]] (''Cauchy and the Creation of Complex Function Theory'', Cambridge University Press, 1997, p.177), [[Augustin-Louis Cauchy]] presented a theorem similar to the above on 27 November 1831, during his self-imposed exile in Turin (then capital of the Kingdom of Piedmont-Sardinia) away from France. However, according to this book, only zeroes were mentioned, not poles. This theorem by Cauchy was only published many years later in 1974 in a hand-written form and so is quite difficult to read. Cauchy published a paper with a discussion on both zeroes and poles in 1855, two years before his death.

	~~== See also ==~~

	* [[Logarithmic derivative]]

	~~==References==~~
	* {{cite book \| last=Rudin \| first=Walter \| title = Real and Complex Analysis (International Series in Pure and Applied Mathematics) \| publisher=McGraw-Hill \| year=1986 \|isbn=978-0-07-054234-1}}
	* {{cite book \| last=Ahlfors \| first=Lars \| title = Complex analysis: an introduction to the theory of analytic functions of one complex variable \| publisher=McGraw-Hill \| year=1979 \|isbn=978-0-07-000657-7}}
	* {{cite book \| last1=Churchill \| first1=Ruel Vance \| last2=Brown \| first2=James Ward \| title = Complex Variables and Applications \| publisher=McGraw-Hill \| year=1989 \|isbn=978-0-07-010905-6}}
	* Backlund, R.-J. (1914) Sur les zéros de la fonction zeta(s) de Riemann, C. R. Acad. Sci. Paris 158, 1979-1982.

	~~{{DEFAULTSORT:Argument Principle}}~~
	~~[[Category:Complex analysis]]~~
	~~[[Category:Theorems in complex analysis]]~~

en>Matt Whyndham: /* Rotation */ Added a new image to demonstrate the result of the functions of (g, R, rpm) for a useful range of values

2014-01-22T19:33:08Z

Rotation: Added a new image to demonstrate the result of the functions of (g, R, rpm) for a useful range of values

Show changes

en>Lyla1205: /* External links */ Category:Space stations.

2012-08-08T13:25:37Z

External links: Category:Space stations.

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