en>Bhny: /* top */ copy edit of bad prose

2014-08-23T13:53:14Z

top: copy edit of bad prose

en>Nickisalaam1 at 13:22, 24 February 2014

2014-02-24T13:22:08Z

en>TAnthony: /* Specific Airway Resistance (sR_aw){{cite web|title=US EPA Glossary of Terms|url=http://www.epa.gov/apti/ozonehealth/glossary.html#S}}{{cite journal|last=Kirkby|first using AWB

2013-12-10T01:05:07Z

/* Specific Airway Resistance (sR<sub>aw</sub>)<ref name=EPA>{{cite web|title=US EPA Glossary of Terms|url=http://www.epa.gov/apti/ozonehealth/glossary.html#S}}</ref><ref name="Kirkby et al">{{cite journal|last=Kirkby|first using AWB

← Older revision		Revision as of 03:05, 10 December 2013
Line 1:		Line 1:
	~~Wilber Berryhill~~ is the ~~title his mothers and~~ [http://~~kard~~.dk/?p=~~24252 real psychic] fathers gave him~~ and ~~he totally digs~~ that ~~title~~. ~~North Carolina is exactly where~~ we'~~ve been living for many years~~ and ~~will never transfer~~. The ~~preferred hobby for him~~ and ~~psychic readings ([~~http://~~Ltreme~~.com/~~index~~.~~php?do=~~/~~profile~~-~~127790~~/~~info~~/ i ~~thought about~~ this]) ~~his kids is fashion~~ and he'll be ~~starting some thing else alongside~~ with it~~. My working day occupation is an info officer but I~~'~~ve already utilized for an additional~~ 1.<br><br>~~my web~~-~~site~~ [~~http~~://~~Www.Publicpledge.com~~/~~blogs~~/~~post~~/~~7034 real psychics~~]		{{Expert-subject\|Mathematics\|date=November 2008}}
			'''Ramanujan summation''' is a technique invented by the mathematician [[Srinivasa Ramanujan]] for assigning a value to infinite [[divergent series]]. Although the Ramanujan summation of a divergent series is not a sum in the traditional sense, it has properties which make it mathematically useful in the study of divergent [[infinite series]], for which conventional summation is undefined.

			== Summation ==

			Ramanujan summation essentially is a property of the partial sums, rather than a property of the entire sum, as that doesn't exist. If we take the [[Euler–Maclaurin formula\|Euler–Maclaurin summation formula]] together with the correction rule using [[Bernoulli number]]s, we see that:

			:<math>\begin{align}
			{} &\frac{1}{2}f\left( 0\right) + f\left( 1\right) + \cdots + f\left( n - 1\right) +
			\frac{1}{2}f\left( n\right) \\
			= &\frac{1}{2}\left[f\left( 0\right) + f\left( n\right)\right] + \sum_{k=1}^{n-1}f\left(k\right)\\
			= &\int_0^n f(x)\,dx + \sum_{k=1}^p\frac{B_{k + 1}}{(k + 1)!}\left[f^{(k)}(n) - f^{(k)}(0)\right] + R_p
			\end{align}</math>

			Ramanujan<ref> Bruce C. Berndt, [http://www.comms.scitech.susx.ac.uk/fft/math/RamanujanNotebooks1.pdf Ramanujan's Notebooks], ''Ramanujan's Theory of Divergent Series'', Chapter 6, Springer-Verlag (ed.), (1939), pp. 133-149.</ref> wrote it for the case ''p'' going to infinity:
			:<math>\sum_{k=1}^{x}f(k) = C + \int_0^x f(t)\,dt + \frac{1}{2}f(x) + \sum_{k=1}^{\infty}\frac{B_{2k}}{(2k)!}f^{(2k - 1)}(x)</math>

			where ''C'' is a constant specific to the series and its analytic continuation and the limits on the integral were not specified by Ramanujan, but presumably they were as given above. Comparing both formulae and assuming that ''R'' tends to 0 as ''x'' tends to infinity, we see that, in a general case, for functions ''f''(''x'') with no divergence at ''x'' = 0:
			:<math>C(a)=\int_0^a f(t)\,dt-\frac{1}{2}f(0)-\sum_{k=1}^{\infty}\frac{B_{2k}}{(2k)!}f^{(2k-1)}(0)</math>

			where Ramanujan assumed <math>\scriptstyle a \,=\, 0</math>. By taking <math>\scriptstyle a \,=\, \infty</math> we normally recover the usual summation for convergent series. For functions ''f''(''x'') with no divergence at ''x'' = 1, we obtain:
			:<math>C(a) = \int_1^a f(t)\,dt+ \frac{1}{2}f(1) - \sum_{k=1}^{\infty}\frac{B_{2k}}{(2k)!}f^{(2k-1)}(1)</math>

			''C''(0) was then proposed to use as the sum of the divergent sequence. It is like a bridge between summation and integration.

			== Sum of divergent series ==

			In the following text, <math>\scriptstyle (\Re)</math> indicates "Ramanujan summation". This formula originally appeared in one of Ramanujan's notebooks, without any notation to indicate that it was a Ramanujan summation.

			For example, the <math>\scriptstyle (\Re)</math> of {{nowrap\|[[Grandi's series\|1 - 1 + 1 - ⋯]]}} is:
			:<math>1 - 1 + 1 - \cdots = \frac{1}{2}\ (\Re)</math>.

			Ramanujan had calculated "sums" of known divergent series. It is important to mention that the Ramanujan sums are not the sums of the series in the usual sense,<ref name="Terry Tao on Ramanujan sums">{{cite web\|title=The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation \|url=http://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/\|accessdate=20 January 2014}}</ref><ref name="Dirichlet vs Zeta">{{cite web\|title=Infinite series are weird \|url=http://skullsinthestars.com/2010/05/25/infinite-series-are-weird-redux/\|accessdate=20 January 2014}}</ref> i.e. the partial sums do not converge to this value, which is denoted by the symbol <math>\scriptstyle (\Re)</math>.
			In particular, the <math>\scriptstyle (\Re)</math> sum of {{nowrap\|[[1 + 2 + 3 + 4 + ···\|1 + 2 + 3 + 4 + ⋯]]}} was calculated as:
			:<math>1+2+3+\cdots = -\frac{1}{12}\ (\Re)</math>

			Extending to positive even powers, this gave:
			:<math>1 + 2^{2k} + 3^{2k} + \cdots = 0\ (\Re)</math>

			and for odd powers the approach suggested a relation with the [[Bernoulli number]]s:
			:<math>1+2^{2k-1}+3^{2k-1}+\cdots = -\frac{B_{2k}}{2k}\ (\Re)</math>

			More recently, the use of ''C''(1) has been proposed as Ramanujan's summation, since then it can be assured that one series <math>\scriptstyle \sum_{k=1}^{\infty}f(k) </math> admits one and only one Ramanujan's summation, defined as the value in 1 of the only solution of the difference equation <math>\scriptstyle R(x) \,-\, R(x \,+\, 1) \,=\, f(x)</math> that verifies the condition <math>\scriptstyle \int_1^2 R(t)\,dt \,=\, 0</math>.<ref> Éric Delabaere, [http://algo.inria.fr/seminars/sem01-02/delabaere2.pdf Ramanujan's Summation], ''Algorithms Seminar 2001–2002'', F. Chyzak (ed.), INRIA, (2003), pp. 83–88.</ref>

			This new definition of Ramanujan's summation (denoted as <math>\scriptstyle \sum_{n \ge 1}^{\Re} f(n)</math>) does not coincide with the earlier defined Ramanujan's summation, ''C''(0), nor with the summation of convergent series, but it has interesting properties, such as:
			If ''R''(''x'') tends to a finite limit when ''x'' → +1, then the series <math>\scriptstyle \sum_{n \ge 1}^{\Re} f(n)</math> is convergent, and we have
			:<math>\sum_{n \ge 1}^{\Re} f(n) = \lim_{N \to \infty}\left[\sum_{n = 1}^{N}f(n) - \int_1^N f(t)\,dt\right]</math>

			In particular we have:
			:<math>\sum_{n \ge 1}^{\Re} \frac{1}{n} = \gamma</math>

			where ''γ'' is the [[Euler–Mascheroni constant]].

			Ramanujan resummation can be extended to integrals for example using the Euler-Maclaurin summation formula one can write

			:<math> \begin{array}{l}
			\int\nolimits_{a}^{\infty }x^{m-s} dx =\frac{m-s}{2} \int\nolimits_{a}^{\infty }x^{m-1-s} dx +\zeta (s-m)-\sum\limits_{i=1}^{a}i^{m-s} +a^{m-s} \\
			-\sum\limits_{r=1}^{\infty }\frac{B_{2r} \Gamma (m-s+1)}{(2r)!\Gamma (m-2r+2-s)} (m-2r+1-s)\int\nolimits_{a}^{\infty }x^{m-2r-s} dx \end{array} </math>

			this is the natural extension to integrals of the Zeta regularization algorithm, this recurrence equation is finite since for <math> m-2r < -1 \qquad \int_{a}^{\infty}dxx^{m-2r}= -\frac{a^{m-2r+1}}{m-2r+1} </math>

			here <math>\scriptstyle I(n,\, \Lambda) \,=\, \int_{0}^{\Lambda}dxx^{n}</math> with <math>\scriptstyle \Lambda \rightarrow \infty</math> the application of this Ramanujan resummation lends to finite results in the renormalization of Quantum Field theories

			== See also ==
			* [[Borel summation]]
			* [[Cesàro summation]]
			* [[Ramanujan's sum]]
			* [[Divergent series]]

			== References ==
			<references/>

			[[Category:Summability methods]]
			[[Category:Srinivasa Ramanujan]]

en>Fredv3b at 18:33, 12 August 2012

2012-08-12T18:33:05Z

New page

Wilber Berryhill is the title his mothers and [http://kard.dk/?p=24252 real psychic] fathers gave him and he totally digs that title. North Carolina is exactly where we've been living for many years and will never transfer. The preferred hobby for him and psychic readings ([http://Ltreme.com/index.php?do=/profile-127790/info/ i thought about this]) his kids is fashion and he'll be starting some thing else alongside with it. My working day occupation is an info officer but I've already utilized for an additional 1.<br><br>my web-site [http://Www.Publicpledge.com/blogs/post/7034 real psychics]

Airway resistance - Revision history

en>Bhny: /* top */ copy edit of bad prose

en>Nickisalaam1 at 13:22, 24 February 2014

en>TAnthony: /* Specific Airway Resistance (sRaw){{cite web|title=US EPA Glossary of Terms|url=http://www.epa.gov/apti/ozonehealth/glossary.html#S}}{{cite journal|last=Kirkby|first using AWB

en>Fredv3b at 18:33, 12 August 2012

en>TAnthony: /* Specific Airway Resistance (sR_aw){{cite web|title=US EPA Glossary of Terms|url=http://www.epa.gov/apti/ozonehealth/glossary.html#S}}{{cite journal|last=Kirkby|first using AWB