Studentized range - Revision history

en>Bearcat: /* Further reading */categorization/tagging using AWB

2014-11-15T09:12:56Z

Further reading: categorization/tagging using AWB

← Older revision		Revision as of 11:12, 15 November 2014
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	~~In [[mathematics]], the '''Jacobi–Anger expansion''' (or '''Jacobi–Anger identity''') is an expansion of exponentials of [[trigonometric function]]s in the basis of their harmonics. It~~ is ~~useful in physics (for example,~~ to ~~convert between [[plane wave]]s~~ and ~~[[cylindrical wave]]s), and in [[signal processing]] (to describe [[frequency modulation\|FM]] signals)~~. ~~This identity~~ is ~~named after~~ the ~~19th-century mathematicians~~ [~~[Carl Gustav Jacob Jacobi\|Carl Jacobi]] and [[Carl Theodor Anger]].~~		Andrew Berryhill is what his wife enjoys to call him and he completely digs that name. Mississippi is the only location I've been residing in but I will psychic solutions by lynne ([http://Www.Familysurvivalgroup.com/easy-methods-planting-looking-backyard/ your domain name]) have to transfer in a year or two. Invoicing is my profession. One of the issues she enjoys most is canoeing and she's been performing it for fairly a while.<br><br>My web blog :: free [http://www.skullrocker.com/blogs/post/10991 psychics online] tarot card readings ([http://chorokdeul.co.kr/index.php?document_srl=324263&mid=customer21 More Information and facts])

	~~The most general identity is given by~~:~~<ref name=Colton_Kress_32>Colton & Kress (1998) p. 32.<~~/~~ref><ref name=Cuyt_et_al_344>Cuyt ''et al.'' (2008) p~~. ~~344~~.</~~ref>~~

	~~:<math>e^{i z \cos \theta}=\sum_{n=~~-~~\infty}^{\infty} i^n\, J_n(z)\, e^{i n \theta}<~~/~~math>~~

	~~and~~

	~~:<math>e^{iz \sin \theta} = \sum_{n=-\infty}^\infty J_n(z~~) ~~e^{~~in~~\theta},</math>~~

	~~where <math>J_n(z)</math>~~ is the '~~'n''-th [[Bessel function]]~~. ~~Using the relation~~ <~~math~~>~~J_{-n}(z) = (-1)^n\, J_{n}(z),~~<~~/math~~> ~~valid for integer ''n'', the expansion becomes~~:~~<ref name=Colton_Kress_32/><ref name=Cuyt_et_al_344/>~~

	~~:<math>e^{i z \cos \theta}=J_0(z)\, +\, 2\, \sum_{n=1}^{\infty}\, i^n\, J_n(z)\, \cos\, (n \theta).</math>~~

	~~The following real-valued variations are often useful as well~~:~~<ref>Abramowitz & Stegun (1965)~~ [http://www.~~math~~.~~sfu.ca~~/~~~cbm~~/~~aands~~/~~page_361.htm p. 361, 9.1.42–45~~]~~</ref>~~

	~~:<math>~~
	~~\begin{align}~~
	~~\cos(z \cos \theta) &= J_0(z)+2 \sum_{n=1}^{\infty}(-1)^n J_{2n}(z) \cos(2n \theta),~~
	\\
	~~\sin~~(~~z \cos \theta) &= -2 \sum_{n=1}^{\infty}(-1)^n J_{2n-1}(z) \cos\left~~[~~\left(2n-1\right) \theta\right],~~
	\\
	~~\cos(z \sin \theta) &= J_0(z)+2 \sum_{n=1}^{\infty} J_{2n}(z) \cos(2n \theta),~~
	\\
	~~\sin(z \sin \theta) &= 2 \sum_{n=1}^{\infty} J_{2n-1}(z) \sin\left[\left(2n-1\right) \theta\right].~~
	~~\end{align}~~
	~~</math>~~

	~~== Notes ==~~
	~~{{reflist}}~~

	~~== References ==~~
	*{{Abramowitz_Stegun_ref2\|9\|355}}
	* {{ citation
	~~\| title=Inverse acoustic and electromagnetic scattering theory~~
	~~\| series=Applied Mathematical Sciences~~
	~~\| volume=93~~
	~~\| last1=Colton~~
	~~\| first1=David~~
	~~\| last2=Kress~~
	~~\| first2=Rainer~~
	~~\| edition=2nd~~
	~~\| year=1998~~
	~~\| isbn=978-3-540-62838-5~~
	}}
	* {{ citation
	~~\| title=Handbook of continued fractions for special functions~~
	~~\| first1=Annie~~
	~~\| last1=Cuyt~~
	~~\| first2=Vigdis~~
	~~\| last2=Petersen~~
	~~\| first3=Brigitte~~
	~~\| last3=Verdonk~~
	~~\| first4=Haakon~~
	~~\| last4=Waadeland~~
	~~\| first5=William B.~~
	~~\| last5=Jones~~
	~~\| publisher=Springer~~
	~~\| year=2008~~
	~~\| isbn=978-1-4020-6948-2~~
	}}

	~~== External links ==~~
	* {{cite web \| last=Weisstein \| first=Eric W. \| title=Jacobi–Anger expansion \| publisher= MathWorld — a Wolfram web resource \| url=http://~~mathworld~~.~~wolfram~~.~~com~~/~~Jacobi-AngerExpansion~~.~~html \| accessdate~~=~~2008-11-11 }}~~

	~~{{DEFAULTSORT:Jacobi-Anger expansion}}~~
	~~[[Category:Special functions]]~~
	~~[[Category:Mathematical identities]~~]

en>Yobot: clean up, References after punctuation per WP:REFPUNC and WP:PAIC using AWB (8748)

2012-12-05T18:23:30Z

clean up, References after punctuation per WP:REFPUNC and WP:PAIC using AWB (8748)

← Older revision		Revision as of 20:23, 5 December 2012
Line 1:		Line 1:
	~~The writer~~'s ~~name~~ is ~~Christy~~. ~~As a woman what she really likes~~ is ~~fashion~~ and ~~she~~'~~s been performing it for fairly a while~~. ~~Since I was eighteen I~~'~~ve been operating as a bookkeeper but quickly my wife and I will start our personal company~~. ~~My wife~~ and ~~I live~~ in ~~Mississippi but now I~~'~~m contemplating other choices~~.<br><br>~~My web site~~: ~~free psychic~~ ([http://~~165~~.~~132~~.39.93/xe/~~visitors~~/~~372912 Read the Full Content~~])		In [[mathematics]], the '''Jacobi–Anger expansion''' (or '''Jacobi–Anger identity''') is an expansion of exponentials of [[trigonometric function]]s in the basis of their harmonics. It is useful in physics (for example, to convert between [[plane wave]]s and [[cylindrical wave]]s), and in [[signal processing]] (to describe [[frequency modulation\|FM]] signals). This identity is named after the 19th-century mathematicians [[Carl Gustav Jacob Jacobi\|Carl Jacobi]] and [[Carl Theodor Anger]].

			The most general identity is given by:<ref name=Colton_Kress_32>Colton & Kress (1998) p. 32.</ref><ref name=Cuyt_et_al_344>Cuyt ''et al.'' (2008) p. 344.</ref>

			:<math>e^{i z \cos \theta}=\sum_{n=-\infty}^{\infty} i^n\, J_n(z)\, e^{i n \theta}</math>

			and

			:<math>e^{iz \sin \theta} = \sum_{n=-\infty}^\infty J_n(z) e^{in\theta},</math>

			where <math>J_n(z)</math> is the ''n''-th [[Bessel function]]. Using the relation <math>J_{-n}(z) = (-1)^n\, J_{n}(z),</math> valid for integer ''n'', the expansion becomes:<ref name=Colton_Kress_32/><ref name=Cuyt_et_al_344/>

			:<math>e^{i z \cos \theta}=J_0(z)\, +\, 2\, \sum_{n=1}^{\infty}\, i^n\, J_n(z)\, \cos\, (n \theta).</math>

			The following real-valued variations are often useful as well:<ref>Abramowitz & Stegun (1965) [http://www.math.sfu.ca/~cbm/aands/page_361.htm p. 361, 9.1.42–45]</ref>

			:<math>
			\begin{align}
			\cos(z \cos \theta) &= J_0(z)+2 \sum_{n=1}^{\infty}(-1)^n J_{2n}(z) \cos(2n \theta),
			\\
			\sin(z \cos \theta) &= -2 \sum_{n=1}^{\infty}(-1)^n J_{2n-1}(z) \cos\left[\left(2n-1\right) \theta\right],
			\\
			\cos(z \sin \theta) &= J_0(z)+2 \sum_{n=1}^{\infty} J_{2n}(z) \cos(2n \theta),
			\\
			\sin(z \sin \theta) &= 2 \sum_{n=1}^{\infty} J_{2n-1}(z) \sin\left[\left(2n-1\right) \theta\right].
			\end{align}
			</math>

			== Notes ==
			{{reflist}}

			== References ==
			*{{Abramowitz_Stegun_ref2\|9\|355}}
			* {{ citation
			\| title=Inverse acoustic and electromagnetic scattering theory
			\| series=Applied Mathematical Sciences
			\| volume=93
			\| last1=Colton
			\| first1=David
			\| last2=Kress
			\| first2=Rainer
			\| edition=2nd
			\| year=1998
			\| isbn=978-3-540-62838-5
			}}
			* {{ citation
			\| title=Handbook of continued fractions for special functions
			\| first1=Annie
			\| last1=Cuyt
			\| first2=Vigdis
			\| last2=Petersen
			\| first3=Brigitte
			\| last3=Verdonk
			\| first4=Haakon
			\| last4=Waadeland
			\| first5=William B.
			\| last5=Jones
			\| publisher=Springer
			\| year=2008
			\| isbn=978-1-4020-6948-2
			}}

			== External links ==
			* {{cite web \| last=Weisstein \| first=Eric W. \| title=Jacobi–Anger expansion \| publisher= MathWorld — a Wolfram web resource \| url=http://mathworld.wolfram.com/Jacobi-AngerExpansion.html \| accessdate=2008-11-11 }}

			{{DEFAULTSORT:Jacobi-Anger expansion}}
			[[Category:Special functions]]
			[[Category:Mathematical identities]]

en>Helpful Pixie Bot: ISBNs (Build KE)

2012-05-08T10:45:33Z

ISBNs (Build KE)

New page

The writer's name is Christy. As a woman what she really likes is fashion and she's been performing it for fairly a while. Since I was eighteen I've been operating as a bookkeeper but quickly my wife and I will start our personal company. My wife and I live in Mississippi but now I'm contemplating other choices.<br><br>My web site: free psychic ([http://165.132.39.93/xe/visitors/372912 Read the Full Content])