Bessel filter - Revision history

en>Dicklyon: dash fix

2014-08-24T19:08:17Z

dash fix

← Older revision		Revision as of 21:08, 24 August 2014
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	~~The author's~~ name ~~is Andera and she thinks it seems quite great~~. ~~Mississippi is exactly where her home is~~ but ~~her husband desires them to move~~. ~~Credit authorising is how he makes cash~~. ~~To perform lacross is something I really enjoy performing~~.<br><br>~~Also visit my web page~~: ~~clairvoyant psychic~~ ([http://~~quietpyramid~~.~~com~~/~~?p=33404 go to website~~])		Eartha precisely what you can call me and I feel comfortable as well as use the full name. My wife need not live in Vermont. I am really fond of to draw 3d graphics but I'm thinking on starting new stuff. For years I've been being employed as a human resources officer. Check out the latest news on his website: http://Www.[http://Www.google.co.uk/search?hl=en&gl=us&tbm=nws&q=Archive&gs_l=news Archive].org/details/miley_cyrus_maxim<br><br>My blog post :: miley cyrus maxim ([http://Www.Archive.org/details/miley_cyrus_maxim Www.Archive.org])

71.167.69.72 at 04:48, 25 February 2014

2014-02-25T04:48:15Z

← Older revision		Revision as of 06:48, 25 February 2014
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	~~In [[mathematics]], '''Riemann~~'s ~~differential equation''', named after [[Bernhard Riemann]],~~ is ~~a generalization of the [[hypergeometric differential equation]], allowing the [[regular singular points]] to occur anywhere on the [[Riemann sphere]], rather than merely at 0, 1,~~ and ∞.		The author's name is Andera and she thinks it seems quite great. Mississippi is exactly where her home is but her husband desires them to move. Credit authorising is how he makes cash. To perform lacross is something I really enjoy performing.<br><br>Also visit my web page: clairvoyant psychic ([http://quietpyramid.com/?p=33404 go to website])

	~~==Definition==~~
	~~The differential equation~~ is ~~given by~~
	~~:<math>\frac{d^2w}{dz^2} + \left[~~
	~~\frac{1-\alpha-\alpha'}{z-a} +~~
	~~\frac{1-\beta-\beta'}{z-b} +~~
	~~\frac{1-\gamma-\gamma'}{z-c} \right] \frac{dw}{dz} </math>~~
	~~::<math>+\left[~~
	~~\frac{\alpha\alpha' (a-b)(a-c)} {z-a}~~
	~~+\frac{\beta\beta' (b-c)(b-a)} {z-b}~~
	~~+\frac{\gamma\gamma' (c-a)(c-b)} {z-c}~~
	~~\right]~~
	~~\frac{w}{(z-a)(z-b)(z-c)}=0.</math>~~

	~~The regular singular points are {{mvar\|a}}, {{mvar\|b}}, and {{mvar\|c}}. The pairs of exponents{{clarification needed\|date=November 2013\|reason=There~~ is no exponentiation yet, and you may not call it “exponents”}} for each are respectively {{math\|''α''; ''α′''}}, {{math\|''β''; ''β′''}}, and {{math\|''γ''; ''γ′''}}. The exponents are subject to the condition

	~~:<math>\alpha+\alpha'+\beta+\beta'+\gamma+\gamma'=1.</math>~~

	~~==Solutions==~~
	~~The solutions are denoted by the ''Riemann P-symbol''~~

	~~:<math>w(z)=P \left\{ \begin{matrix} a & b & c & \; \\~~
	~~\alpha & \beta & \gamma & z \\~~
	~~\alpha' & \beta' & \gamma' & \;~~
	~~\end{matrix} \right\}</math>~~

	~~The standard [[hypergeometric function]] may be expressed as~~

	~~:<math>\;_2F_1(a,b;c;z) =~~
	~~P \left\{ \begin{matrix} 0 & \infty & 1 & \; \\~~
	~~0 & a & 0 & z \\~~
	~~1-c & b & c-a-b & \;~~
	~~\end{matrix} \right\}</math>~~

	~~The P-functions obey a number of identities; one of~~ them ~~allows a general P-function~~ to ~~be expressed in terms of the hypergeometric function~~. It is

	~~:<math>P \left\{ \begin{matrix} a & b & c & \; \\~~
	~~\alpha & \beta & \gamma & z \\~~
	~~\alpha' & \beta' & \gamma' & \;~~
	~~\end{matrix} \right\} =~~
	~~\left(\frac{z-a}{z-b}\right)^\alpha~~
	~~\left(\frac{z-c}{z-b}\right)^\gamma~~
	~~P \left\{ \begin{matrix} 0 & \infty & 1 & \; \\~~
	~~0 & \alpha+\beta+\gamma & 0 & \;\frac{(z-a)(c-b)}{(z-b)(c-a)} \\~~
	~~\alpha'-\alpha & \alpha+\beta'+\gamma & \gamma'-\gamma & \;~~
	~~\end{matrix} \right\}~~
	~~</math>~~

	~~In other words, one may write the solutions in terms of the hypergeometric function as~~

	~~:<math>w(z)=~~
	~~\left(\frac{z-a}{z-b}\right)^\alpha~~
	~~\left(\frac{z-c}{z-b}\right)^\gamma~~
	~~\;_2F_1 \left(~~
	~~\alpha+\beta +\gamma,~~
	~~\alpha+\beta'+\gamma;~~
	~~1+\alpha-\alpha';~~
	~~\frac{(z-a)(c-b)}{(z-b)(c-a)} \right)~~
	~~</math>~~

	~~The full complement of [[Ernst Kummer\|Kummer]]'s 24 solutions may be obtained in this way; see the article [[hypergeometric differential equation]] for a treatment of Kummer's solutions~~.

	~~==Fractional linear transformations==~~
	The P-function possesses a simple symmetry under the action of [[fractional linear transformation]]s known as [[Möbius transformation]]s (that are the [[conformal map\|conformal remappings]] of the Riemann sphere), or equivalently, under the action of the group {{math\|[[general linear group\|GL]](2, '''C''')}}. ~~Given arbitrary [[complex number]]s {{mvar\|A}}, {{mvar\|B}}, {{mvar\|C}}, {{mvar\|D}} such that {{math\|''AD'' − ''BC'' ≠ 0}}, define the quantities~~

	:<~~math~~>~~u=\frac{Az+B}{Cz+D}~~
	~~\quad \text{ and } \quad~~
	~~\eta=\frac{Aa+B}{Ca+D}~~<~~/math~~>

	~~and~~

	:~~<math>\zeta=\frac{Ab+B}{Cb+D}~~
	~~\quad \text{ and } \quad~~
	~~\theta=\frac{Ac+B}{Cc+D}</math>~~

	~~then one has the simple relation~~

	~~:<math>P \left\{ \begin{matrix} a & b & c & \; \\~~
	~~\alpha & \beta & \gamma & z \\~~
	~~\alpha' & \beta' & \gamma' & \;~~
	~~\end{matrix} \right\}~~
	~~=P \left\{ \begin{matrix}~~
	~~\eta & \zeta & \theta & \; \\~~
	~~\alpha & \beta & \gamma & u \\~~
	~~\alpha' & \beta' & \gamma' & \;~~
	~~\end{matrix} \right\}</math>~~

	~~expressing the symmetry.~~

	~~==See also==~~
	*[[Complex differential equation]]

	~~==References==~~
	* Milton Abramowitz and Irene A. Stegun, eds., ''[[Abramowitz and Stegun\|Handbook of Mathematical Functions]] with Formulas, Graphs, and Mathematical Tables'' (~~Dover: New York, 1972)~~
	** [http://~~www.math.sfu~~.ca/~~~cbm/aands/page_556.htm Chapter 15] Hypergeometric Functions~~
	***[http://www.math.sfu.ca/~cbm/aands/page_564.htm Section 15.6] Riemann's Differential Equation

	~~[[Category:Hypergeometric functions]]~~
	~~[[Category:Ordinary differential equations]~~]

en>Hypersonic09: Linked to "Bessel functions" page in introduction section.

2013-09-09T04:33:43Z

Linked to "Bessel functions" page in introduction section.

← Older revision		Revision as of 06:33, 9 September 2013
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	~~I would like to introduce myself~~ to you, ~~I am Andrew~~ and ~~my spouse doesn~~'~~t like it at all~~. ~~Ohio is exactly where my house is but my spouse wants us~~ to ~~move~~. ~~What me and my family members adore is doing ballet but I~~'~~ve been using on new issues recently~~. ~~My working day occupation~~ is ~~an invoicing officer but I~~'~~ve currently utilized~~ for ~~an additional one~~.<br><br>~~Take~~ a ~~look at my blog post~~; [http://www.~~facequan~~.~~com~~/~~space~~.~~php?uid=285656&do=blog&id=1228443 psychics online~~]		In [[mathematics]], '''Riemann's differential equation''', named after [[Bernhard Riemann]], is a generalization of the [[hypergeometric differential equation]], allowing the [[regular singular points]] to occur anywhere on the [[Riemann sphere]], rather than merely at 0, 1, and ∞.

			==Definition==
			The differential equation is given by
			:<math>\frac{d^2w}{dz^2} + \left[
			\frac{1-\alpha-\alpha'}{z-a} +
			\frac{1-\beta-\beta'}{z-b} +
			\frac{1-\gamma-\gamma'}{z-c} \right] \frac{dw}{dz} </math>
			::<math>+\left[
			\frac{\alpha\alpha' (a-b)(a-c)} {z-a}
			+\frac{\beta\beta' (b-c)(b-a)} {z-b}
			+\frac{\gamma\gamma' (c-a)(c-b)} {z-c}
			\right]
			\frac{w}{(z-a)(z-b)(z-c)}=0.</math>

			The regular singular points are {{mvar\|a}}, {{mvar\|b}}, and {{mvar\|c}}. The pairs of exponents{{clarification needed\|date=November 2013\|reason=There is no exponentiation yet, and you may not call it “exponents”}} for each are respectively {{math\|''α''; ''α′''}}, {{math\|''β''; ''β′''}}, and {{math\|''γ''; ''γ′''}}. The exponents are subject to the condition

			:<math>\alpha+\alpha'+\beta+\beta'+\gamma+\gamma'=1.</math>

			==Solutions==
			The solutions are denoted by the ''Riemann P-symbol''

			:<math>w(z)=P \left\{ \begin{matrix} a & b & c & \; \\
			\alpha & \beta & \gamma & z \\
			\alpha' & \beta' & \gamma' & \;
			\end{matrix} \right\}</math>

			The standard [[hypergeometric function]] may be expressed as

			:<math>\;_2F_1(a,b;c;z) =
			P \left\{ \begin{matrix} 0 & \infty & 1 & \; \\
			0 & a & 0 & z \\
			1-c & b & c-a-b & \;
			\end{matrix} \right\}</math>

			The P-functions obey a number of identities; one of them allows a general P-function to be expressed in terms of the hypergeometric function. It is

			:<math>P \left\{ \begin{matrix} a & b & c & \; \\
			\alpha & \beta & \gamma & z \\
			\alpha' & \beta' & \gamma' & \;
			\end{matrix} \right\} =
			\left(\frac{z-a}{z-b}\right)^\alpha
			\left(\frac{z-c}{z-b}\right)^\gamma
			P \left\{ \begin{matrix} 0 & \infty & 1 & \; \\
			0 & \alpha+\beta+\gamma & 0 & \;\frac{(z-a)(c-b)}{(z-b)(c-a)} \\
			\alpha'-\alpha & \alpha+\beta'+\gamma & \gamma'-\gamma & \;
			\end{matrix} \right\}
			</math>

			In other words, one may write the solutions in terms of the hypergeometric function as

			:<math>w(z)=
			\left(\frac{z-a}{z-b}\right)^\alpha
			\left(\frac{z-c}{z-b}\right)^\gamma
			\;_2F_1 \left(
			\alpha+\beta +\gamma,
			\alpha+\beta'+\gamma;
			1+\alpha-\alpha';
			\frac{(z-a)(c-b)}{(z-b)(c-a)} \right)
			</math>

			The full complement of [[Ernst Kummer\|Kummer]]'s 24 solutions may be obtained in this way; see the article [[hypergeometric differential equation]] for a treatment of Kummer's solutions.

			==Fractional linear transformations==
			The P-function possesses a simple symmetry under the action of [[fractional linear transformation]]s known as [[Möbius transformation]]s (that are the [[conformal map\|conformal remappings]] of the Riemann sphere), or equivalently, under the action of the group {{math\|[[general linear group\|GL]](2, '''C''')}}. Given arbitrary [[complex number]]s {{mvar\|A}}, {{mvar\|B}}, {{mvar\|C}}, {{mvar\|D}} such that {{math\|''AD'' − ''BC'' ≠ 0}}, define the quantities

			:<math>u=\frac{Az+B}{Cz+D}
			\quad \text{ and } \quad
			\eta=\frac{Aa+B}{Ca+D}</math>

			and

			:<math>\zeta=\frac{Ab+B}{Cb+D}
			\quad \text{ and } \quad
			\theta=\frac{Ac+B}{Cc+D}</math>

			then one has the simple relation

			:<math>P \left\{ \begin{matrix} a & b & c & \; \\
			\alpha & \beta & \gamma & z \\
			\alpha' & \beta' & \gamma' & \;
			\end{matrix} \right\}
			=P \left\{ \begin{matrix}
			\eta & \zeta & \theta & \; \\
			\alpha & \beta & \gamma & u \\
			\alpha' & \beta' & \gamma' & \;
			\end{matrix} \right\}</math>

			expressing the symmetry.

			==See also==
			*[[Complex differential equation]]

			==References==
			* Milton Abramowitz and Irene A. Stegun, eds., ''[[Abramowitz and Stegun\|Handbook of Mathematical Functions]] with Formulas, Graphs, and Mathematical Tables'' (Dover: New York, 1972)
			** [http://www.math.sfu.ca/~cbm/aands/page_556.htm Chapter 15] Hypergeometric Functions
			***[http://www.math.sfu.ca/~cbm/aands/page_564.htm Section 15.6] Riemann's Differential Equation

			[[Category:Hypergeometric functions]]
			[[Category:Ordinary differential equations]]

en>BattyBot: changed {{no footnotes}} to {{more footnotes}} & general fixes using AWB (8097)

2012-07-17T02:49:03Z

changed {{no footnotes}} to {{more footnotes}} & general fixes using AWB (8097)

New page

I would like to introduce myself to you, I am Andrew and my spouse doesn't like it at all. Ohio is exactly where my house is but my spouse wants us to move. What me and my family members adore is doing ballet but I've been using on new issues recently. My working day occupation is an invoicing officer but I've currently utilized for an additional one.<br><br>Take a look at my blog post; [http://www.facequan.com/space.php?uid=285656&do=blog&id=1228443 psychics online]