Rectified tesseract - Revision history

en>Tomruen: 4-polytope

2014-12-27T13:39:50Z

4-polytope

← Older revision		Revision as of 15:39, 27 December 2014
Line 1:		Line 1:
	~~{{about\|reflection in number theory and calculus\|reflection formulas in geometry\|Reflection (mathematics)}}~~		Nice to satisfy you, my name is Numbers Held although I don't really like becoming known as like that. Hiring is his occupation. Years ago we moved to North Dakota. What I adore doing is playing baseball but I haven't made a dime with it.<br><br>my page: [http://faculty.jonahmancini.com/groups/solid-advice-for-dealing-with-a-candida-albicans/members/ std home test]
	~~In [[mathematics]]~~, ~~a '''reflection formula''' or '''reflection relation''' for a [[function (mathematics)\|function]] ''f''~~ is ~~a relationship between~~ '~~'f''(''a'' − ''x'') and ''f''(''x'')~~. It is ~~a special case of a [[functional equation]], and it is very common in the literature~~ to ~~use the term "functional equation" when "reflection formula"~~ is ~~meant.~~

	Reflection formulas are useful for [[numerical analysis\|numerical computation]] of [[special function]]s. In effect, an approximation that has greater accuracy or only converges on one side of a reflection point (typically in the positive half of the [[complex plane]]) can be employed for all arguments.

	~~== Known formulae ==~~
	~~The [[even and odd functions]] satisfy simple reflection relations around '~~'a~~'' = 0~~. ~~For all even functions,~~

	:<~~math~~>~~f(-x) = f(x),\,\!~~<~~/math~~>

	~~and for all odd functions,~~

	:~~<math>f(-x) = -f(x)~~.~~\,\!<~~/~~math>~~

	~~A famous relationship is '''Euler's reflection formula'''~~

	~~:<math>\Gamma(z)\Gamma(1~~-~~z) = \frac{\pi}{\sin{(\pi z)}}\!</math>~~

	~~for the [[Gamma function]] Γ(''z''), due to [[Leonhard Euler]].~~

	~~There is also a reflection formula~~ for ~~the general ''n''~~-~~th order [[polygamma function]] ψ<sup>(n)</sup>(''z''),~~

	~~:<math>\psi^{(n)} (1~~-~~z)+(~~-~~1)^{n+1}\psi^{(n)} (z) = (~~-~~1)^n \pi \frac{d^n}{d z^n} \cot{(\pi z)} \,</math>~~

	~~which springs trivally from the fact that the polygamma functions are defined as the derivations of the <math>\ln \Gamma</math> and thus its reflection formula is inherited to them.~~

	~~The [[Riemann zeta function]] ζ(''z'') satisfies~~

	~~:<math>\frac{\zeta(1~~-~~z)}{\zeta(z)} = \frac{2\, \Gamma(z)}{(2\pi)^{z}} \cos\left(\frac{\pi z}{2}\right),\,\!<~~/~~math>~~

	~~and the [[Riemann Xi function]] ξ(''z'') satisfies~~

	~~:<math>\xi(z) = \xi(1-z). \,\!<~~/~~math>~~

	~~==References==~~
	* {{MathWorld\|urlname=ReflectionRelation\|title=Reflection Relation}}
	* {{MathWorld\|urlname=PolygammaFunction\|title=Polygamma Function}}

	~~[[Category:Calculus]~~]

en>Tomruen at 18:31, 20 October 2013

2013-10-20T18:31:38Z

← Older revision		Revision as of 20:31, 20 October 2013
Line 1:		Line 1:
	~~Emilia Shryock~~ is ~~my name but you can contact me something you like~~. ~~Managing individuals~~ is ~~what I do~~ and the ~~wage~~ has ~~been truly fulfilling~~. ~~South Dakota~~ is ~~exactly where I~~'~~ve always been living~~. ~~To do aerobics~~ is a ~~thing that I~~'~~m totally addicted to.~~<br><br>~~Also visit my homepage at home std testing~~ (~~[http~~://~~www~~.~~videokeren.com~~/~~user/FJWW click through~~ the ~~up coming web page~~])		{{about\|reflection in number theory and calculus\|reflection formulas in geometry\|Reflection (mathematics)}}
			In [[mathematics]], a '''reflection formula''' or '''reflection relation''' for a [[function (mathematics)\|function]] ''f'' is a relationship between ''f''(''a'' − ''x'') and ''f''(''x''). It is a special case of a [[functional equation]], and it is very common in the literature to use the term "functional equation" when "reflection formula" is meant.

			Reflection formulas are useful for [[numerical analysis\|numerical computation]] of [[special function]]s. In effect, an approximation that has greater accuracy or only converges on one side of a reflection point (typically in the positive half of the [[complex plane]]) can be employed for all arguments.

			== Known formulae ==
			The [[even and odd functions]] satisfy simple reflection relations around ''a'' = 0. For all even functions,

			:<math>f(-x) = f(x),\,\!</math>

			and for all odd functions,

			:<math>f(-x) = -f(x).\,\!</math>

			A famous relationship is '''Euler's reflection formula'''

			:<math>\Gamma(z)\Gamma(1-z) = \frac{\pi}{\sin{(\pi z)}}\!</math>

			for the [[Gamma function]] Γ(''z''), due to [[Leonhard Euler]].

			There is also a reflection formula for the general ''n''-th order [[polygamma function]] ψ<sup>(n)</sup>(''z''),

			:<math>\psi^{(n)} (1-z)+(-1)^{n+1}\psi^{(n)} (z) = (-1)^n \pi \frac{d^n}{d z^n} \cot{(\pi z)} \,</math>

			which springs trivally from the fact that the polygamma functions are defined as the derivations of the <math>\ln \Gamma</math> and thus its reflection formula is inherited to them.

			The [[Riemann zeta function]] ζ(''z'') satisfies

			:<math>\frac{\zeta(1-z)}{\zeta(z)} = \frac{2\, \Gamma(z)}{(2\pi)^{z}} \cos\left(\frac{\pi z}{2}\right),\,\!</math>

			and the [[Riemann Xi function]] ξ(''z'') satisfies

			:<math>\xi(z) = \xi(1-z). \,\!</math>

			==References==
			* {{MathWorld\|urlname=ReflectionRelation\|title=Reflection Relation}}
			* {{MathWorld\|urlname=PolygammaFunction\|title=Polygamma Function}}

			[[Category:Calculus]]

en>Tomruen: order

2012-06-26T21:35:48Z

order

New page

Emilia Shryock is my name but you can contact me something you like. Managing individuals is what I do and the wage has been truly fulfilling. South Dakota is exactly where I've always been living. To do aerobics is a thing that I'm totally addicted to.<br><br>Also visit my homepage at home std testing ([http://www.videokeren.com/user/FJWW click through the up coming web page])