en>AnomieBOT: Dating maintenance tags: {{Elucidate}}

2015-01-06T22:58:00Z

Dating maintenance tags: {{Elucidate}}

en>Eric Kvaalen: Not sure whether H^p is a subspace of L^p when p<1.

2013-12-12T19:50:23Z

Not sure whether H^p is a subspace of L^p when p<1.

← Older revision		Revision as of 21:50, 12 December 2013
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	~~I am Carri from Saint Clairsville. I love~~ to ~~play Lap Steel Guitar~~. ~~Other hobbies are Airsoft~~.<br><br>~~My website~~ ... [http://~~www~~.~~huajuckclub~~.~~com~~/~~modules~~.~~php?name~~=~~Your_Account~~&op=~~userinfo~~&~~username~~=~~IUwa how to get free fifa 15 coins~~]		{{Otheruses4\|the Dirichlet beta function\|other beta functions\|Beta function (disambiguation)}}
			In [[mathematics]], the '''Dirichlet beta function''' (also known as the '''Catalan beta function''') is a [[special function]], closely related to the [[Riemann zeta function]]. It is a particular [[Dirichlet L-function]], the L-function for the alternating [[Dirichlet character\|character]] of period four.

			==Definition==
			The Dirichlet beta function is defined as

			:<math>\beta(s) = \sum_{n=0}^\infty \frac{(-1)^n} {(2n+1)^s},</math>

			or, equivalently,

			:<math>\beta(s) = \frac{1}{\Gamma(s)}\int_0^{\infty}\frac{x^{s-1}e^{-x}}{1 + e^{-2x}}\,dx.</math>

			In each case, it is assumed that Re(''s'') > 0.

			Alternatively, the following definition, in terms of the [[Hurwitz zeta function]], is valid in the whole complex ''s''-plane:

			:<math>\beta(s) = 4^{-s} \left( \zeta\left(s,{1 \over 4}\right)-\zeta\left( s, {3 \over 4}\right) \right).</math> [http://engineeringandmathematics.blogspot.co.nz/2012/09/dirichlet-beta-hurwitz-zeta-relation.html proof]

			Another equivalent definition, in terms of the [[Lerch transcendent]], is:

			:<math>\beta(s) = 2^{-s} \Phi\left(-1,s,{{1} \over {2}}\right),</math>

			which is once again valid for all complex values of ''s''.

			==Functional equation==
			The [[functional equation (L-function)\|functional equation]] extends the beta function to the left side of the [[complex plane]] Re(''s'')<0. It is given by
			:<math>\beta(s)=\left(\frac{\pi}{2}\right)^{s-1} \Gamma(1-s)
			\cos \frac{\pi s}{2}\,\beta(1-s)</math>

			where Γ(''s'') is the [[gamma function]].

			==Special values==
			Some special values include:

			:<math>\beta(0)= \frac{1}{2}, </math>

			:<math>\beta(1)\;=\;\tan^{-1}(1)\;=\;\frac{\pi}{4}, </math>

			:<math>\beta(2)\;=\;G,</math>

			where ''G'' represents [[Catalan's constant]], and

			:<math>\beta(3)\;=\;\frac{\pi^3}{32},</math>

			:<math>\beta(4)\;=\;\frac{1}{768}(\psi_3(\frac{1}{4})-8\pi^4),</math>

			:<math>\beta(5)\;=\;\frac{5\pi^5}{1536},</math>

			:<math>\beta(7)\;=\;\frac{61\pi^7}{184320},</math>

			where <math>\psi_3(1/4)</math> in the above is an example of the [[polygamma function]]. More generally, for any positive integer ''k'':

			:<math>\beta(2k+1)={{{({-1})^k}{E_{2k}}{\pi^{2k+1}} \over {4^{k+1}}(2k)!}}, </math>

			where <math> \!\ E_{n} </math> represent the [[Euler number]]s. For integer ''k'' ≥ 0, this extends to:

			:<math>\beta(-k)={{E_{k}} \over {2}}.</math>

			Hence, the function vanishes for all odd negative integral values of the argument.
			{{Table
			\|type=class="wikitable"
			\|hdrs=s!!approximate value β(s)!![[OEIS]]
			\|row1=1/5{{!!}}0.5737108471859466493572665{{!!}}
			\|row2=1/4{{!!}}0.5907230564424947318659591{{!!}}
			\|row3=1/3{{!!}}0.6178550888488520660725389{{!!}}
			\|row4=1/2{{!!}}0.6676914571896091766586909{{!!}}{{OEIS link\|A195103}}
			\|row5=1{{!!}}0.7853981633974483096156608{{!!}}{{OEIS link\|A003881}}
			\|row6=2{{!!}}0.9159655941772190150546035{{!!}}{{OEIS link\|A006752}}
			\|row7=3{{!!}}0.9689461462593693804836348{{!!}}{{OEIS link\|A153071}}
			\|row8=4{{!!}}0.9889445517411053361084226{{!!}}{{OEIS link\|A175572}}
			\|row9=5{{!!}}0.9961578280770880640063194{{!!}}{{OEIS link\|A175571}}
			\|row10=6{{!!}}0.9986852222184381354416008{{!!}}{{OEIS link\|A175570}}
			\|row11=7{{!!}}0.9995545078905399094963465
			\|row12=8{{!!}}0.9998499902468296563380671
			\|row13=9{{!!}}0.9999496841872200898213589
			\|row14=10{{!!}}0.9999831640261968774055407
			}}

			There are zeros at -1; -3; -5; -7 etc.

			==See also==
			* [[Hurwitz zeta function]]

			==References==
			* {{cite journal
			\|first1=M. L.
			\|last1=Glasser
			\|title=The evaluation of lattice sums. I. Analytic procedures
			\|year=1972
			\|journal=J. Math. Phys.
			\|doi=10.1063/1.1666331
			\|volume=14
			\|page=409
			}}
			* J. Spanier and K. B. Oldham, ''An Atlas of Functions'', (1987) Hemisphere, New York.
			* {{MathWorld\|title=Dirichlet Beta Function\|urlname=DirichletBetaFunction}}

			[[Category:Zeta and L-functions]]

en>Giuseppe Negro: /* References */ added link to David Gilbarg's page

2012-05-19T21:22:38Z

References: added link to David Gilbarg's page

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I am Carri from Saint Clairsville. I love to play Lap Steel Guitar. Other hobbies are Airsoft.<br><br>My website ... [http://www.huajuckclub.com/modules.php?name=Your_Account&op=userinfo&username=IUwa how to get free fifa 15 coins]

Poisson kernel - Revision history

en>AnomieBOT: Dating maintenance tags: {{Elucidate}}

en>Eric Kvaalen: Not sure whether H^p is a subspace of L^p when p<1.

en>Giuseppe Negro: /* References */ added link to David Gilbarg's page