Title: Dirichlet Beta Function -- from Wolfram MathWorld
Open Graph Title: Dirichlet Beta Function -- from Wolfram MathWorld
X Title: Dirichlet Beta Function -- from Wolfram MathWorld
Description: The Dirichlet beta function is defined by the sum beta(x) = sum_(n=0)^(infty)(-1)^n(2n+1)^(-x) (1) = 2^(-x)Phi(-1,x,1/2), (2) where Phi(z,s,a) is the Lerch transcendent. The beta function can be written in terms of the Hurwitz zeta function zeta(x,a) by beta(x)=1/(4^x)[zeta(x,1/4)-zeta(x,3/4)]. (3) The beta function can be defined over the whole complex plane using analytic continuation, beta(1-z)=(2/pi)^zsin(1/2piz)Gamma(z)beta(z), (4) where Gamma(z) is the gamma function....
Open Graph Description: The Dirichlet beta function is defined by the sum beta(x) = sum_(n=0)^(infty)(-1)^n(2n+1)^(-x) (1) = 2^(-x)Phi(-1,x,1/2), (2) where Phi(z,s,a) is the Lerch transcendent. The beta function can be written in terms of the Hurwitz zeta function zeta(x,a) by beta(x)=1/(4^x)[zeta(x,1/4)-zeta(x,3/4)]. (3) The beta function can be defined over the whole complex plane using analytic continuation, beta(1-z)=(2/pi)^zsin(1/2piz)Gamma(z)beta(z), (4) where Gamma(z) is the gamma function....
X Description: The Dirichlet beta function is defined by the sum beta(x) = sum_(n=0)^(infty)(-1)^n(2n+1)^(-x) (1) = 2^(-x)Phi(-1,x,1/2), (2) where Phi(z,s,a) is the Lerch transcendent. The beta function can be written in terms of the Hurwitz zeta function zeta(x,a) by beta(x)=1/(4^x)[zeta(x,1/4)-zeta(x,3/4)]. (3) The beta function can be defined over the whole complex plane using analytic continuation, beta(1-z)=(2/pi)^zsin(1/2piz)Gamma(z)beta(z), (4) where Gamma(z) is the gamma function....
Opengraph URL: https://mathworld.wolfram.com/DirichletBetaFunction.html
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| DC.Title | Dirichlet Beta Function |
| DC.Creator | Weisstein, Eric W. |
| DC.Description | The Dirichlet beta function is defined by the sum beta(x) = sum_(n=0)^(infty)(-1)^n(2n+1)^(-x) (1) = 2^(-x)Phi(-1,x,1/2), (2) where Phi(z,s,a) is the Lerch transcendent. The beta function can be written in terms of the Hurwitz zeta function zeta(x,a) by beta(x)=1/(4^x)[zeta(x,1/4)-zeta(x,3/4)]. (3) The beta function can be defined over the whole complex plane using analytic continuation, beta(1-z)=(2/pi)^zsin(1/2piz)Gamma(z)beta(z), (4) where Gamma(z) is the gamma function.... |
| DC.Date.Modified | 2014-06-05 |
| DC.Subject | 11M |
| DC.Rights | Copyright 1999-2026 Wolfram Research, Inc. See https://mathworld.wolfram.com/about/terms.html for a full terms of use statement. |
| DC.Format | text/html |
| DC.Identifier | https://mathworld.wolfram.com/DirichletBetaFunction.html |
| DC.Language | en |
| DC.Publisher | Wolfram Research, Inc. |
| DC.Relation.IsPartOf | https://mathworld.wolfram.com/ |
| DC.Type | Text |
| Last-Modified | 2014-06-05 |
| og:image | https://mathworld.wolfram.com/images/socialmedia/share/ogimage_DirichletBetaFunction.png |
| og:type | website |
| twitter:card | summary_large_image |
| twitter:image:src | https://mathworld.wolfram.com/images/socialmedia/share/ogimage_DirichletBetaFunction.png |
| None | ie=edge |
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