Podcast Beta
Questions and Answers
What is the Dirichlet beta function (also known as the Catalan beta function)?
The Dirichlet beta function is defined as $\beta(s)=\sum_{n=0}^{\infty}\frac{(-1)^n},{(2n+1)^s}$ or, equivalently, $\beta(s) = \frac{1},{\Gamma(s)}\int_{0}^{\infty}\frac{x^{s-1}e^{-x}},{1+e^{-2x}},dx$. Additionally, it can be expressed as $\beta(s)=4^{-s}\left(\zeta\left(s,\frac{1},{4}\right)-\zeta\left(s,\frac{3},{4}\right)\right)$ and $\beta(s)=2^{-s}\Phi\left(-1,s,\frac{1},{2}\right)$, valid for all complex values of s. It can also be written in terms of the polylogarithm function as $\beta(s)=\frac{i},{2}\left(\text{Li}_s(-i)-\text{Li}_s(i)\right)$.
What is the assumption for the Dirichlet beta function when expressed in the first two definitions?
In the first two definitions, it is assumed that Re(s) > 0.
What is another equivalent definition of the Dirichlet beta function, in terms of a specific transcendent function?
Another equivalent definition, in terms of the Lerch transcendent, is $\beta(s)=2^{-s}\Phi\left(-1,s,\frac{1},{2}\right)$, which is valid for all complex values of s.
How can the Dirichlet beta function be expressed in terms of the polylogarithm function?
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In which complex s-plane is the Dirichlet beta function valid, according to an alternative definition?
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