in mathematics, catalan's constant g, which occasionally appears in estimates in combinatorics, is defined by :g = \beta(2) = \sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2n+1)^2} = \frac{1}{1^2} - \frac{1}{3^2} + \frac{1}{5^2} - \frac{1}{7^2} + \cdots \! where Î² is the dirichlet beta function. its numerical value  is approximately  :g = 0.915 965 594 177 219 015 054 603 514 932 384 110 774 â¦ it is not known whether g is irrational, let alone transcendental. catalan's constant was named after eugÃ¨ne charles catalan. ==integral identities== some identities include :g = \int_0^1 \int_0^1 \frac{1}{1+x^2 y^2} \,dx\, dy \! :g = -\int_0^1 \frac{\ln t}{1 + t^2} \,dt \! :g = \int_{0}^{\pi/4} \frac{t}{\sin t \cos t} \;dt \! :g = \frac{1}{4} \int_{-\pi/2}^{\pi/2} \frac{t}{\sin t} \;dt \! :g = \int_0^{\pi/4} \ln ( \cot(t) ) \,dt \! :g = \int_0^\infty \arctan (e^{-t}) \,dt \! along with : g = \frac{1}{2} \int_0^1 \mathrm{k}(t)\,dt \! where k(t) is a complete elliptic integral of the first kind. and with gamma function Î(x+1) = x! : g = \frac{\pi}{2} \int_0^\tfrac12\gamma(1+x)\gamma(1-x)\,dx the integral: : g = \text{ti}_2(1)=\int_0^1 \frac{\arctan t}{t}\,dt. \! is a known special function, called the inverse tangent integral, and was extensively studied by ramanujan ==uses== g appears in combinatorics, as well as in values of the second polygamma function, also called the trigamma function, at fractional arguments: : \psi_1 \left(\tfrac14\right) = \pi^2 + 8g : \psi_1 \left(\tfrac34\right) = \pi^2 - 8g. simon plouffe gives an infinite collection of identities between the trigamma function, &pi;2 and catalan's constant; these are expressible as paths on a graph. it also appears in connection with the hyperbolic secant distribution. ==relation to other special function== catalan's constant occurs frequently in relation to the clausen function, the inverse tangent integral, the inverse sine integral, barnes g-function, as well as integrals and series summable in terms of the aforementioned functions. as a particular example, by first expressing the inverse tangent integral in its closed form â in terms of clausen functions - and then expressing those clausen functions in terms of the barnes g-function, the following expression is easily obtained (n.b. all the relevant relations for this derivation have been added to the page for the clausen function): :g=4\pi \log\left( \frac{ g(\tfrac{3}{8}) g(\tfrac{7}{8}) }{ g(\tfrac{1}{8}) g(\tfrac{5}{8}) } \right) +4 \pi \log \left( \frac{ \gamma(\tfrac{3}{8}) }{ \gamma(\tfrac{1}{8}) } \right) +\frac{\pi}{2} \log \left( \frac{1+\sqrt{2} }{2 \, (2-\sqrt{2})} \right). if one defines the lerch transcendent, \phi(z, s, \alpha), (related to the lerch zeta function) by, :\phi(z, s, \alpha) = \sum_{n=0}^\infty
Combinatorics Mathematical_constants