Catalan's constant

'''Catalan's constant''' ''K'', which occasionally appears in estimates in combinatorics, is defined by :

\Kappa = \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} + ...

or equivalently :

K = -\int_{0}^{1} \frac{\ln(t)}{1 + t^2} \mbox{ d} t.

along with :

K = \frac{1}{2}\int_0^1 \mathrm{K}(x)\,dx

K = \int_0^1 \frac{\tan^{-1}x}{x}dx

where K(x) is a complete elliptic integral of the first kind, and has nothing to do with the constant itself. ==Uses== K appears in combinatorics, as well as in values of the second polygamma function, also called the trigamma function, at fractional arguments: :

\psi_{1}\left(\frac{1}{4}\right) = \pi^2 + 8K

\psi_{1}\left(\frac{3}{4}\right) = \pi^2 - 8K

Its numerical value is approximately :''K'' = .915 965 594 177 219 015 054 603 514 932 384 110 774 ... It also appears in connection with the hyperbolic secant distribution. It is not known whether ''K'' is rational number or irrational number. ==See Also== * polygamma function * trigamma function * elliptic integral ===External links=== [http://mathworld.wolfram.com/CatalansConstant.html| Catalan's Constant -- from MathWorld] Mathematical constants Real numbers

See other meanings of words starting from letter:

C

CA | CB | CD | CE | CF | CG | CH | CI | CJ | CK | CL | CM | CN | CO | CP | CR | CS | CT | CU | CW | CX | CY | CZ |

Words begining with Catalan\'s_constant:

Catalan's_constant