Catalan's constant K, which occasionally appears in estimates in combinatorics, is defined by Combinatorics is a branch of mathematics that studies finite collections of objects that satisfy specified criteria. ...

or equivalently

along with

where K(x) is a complete elliptic integral of the first kind, and has nothing to do with the constant itself. In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse and were first studied by Fagnano and Leonhard Euler. ...

Uses

K appears in combinatorics, as well as in values of the second polygamma function, also called the trigamma function, at fractional arguments: Combinatorics is a branch of mathematics that studies finite collections of objects that satisfy specified criteria. ... In mathematics, the polygamma function of order m is defined as the m+1 th derivative of the logarithm of the gamma function: Here is the digamma function and is the gamma function. ... In mathematics, the trigamma function, denoted ψ1(z), is the second of the polygamma functions, and is defined by . It follows from this definition that where ψ(z) is the digamma function. ...

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. In probability theory and statistics, the hyperbolic secant distribution is a continuous probability distribution whose probability density function and characteristic function are proportional to the hyperbolic secant function. ...

It is not known whether K is rational or irrational. In mathematics, a rational number (or informally fraction) is a ratio of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... In mathematics, an irrational number is any real number that is not a rational number, i. ...

See Also

polygamma function In mathematics, the polygamma function of order m is defined as the m+1 th derivative of the logarithm of the gamma function: Here is the digamma function and is the gamma function. ...

elliptic integral In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse and were first studied by Fagnano and Leonhard Euler. ...

External links

Catalan's Constant -- from MathWorld (http://mathworld.wolfram.com/CatalansConstant.html|)