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Let Gamma(z) be the gamma function and n!! denote a double factorial, then [(Gamma(m+1/2))/(Gamma(m))]^2[1/m+(1/2)^21/(m+1)+((1·3)/(2·4))^21/(m+2)+...]_()_(n) ...

A fractional clique of a graph G is a nonnegative real function on the vertices of G such that sum of the values on the vertices of any independent set is at most one. The ...

A necessary and sufficient condition that there should exist at least one nondecreasing function alpha(t) such that mu_n=int_(-infty)^inftyt^ndalpha(t) for n=0, 1, 2, ..., ...

A function of the coordinates which is constant along a trajectory in phase space. The number of degrees of freedom of a dynamical system such as the Duffing differential ...

Laplace's integral is one of the following integral representations of the Legendre polynomial P_n(x), P_n(x) = 1/piint_0^pi(du)/((x+sqrt(x^2-1)cosu)^(n+1))du (1) = ...

A smooth curve is a curve which is a smooth function, where the word "curve" is interpreted in the analytic geometry context. In particular, a smooth curve is a continuous ...

A function f is topologically transitive if, given any two intervals U and V, there is some positive integer k such that f^k(U) intersection V!=emptyset. Vaguely, this means ...

Although Bessel functions of the second kind are sometimes called Weber functions, Abramowitz and Stegun (1972) define a separate Weber function as ...

A.k.a. the pigeonhole principle. Given n boxes and m>n objects, at least one box must contain more than one object. This statement has important applications in number theory ...

Let H be a two-dimensional distribution function with marginal distribution functions F and G. Then there exists a copula C such that H(x,y)=C(F(x),G(y)). Conversely, for any ...