Search Results for ""

J_m(x)=(x^m)/(2^(m-1)sqrt(pi)Gamma(m+1/2))int_0^1cos(xt)(1-t^2)^(m-1/2)dt, where J_m(x) is a Bessel function of the first kind and Gamma(z) is the gamma function. Hankel's ...

The symbol defined by (v,n) = (2^(-2n){(4v^2-1)(4v^2-3^2)...[4v^2-(2n-1)^2]})/(n!) (1) = ((-1)^ncos(piv)Gamma(1/2+n-v)Gamma(1/2+n+v))/(pin!), (2) where Gamma(z) is the gamma ...

J_n(z) = 1/(2pi)int_(-pi)^pie^(izcost)e^(in(t-pi/2))dt (1) = (i^(-n))/piint_0^pie^(izcost)cos(nt)dt (2) = 1/piint_0^picos(zsint-nt)dt (3) for n=0, 1, 2, ..., where J_n(z) is ...

Let alpha_(n+1) = (2alpha_nbeta_n)/(alpha_n+beta_n) (1) beta_(n+1) = sqrt(alpha_nbeta_n), (2) then H(alpha_0,beta_0)=lim_(n->infty)a_n=1/(M(alpha_0^(-1),beta_0^(-1))), (3) ...

Let X be a metric space, A be a subset of X, and d a number >=0. The d-dimensional Hausdorff measure of A, H^d(A), is the infimum of positive numbers y such that for every ...

For n>=3, there exist no additive finite and invariant measures for the group of displacements in R^n.

The havercosine, also called the haversed cosine, is a little-used trigonometric function defined by havercosz = vercosz (1) = 1/2(1+cosz), (2) where vercosz is the vercosine ...

An absolutely continuous measure on partialD whose density has the form exp(x+y^_), where x and y are real-valued functions in L^infty, ||y||_infty<pi/2, exp is the ...

A function S_n(z) which satisfies the recurrence relation S_(n-1)(z)-S_(n+1)(z)=2S_n^'(z) together with S_1(z)=-S_0^'(z) is called a hemicylindrical function.

The Machin-like formula 1/4pi=2cot^(-1)(2)-cot^(-1)(7). The other 2-term Machin-like formulas are Euler's Machin-like formula, Hutton's formula, and Machin's formula.