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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 ...

A pyramidal number of the form n(n+1)(5n-2)/6, The first few are 1, 8, 26, 60, 115, ... (OEIS A002413). The generating function for the heptagonal pyramidal numbers is ...

The bias of a series is defined as Q[a_i,a_(i+1),a_(i+2)]=(a_ia_(i+2)-a_(i+1)^2)/(a_ia_(i+1)a_(i+2)). A series is geometric iff Q=0. A series is artistic iff the bias is ...

An odd composite number N is called a Somer-Lucas d-pseudoprime (with d>=1) if there exists a nondegenerate Lucas sequence U(P,Q) with U_0=0, U_1=1, D=P^2-4Q, such that ...

By the definition of the trigonometric functions, cos0 = 1 (1) cot0 = infty (2) csc0 = infty (3) sec0 = 1 (4) sin0 = 0 (5) tan0 = 0. (6)

The minimal residue of a (mod m) is the value b or b-m, whichever is smaller in absolute value, where a=b (mod m). If m=2b (so that b=|b-m|), then the minimal residue is ...

A doubly periodic function with periods 2omega_1 and 2omega_2 such that f(z+2omega_1)=f(z+2omega_2)=f(z), (1) which is analytic and has no singularities except for poles in ...

For F_n the nth Fibonacci number, F_(n-1)F_(n+1)-F_n^2=(-1)^n. This identity was also discovered by Simson (Coxeter and Greitzer 1967, p. 41; Coxeter 1969, pp. 165-168; Wells ...

The Kuen surface is a special case of Enneper's negative curvature surfaces which can be given parametrically by x = (2(cosu+usinu)sinv)/(1+u^2sin^2v) (1) = ...

The average distance between two points chosen at random inside a unit cube (the n=3 case of hypercube line picking), sometimes known as the Robbins constant, is Delta(3) = ...