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One of the four divisions of observations which have been grouped into four equal-sized sets based on their statistical rank. The quartile including the top statistically ...

The decimal expansion of a number is its representation in base-10 (i.e., in the decimal system). In this system, each "decimal place" consists of a digit 0-9 arranged such ...

Machin-like formulas have the form mcot^(-1)u+ncot^(-1)v=1/4kpi, (1) where u, v, and k are positive integers and m and n are nonnegative integers. Some such formulas can be ...

Ahmed's integral is the definite integral int_0^1(tan^(-1)(sqrt(x^2+2)))/(sqrt(x^2+2)(x^2+1))dx=5/(96)pi^2 (OEIS A096615; Ahmed 2002; Borwein et al. 2004, pp. 17-20). This is ...

mu_i(epsilon), sometimes denoted P_i(epsilon), is the probability that element i is populated, normalized such that sum_(i=1)^Nmu_i(epsilon)=1.

j_n(z)=(z^n)/(2^(n+1)n!)int_0^picos(zcostheta)sin^(2n+1)thetadtheta, where j_n(z) is a spherical Bessel function of the first kind.

Let C denote a chain complex, a portion of which is shown below: ...->C_(n+1)->C_n->C_(n-1)->.... Let H_n(C)=kerpartial_n/Impartial_(n+1) denotes the nth homology group. Then ...

The two functions theta(x) and psi(x) defined below are known as the Chebyshev functions. The function theta(x) is defined by theta(x) = sum_(k=1)^(pi(x))lnp_k (1) = ...

The cotangent function cotz is the function defined by cotz = 1/(tanz) (1) = (i(e^(iz)+e^(-iz)))/(e^(iz)-e^(-iz)) (2) = (i(e^(2iz)+1))/(e^(2iz)-1), (3) where tanz is the ...

A prime gap of length n is a run of n-1 consecutive composite numbers between two successive primes. Therefore, the difference between two successive primes p_k and p_(k+1) ...