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Let E(k) and K(k) be complete elliptic integrals of the first and second kinds, with E^'(k) and K^'(k) the complementary integrals. Then ...

Legendre's conjecture asserts that for every n there exists a prime p between n^2 and (n+1)^2 (Hardy and Wright 1979, p. 415; Ribenboim 1996, pp. 397-398). It is one of ...

The Legendre differential equation is the second-order ordinary differential equation (1-x^2)(d^2y)/(dx^2)-2x(dy)/(dx)+l(l+1)y=0, (1) which can be rewritten ...

The (associated) Legendre function of the first kind P_n^m(z) is the solution to the Legendre differential equation which is regular at the origin. For m,n integers and z ...

Because the Legendre polynomials form a complete orthogonal system over the interval [-1,1] with respect to the weighting function w(x)=1, any function f(x) may be expanded ...

The associated Legendre polynomials P_l^m(x) and P_l^(-m)(x) generalize the Legendre polynomials P_l(x) and are solutions to the associated Legendre differential equation, ...

Legendre-Gauss quadrature is a numerical integration method also called "the" Gaussian quadrature or Legendre quadrature. A Gaussian quadrature over the interval [-1,1] with ...

The function defined by chi_nu(z)=sum_(k=0)^infty(z^(2k+1))/((2k+1)^nu). (1) It is related to the polylogarithm by chi_nu(z) = 1/2[Li_nu(z)-Li_nu(-z)] (2) = ...

A prime factorization algorithm in which a sequence of trial divisors is chosen using a quadratic sieve. By using quadratic residues of N, the quadratic residues of the ...

Gamma functions of argument 2z can be expressed in terms of gamma functions of smaller arguments. From the definition of the beta function, ...