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Just as many interesting integer sequences can be defined and their properties studied, it is often of interest to additionally determine which of their elements are prime. ...

The constant pi, denoted pi, is a real number defined as the ratio of a circle's circumference C to its diameter d=2r, pi = C/d (1) = C/(2r) (2) pi has decimal expansion ...

Goldbach's original conjecture (sometimes called the "ternary" Goldbach conjecture), written in a June 7, 1742 letter to Euler, states "at least it seems that every number ...

The Barnes G-function is an analytic continuation of the G-function defined in the construction of the Glaisher-Kinkelin constant G(n)=([Gamma(n)]^(n-1))/(H(n-1)) (1) for ...

The Bessel functions of the first kind J_n(x) are defined as the solutions to the Bessel differential equation x^2(d^2y)/(dx^2)+x(dy)/(dx)+(x^2-n^2)y=0 (1) which are ...

A polynomial given by Phi_n(x)=product_(k=1)^n^'(x-zeta_k), (1) where zeta_k are the roots of unity in C given by zeta_k=e^(2piik/n) (2) and k runs over integers relatively ...

A Fourier series is an expansion of a periodic function f(x) in terms of an infinite sum of sines and cosines. Fourier series make use of the orthogonality relationships of ...

Generally speaking, a Green's function is an integral kernel that can be used to solve differential equations from a large number of families including simpler examples such ...

A two-dimensional grid graph, also known as a rectangular grid graph or two-dimensional lattice graph (e.g., Acharya and Gill 1981), is an m×n lattice graph that is the graph ...

The Hermite polynomials H_n(x) are set of orthogonal polynomials over the domain (-infty,infty) with weighting function e^(-x^2), illustrated above for n=1, 2, 3, and 4. ...