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For a single variate X having a distribution P(x) with known population mean mu, the population variance var(X), commonly also written sigma^2, is defined as ...

Jackson's theorem is a statement about the error E_n(f) of the best uniform approximation to a real function f(x) on [-1,1] by real polynomials of degree at most n. Let f(x) ...

For p an odd prime and a positive integer a which is not a multiple of p, a^((p-1)/2)=(a/p) (mod p), where (a|p) is the Legendre symbol.

Let f(x) be integrable in [-1,1], let (1-x^2)f(x) be of bounded variation in [-1,1], let M^' denote the least upper bound of |f(x)(1-x^2)| in [-1,1], and let V^' denote the ...

Abstract Algebra

Gaussian curvature, sometimes also called total curvature (Kreyszig 1991, p. 131), is an intrinsic property of a space independent of the coordinate system used to describe ...

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 ...

A matrix is a concise and useful way of uniquely representing and working with linear transformations. In particular, every linear transformation can be represented by a ...

Apéry's numbers are defined by A_n = sum_(k=0)^(n)(n; k)^2(n+k; k)^2 (1) = sum_(k=0)^(n)([(n+k)!]^2)/((k!)^4[(n-k)!]^2) (2) = _4F_3(-n,-n,n+1,n+1;1,1,1;1), (3) where (n; k) ...

The dilogarithm Li_2(z) is a special case of the polylogarithm Li_n(z) for n=2. Note that the notation Li_2(x) is unfortunately similar to that for the logarithmic integral ...