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The word quadrature has (at least) three incompatible meanings. Integration by quadrature either means solving an integral analytically (i.e., symbolically in terms of known ...

The prime number theorem gives an asymptotic form for the prime counting function pi(n), which counts the number of primes less than some integer n. Legendre (1808) suggested ...

The Gegenbauer polynomials C_n^((lambda))(x) are solutions to the Gegenbauer differential equation for integer n. They are generalizations of the associated Legendre ...

Seeks to obtain the best numerical estimate of an integral by picking optimal abscissas x_i at which to evaluate the function f(x). The fundamental theorem of Gaussian ...

A function I_n(x) which is one of the solutions to the modified Bessel differential equation and is closely related to the Bessel function of the first kind J_n(x). The above ...

Solutions to the associated Laguerre differential equation with nu!=0 and k an integer are called associated Laguerre polynomials L_n^k(x) (Arfken 1985, p. 726) or, in older ...

The Laguerre polynomials are solutions L_n(x) to the Laguerre differential equation with nu=0. They are illustrated above for x in [0,1] and n=1, 2, ..., 5, and implemented ...

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. ...

A Lucas polynomial sequence is a pair of generalized polynomials which generalize the Lucas sequence to polynomials is given by W_n^k(x) = ...

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 ...