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In numerical analysis, Gauss–Jacobi quadrature is a method of numerical quadrature based on Gaussian quadrature. Gauss–Jacobi quadrature can be used to approximate integrals of the form
where ƒ is a smooth function on and α, β > −1. The interval can be replaced by any other interval by a linear transformation. Thus, Gauss–Jacobi quadrature can be used to approximate integrals with singularities at the end points. Gauss–Legendre quadrature is a special case of Gauss–Jacobi quadrature with α = β = 0. Similarly, the Chebyshev–Gauss quadrature of the first kind arises when one takes α = β = −0.5. More generally, the special case α = β turns Jacobi polynomials into Gegenbauer polynomials, in which case the technique is sometimes called Gauss–Gegenbauer quadrature.
Gauss–Jacobi quadrature uses ω = as the weight function. The corresponding sequence of orthogonal polynomials consist of Jacobi polynomials. Thus, the Gauss–Jacobi quadrature rule on n points has the form
where x1, …, xn are the roots of the Jacobi polynomial of degree n. The weights λ1, …, λn are given by the formula