Gaussian quadrature

where f,w are continuous functions and the weight function w(x) is positive. Then

defines an inner product on the space of continuous functions, and an orthonormal sequence pn(x) of polynomials may be constructed such that pn(x) has degree n. Gauss then showed that pn + 1(x) has n + 1 distinct real roots x0,…,xn in (a,b) and that there are unique solutions W0,…,Wn to the equations

It is then the case that

exactly equals I(f) for polynomials of degree 2n + 1 or less, and more generally for continuous functions Qn(f) tends to I(f) as n→∞.

• Cauchy's integral formula
• Cauchy's Integral Theorem
• Cauchy's mean value theorem
• Cauchy's Residue Theorem
• Cauchy's test
• Cauchy's Theorem
• Cauchy's Theorem
• Cauchy–Riemann equations
• Cauchy–Schwarz inequality
• Cauchy–Schwarz inequality for integrals
• Cauchy–Schwarz inequality for sums
• causation
• Cavalieri, Bonaventura
• Cayley, Arthur
• Cayley Representation Theorem
• Cayley's Theorem
• Cayley table
• Cayley–Hamilton Theorem
• cdf
• ceiling
• Celsius
• centesimal
• centi-
• centile
• central angle