By V. I. Krylov

The three-part remedy starts with thoughts and theorems encountered within the conception of quadrature. the second one half is dedicated to the matter of calculation of convinced integrals. This part considers 3 uncomplicated issues: the speculation of the development of mechanical quadrature formulation for sufficiently delicate integrand services, the matter of accelerating the precision of quadratures, and the convergence of the quadrature strategy. the ultimate half explores tools for the calculation of indefinite integrals, and the textual content concludes with invaluable appendixes.

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**Sample text**

X) is a polynomial of degree n of the form H. (x) = 2nxn +. The polynomials H. (x) satisfy the following relationship: for m#n 0 J00 e-x2Hn (x) Hm (x) dx = 2n\/-,, n! for m = n. ) form an orthogonal system on e_ , 2. 1) by Hn (x) = 2 2 H,, l `1. = 00 0Hm(n)(x)dx. f-W -oo Form < n, Hr(n) (x) = 0 and thus I = 0. If m = n then 00 I= 2nn! 0o 95 dx = 21-n! oo An m e-" dx = 24n1 c. system is formed by the polynomials H. 4) 2or 4. 5. CHEBYSHEV-LAGUERRE POLYNOMIALS The Chebyshev-Laguerre polynomials are orthogonal on the half-line 0 S x < o with respect to the weight function p (x) = xae-".

X) its expression in terms of 40n: (-1)n J 00 0-(a)Lm(a)(x)dx 0 oo (-1)n0n(n-1)Lm(a)(x) + I (-1)n-1 = (-1)n-1 J 0n(n-1)[Lm(a)(x)]' dx = 0 0 oM S6n(n-1)[Lm(a)(x)], dx. 0 The term which does not involve the integral vanishes because a> -1. Carrying out the integration by parts n times we obtain 1= J0 0. [Lm(a)(x)](n)dx. Form < n, we have [L m(a) (x)] (n) = 0 and therefore 1= 0. When m = n, 1 = a! f cdx = n! f xa+nedx = n! I' (a + n + 1). 0 The orthonormal Chebyshev-Laguerre polynomials are Ln(a)(x) ln(a)(x)= 1.

0 The orthonormal Chebyshev-Laguerre polynomials are Ln(a)(x) ln(a)(x)= 1. ]P (a + n + 1)] 2. 5) REFERENCES V. L. Goncharov, Theory of Interpolation and Approximation of Functions, Gostekhizdat, Moscow, 1954, Chap. 3, 4 (Russian). D. Jackson, Fourier Series and Orthogonal Polynomials, Cams Monograph No. 6, Math. Assoc. , 1941, Chap. 2, 7-10. 36 Preliminary Information A. N. Korkin and E. I. " Collected works of E. I. Zolotarev, Vol. 1, pp. 138-153. I. P. Natanson, Constructive Theory of Functions, Gostekhizdat, Moscow, 1949 (Russian).