Differentiation formulas for analytic functions

Error Estimates for Gauss Quadrature Formulas for Analytic Functions

1. Introduction. The estimation of quadrature errors for analytic functions has been considered by Davis and Rabinowitz [1]. An estimate for the error of the Gaussian quadrature formula for analytic functions was obtained by Davis [2]. McNamee [3] has also discussed the estimation of error of the Gauss-Legendre quadrature for analytic functions. Convergence of the Gaussian quadratures was discu...

Complements of Subanalytic Sets and Existential Formulas for Analytic Functions

We show that the complement of a subanalytic set defined by real analytic functions from any subalgebra closed under differentiation is a subanalytic set defined by the functions from the same subalgebra. This result has an equivalent formulation in logic: Consider an expression built from functions as above using equalities and inequalities as well as existential and universal quantifiers. Suc...

Several Differentiation Formulas of Special Functions . Part

(Def. 2) cosec = 1 the function sin . For simplicity, we follow the rules: x, a, b, c are real numbers, n is a natural number, Z is an open subset of R, and f , f1, f2 are partial functions from R to R. One can prove the following propositions: (1) If (the function cos)(x) 6= 0, then sec is differentiable in x and (sec)′(x) = (the function sin)(x) (the function cos)(x)2 . (2) If (the function s...

Several Differentiation Formulas of Special Functions

Several Higher Differentiation Formulas of Special Functions

The notation and terminology used in this paper are introduced in the following articles: [16], [13], [2], [3], [5], [1], [7], [9], [12], [10], [8], [18], [14], [11], [6], [15], and [17]. For simplicity, we use the following convention: x, r, a, x0, p are real numbers, n, i, m are elements of N, Z is an open subset of R, and f , f1, f2 are partial functions from R to R. Next we state a number o...