Numerical differentiation of analytic functions using GAG quadrature rules on the semicircle

Abstract

As stated in [1], until 1991, the numerical differentiation of analytical functions was discussed in several papers. The corresponding formulas for differentiation were mostly obtained by Cauchy’s integral formula and the application of the trapezoidal rule. In paper [1], new formulas for numerical differentiation were introduced using Gaussian quadrature rules on the semicircle. In [2] generalized averaged Gaussian (GAG) quadrature rule on the semicircle are introduced with aim of better estimate of the error of Gaussian quadrature rule on the semicircle. In this paper we give formula for numerical differentiation of analytic function using GAG quadrature rules on the semicircle. Numerical examples with comparison of our results with results from papers published after 1991 are included. References [1] F. Calio’, M. Frontini, G. V. Milovanovi´c. Numerical differentiation of analytic functions using quadratures on the semicircle. Comput. Math. Appl., 1991, 22(10), 99{106. [2] M. P. Stani´c, T. V. Tomovi´c Mladenovi´c, A. S. Milosavljevi´c. Generalized averaged Gaussian quadrature rules on the semicircle. Numer. Algor., 2025, https://doi.org/10.1007/s11075-025-02231-5