A Mixed Quadrature Rule Using Birkhoff-Young Rule Through Richardson Extrapolation for Numerical Integration of Analytic Functions

This study introduces a novel high-precision quadrature rule, achieved by using two lower-precision quadrature rules. The focus is on facilitating the approximate evaluation of integrals over line segments in the complex plane, particularly for analytic functions. The versatility of the newly developed quadrature rule is demonstrated through its application to various mathematical scenarios. To assess the efficacy of the proposed quadrature rule, an asymptotic error estimate is provided. Numerical verification is then conducted to validate the accuracy and efficiency of the rule. The results from these numerical experiments highlight the superior precision of our quadrature rule when applied to the numerical integration of functions over complex line segments. This study significantly contributes to the advancement of numerical integration techniques, presenting a promising avenue for achieving heightened accuracy in the evaluation of integrals over complex domains, particularly in the context of analytic functions.