Analysis of numerical integration error for Bessel integral identity in fast multipole method for 2D Helmholtz equation

In 2D fast multipole method for scattering problems, square quadrature rule is used to discretize the Bessel integral identity for diagonal expansion of 2D Helmholtz kernel, and numerical integration error is introduced. Taking advantage of the relationship between Euler-Maclaurin formula and trapezoidal quadrature rule, and the relationship between trapezoidal and square quadrature rule, sharp computable bound with analytical form on the error of numerical integration of Bessel integral identity by square quadrature rule is derived in this paper. Numerical experiments are presented at the end to demonstrate the accuracy of the sharp computable bound on the numerical integration error.