Numerical quadrature formulas through the theory of analytic functions

In this article, we discuss some examples on the use of the theory of analytic functions to construct accurate and efficient quadrature formulas for numerical evaluation of integrals, and also examine the errors incurred associated with some quadrature formulas. Though some integrals treated in this article are real, their behaviours are best understood only if we consider the variables as complex. This is analogous to the example that we can understand why the simple real function 1/(1 + x²) converges for x < 1 only if we move into the complex plane. In the first section, we examine the deterioration of accuracy when the integrand has poles near the interval of integration and we derive appropriate modified quadrature formulas to improve accuracy. The integrals with rational and Poisson-type kernels are discussed as examples. In the second section, we show the derivation of some algorithms which perform the numerical inversion of Laplace transform. Three inversion algorithms are presented, namely, Fourier series expansion method, Laguerre polynomial expansion method and kernel approximation method. Inherent difficulties in numerical Laplace inversion are also discussed.