Analytical and Numerical Methods For Slowly Convergent Integrals and Applications

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Slowly convergent integrals play a major role in science and engineering. They arise when solving the simplest to the most complicated problems through iterative, discretization or other expansion methods. Slowly convergent integrals must be evaluated in Fourier series to obtain a function approximation scheme that converges faster than the standard Taylor series. Many special functions, which play an extremely important role in applied mathematics and physics, have slowly convergent integral representations. Traditional quadrature rules have failed to provide accurate approximations to slowly convergent integrals, therefore new techniques are highly sought after and desired. The most challenging application concerns the computation of molecular multi-centre integrals needed for the computation of energy components, such as nuclear attraction energy. The computation of these integrals takes as much as 40% of the wall time of molecular structure calculations. Accordingly, any reduction in the calculation time will significantly impact the performance of any software used for molecular structure calculations.***In this research program, we will develop new methods specifically for slowly convergent integrals that will overcome the difficulties that existing methods face. We will introduce analytic developments of a class of spherical Bessel integrals required millions of times for the computation of molecular multi-centre integrals. Asymptotic series representations for integrals will also be used. For the summation of the asymptotic series, we will use sequence transformations and convergence acceleration methods. In the case of integrals for which the analytic development will be fruitless, such as those involved in the four-centre molecular integrals, we will focus our efforts on specifically tailoring a quadrature rule to these integrals. The optimality of the double exponential transformation for the trapezoidal rule clearly indicates the tremendous potential in applying it to this kind of integrals. The trapezoidal rule, recognized as a by-product of Sinc numerical methods, has been shown to have extremely promising properties as a general-purpose integrator. The meta-optimality and exponential convergence rate of the trapezoidal rule is derived for integrands that observe a double exponential decay rate at the endpoints. ***The developed methods will be applied to molecular multi-centre integrals and to other challenging integrals including the incomplete Bessel functions, tail probability distributions, and the generalized hyperbolic distribution. Through these applications, we will assess the proposed methods and will perform comparisons with regard to accuracy and efficiency with the existing ones. The results of this research program will be useful in applied mathematics; molecular physics and chemistry, where oscillatory integrals are prevalent.**