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.