Analytical and Numerical Methods For Slowly Convergent Integrals and Applications

缓慢收敛的积分在科学和工程中扮演着重要的角色。当通过迭代、离散化或其他展开方法解决最简单到最复杂的问题时，它们就会出现。要获得比标准泰勒级数更快收敛的函数逼近方案，必须用傅里叶级数来计算缓慢收敛的积分。许多特殊函数在应用数学和物理中起着极其重要的作用，它们具有缓慢收敛的积分表示。传统的求积规则不能提供对缓慢收敛的积分的精确逼近，因此新的技术备受追捧和期望。最具挑战性的应用涉及计算能量分量(如核吸引能)所需的分子多中心积分的计算。这些积分的计算占用了分子结构计算的40%左右的时间。因此，计算时间的任何减少都将显著影响用于分子结构计算的任何软件的性能。

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.