We suggest 3 families of conformal deformations and changes of variables for evaluation of integrals arising in applications of the Fourier analysis to fractional partial differential equations and evaluation of special functions, probability distribution functions, cumulative probability distribution functions and quantiles of stable distributions. For the error tolerance E-15, hypergeometric functions can be calculated much faster (in Matlab implementation) than using SFT in Matlab, Python and Mathematica; even when the index \(\alpha \) of the stable distribution is small or close to 1, the same error tolerance can be satisfied in 0.005–0.1 msec. For the calculation of quantiles in wide regions in the tails using the Newton or bisection method, it suffices to precompute several hundred values of the characteristic exponent at points of an appropriate grid (conformal principal components) and use these values in formulas for cpdf and pdf. The same three families can be used to evaluate more general distributions and solutions of boundary problems for fractional partial differential equations more general than the ones related to stable distributions. The methods of the paper are applicable to other classes of integrals, highly oscillatory ones especially.

中文翻译：

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保形加速方法和稳定分布的有效评估

我们建议使用3个共形形变和变量变化类来评估在将傅里叶分析应用于分数阶偏微分方程以及评估特殊函数，概率分布函数，累积概率分布函数和稳定分位数时产生的积分。对于E-15的容错性，与在Matlab，Python和Mathematica中使用SFT相比，超几何函数的计算速度（在Matlab实现中）要快得多；即使索引\（\ alpha \）稳定分布的误差很小或接近于1，相同的误差容限可以在0.005–0.1毫秒内得到满足。为了使用牛顿法或二分法在尾部的较宽区域中计算分位数，只需预先计算适当网格（共形主成分）上的特征指数的数百个值，然后将这些值用于cpdf和pdf的公式即可。相同的三个族可以用来评估分数阶偏微分方程的广义分布和边界问题的解，而不是与稳定分布有关的解。本文的方法适用于其他类别的积分，尤其是高度振荡的积分。