In López, Pagola and Perez (2009) we introduced a modification of the Laplace's method for deriving asymptotic expansions of Laplace integrals which simplifies the computations, giving explicit formulas for the coefficients of the expansion. On the other hand, motivated by the approximation of special functions with two asymptotic parameters, Nemes has generalized Laplace's method by considering Laplace integrals with two asymptotic parameters of a different asymptotic order. Nemes considers a linear dependence of the phase function on the two asymptotic parameters. In this paper, we investigate if the simplifying ideas introduced in López, Pagola and Perez (2009) for Laplace integrals with one large parameter may be also applied to the more general Laplace integrals considered in Nemes's theory. We show in this paper that the answer is yes, but moreover, we show that those simplifying ideas can be applied to more general Laplace integrals where the phase function depends on the large variable in a more general way, not necessarily in a linear form. We derive new asymptotic expansions for this more general kind of integrals with simple and explicit formulas for the coefficients of the expansion. Our theory can be applied to special functions with two or more large parameters of a different asymptotic order. We give some examples of special functions that illustrate the theory.

中文翻译：

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拉普拉斯积分法的推广


在 López、Pagola 和 Perez (2009) 中，我们引入了对拉普拉斯方法的修改，用于导出拉普拉斯积分的渐近展开式，这简化了计算，给出了展开系数的明确公式。另一方面，受具有两个渐近参数的特殊函数逼近的启发，Nemes 通过考虑具有不同渐近阶数的两个渐近参数的拉普拉斯积分，推广了拉普拉斯方法。 Nemes 考虑相位函数对两个渐近参数的线性相关性。在本文中，我们研究了 López、Pagola 和 Perez (2009) 中针对具有一个大参数的拉普拉斯积分引入的简化思想是否也可以应用于 Nemes 理论中考虑的更一般的拉普拉斯积分。我们在本文中表明答案是肯定的，但此外，我们表明这些简化思想可以应用于更一般的拉普拉斯积分，其中相位函数以更一般的方式依赖于大变量，不一定以线性形式。我们用简单而明确的展开式系数公式导出了这种更一般类型的积分的新渐近展开式。我们的理论可以应用于具有不同渐近阶数的两个或多个大参数的特殊函数。我们给出了一些特殊函数的例子来说明该理论。