Asymptotic expansions are obtained for contour integrals of the form $$\begin{aligned} \int _a^b \exp \left( - zp(t) + z^{\nu /\mu } r(t) \right) q(t)\mathrm{d}t, \end{aligned}$$ ∫ a b exp - z p ( t ) + z ν / μ r ( t ) q ( t ) d t , in which z is a large real or complex parameter; p ( t ), q ( t ), and r ( t ) are analytic functions of t ; and the positive constants $$\mu $$ μ and $$\nu $$ ν are related to the local behavior of the functions p ( t ) and r ( t ) near the endpoint a . Our main theorem includes as special cases several important asymptotic methods for integrals such as those of Laplace, Watson, Erdélyi, and Olver. Asymptotic expansions similar to ours were derived earlier by Dingle using formal, nonrigorous methods. The results of the paper also serve to place Dingle’s investigations on a rigorous mathematical foundation. The new results have potential applications in the asymptotic theory of special functions in transition regions, and we illustrate this by two examples.

中文翻译：

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拉普拉斯方法的扩展

对形如 $$\begin{aligned} \int _a^b \exp \left( - zp(t) + z^{\nu /\mu } r(t) \right) q 的轮廓积分获得渐近展开式(t)\mathrm{d}t, \end{aligned}$$ ∫ ab exp - zp ( t ) + z ν / μ r ( t ) q ( t ) dt ，其中 z 是一个大的实数或复数参数; p ( t )、q ( t ) 和 r ( t ) 是 t 的解析函数；正常数 $$\mu $$ μ 和 $$\nu $$ ν 与函数 p ( t ) 和 r ( t ) 在端点 a 附近的局部行为有关。我们的主要定理包括作为特殊情况的几个重要的积分渐近方法，例如 Laplace、Watson、Erdélyi 和 Olver 的方法。与我们类似的渐近展开式是由 Dingle 早先使用正式的、非严格的方法推导出来的。论文的结果也有助于将丁格尔的研究建立在严格的数学基础上。