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首页> 外文期刊>Advances in Pure Mathematics >The Theory of Higher-Order Types of Asymptotic Variation for Differentiable Functions. Part II: Algebraic Operations and Types of Exponential Variation
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The Theory of Higher-Order Types of Asymptotic Variation for Differentiable Functions. Part II: Algebraic Operations and Types of Exponential Variation

机译：可微函数的渐近变化的高阶类型理论。第二部分：代数运算和指数变化的类型

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In this second part, we thoroughly examine the types of higher-order asymptotic variation of a function obtained by all possible basic algebraic operations on higher-order varying functions. The pertinent proofs are somewhat demanding except when all the involved functions are regularly varying. Next, we give an exposition of three types of exponential variation with an exhaustive list of various asymptotic functional equations satisfied by these functions and detailed results concerning operations on them. Simple applications to integrals of a product and asymptotic behavior of sums are given. The paper concludes with applications of higher-order regular, rapid or exponential variation to asymptotic expansions for an expression of type f(x+r(x)).
机译：在第二部分中，我们彻底检查了函数的高阶渐近变化的类型，该函数是通过对高阶变化函数进行所有可能的基本代数运算而获得的。有关的证明有些苛刻，除非所有涉及的功能都定期变化。接下来，我们将对三种类型的指数变化进行说明，并详尽列出这些函数满足的各种渐近函数方程，并详细说明关于它们的运算。给出了乘积积分和和的渐近行为的简单应用。本文以f（x + r（x））类型的渐近展开式的高阶正则，快速或指数变化的应用为结论。

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