The Theory of Higher-Order Types of Asymptotic Variation for Differentiable Functions. Part II: Algebraic Operations and Types of Exponential Variation  

作　　者：Antonio Granata Antonio Granata(Department of Mathematics and Computer Science, University of Calabria, Rende (Cosenza), Italy)

机构地区：[1]Department of Mathematics and Computer Science, University of Calabria, Rende (Cosenza), Italy

出　　处：《Advances in Pure Mathematics》2016年第12期817-867,共52页理论数学进展（英文）

摘　　要：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)).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)).

关 键 词：Higher-Order Regularly-Varying Functions Higher-Order Rapidly-Varying Functions Smoothly-Varying Functions Exponentially-Varying Functions Asymptotic Functional Equations Asymptotic Expansions 

分 类 号：O17[理学—数学]