积分、级数与乘积表第七版概览

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"《积分、级数与乘积表_7th》是关于数学中积分、级数和乘积计算的重要参考书籍。本书由I.S.格拉代什坦和I.M.李日茨基撰写,并由阿兰·杰弗里和丹尼尔·兹威林格编辑。书中详细介绍了各种数学概念,包括有限求和、数值级数和无穷乘积,以及函数级数的定义和定理。" 本文主要涉及以下几个数学知识点: 1. **有限求和** (Finite Sums): 这一章节涵盖了等差数列(Progressions)和自然数的幂次之和(Sums of powers of natural numbers)。例如,前n个自然数的和可以用公式1+2+...+n = n*(n+1)/2表示,而平方和有更复杂的公式,如1^2+2^2+...+n^2 = n*(n+1)*(2n+1)/6。 2. **自然数的倒数之和** (Sums of reciprocals of natural numbers): 如调和级数1 + 1/2 + 1/3 + ...,它是一个发散级数,其和趋向于无穷大。 3. **自然数倒数的乘积之和** (Sums of products of reciprocals of natural numbers):讨论了更复杂形式的求和,如两数倒数的和或更多数的倒数的乘积之和。 4. **二项式系数之和** (Sums of the binomial coefficients): 这涉及到组合数学中的二项式定理,如(n choose k)的求和规律。 5. **数值级数与无穷乘积** (Numerical Series and Infinite Products): 讨论了级数的收敛性,包括收敛测试(如比值测试、根值测试)和各种级数的例子。无穷乘积则是函数展开的一种形式,例如欧拉的无穷乘积公式。 6. **函数级数** (Functional Series): 定义了幂级数、泰勒级数和傅立叶级数等,以及相关的定理,如一致收敛、柯西准则等,这些理论在微积分和函数分析中具有重要应用。 该书还包含大量的实例,帮助读者理解和应用这些理论,对于数学研究者和工程技术人员来说,是一本不可或缺的工具书。通过深入学习和掌握这些内容,可以提升在实际问题中处理积分、级数和乘积计算的能力。

For macroscopically anisotropic media in which the variations in the phase stiffness tensor are small, formal solutions to the boundary-value problem have been developed in the form of perturbation series (Dederichs and Zeller, 1973; Gubernatis and Krumhansl, 1975 ; Willis, 1981). Due to the nature of the integral operator, one must contend with conditionally convergent integrals. One approach to this problem is to carry out a “renormalization” procedure which amounts to identifying physically what the conditionally convergent terms ought to contribute and replacing them by convergent terms that make this contribution (McCoy, 1979). For the special case of macroscopically isotropic media, the first few terms of this perturbation expansion have been explicitly given in terms of certain statistical correlation functions for both three-dimensional media (Beran and Molyneux, 1966 ; Milton and Phan-Thien, 1982) and two-dimensional media (Silnutzer, 1972 ; Milton, 1982). A drawback of all of these classical perturbation expansions is that they are only valid for media in which the moduli of the phases are nearly the same, albeit applicable for arbitrary volume fractions. In this paper we develop new, exact perturbation expansions for the effective stiffness tensor of macroscopically anisotropic composite media consisting of two isotropic phases by introducing an integral equation for the so-called “cavity” strain field. The expansions are not formal but rather the nth-order tensor coefficients are given explicitly in terms of integrals over products of certain tensor fields and a determinant involving n-point statistical correlation functions that render the integrals absolutely convergent in the infinite-volume limit. Thus, no renormalization analysis is required because the procedure used to solve the integral equation systematically leads to absolutely convergent integrals. Another useful feature of the expansions is that they converge rapidly for a class of dispersions for all volume fractions, even when the phase moduli differ significantly.

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