无界函数的反常积分的计算

运用牛顿‐莱布尼茨公式和无界函数的反常积分的定义证明无界函数的反常积分的计算定理,运用这个定理计算无界函数的反常积分简单快捷,通过举例说明这个定理的应用,并指出多种大学数学参考书中存在的一个共同错误。     

定积分计算探讨

综合利用傅里叶级数法,参数求和法,参数展开法,得到了一些定积分的值.     

Extrapolation Methods for Oscillatory Infinite Integrals

The non-linear transformations for accelerating the convergenceof slowly convergent infinite integrals due to Levin & Sidi(1975) are modified in two ways. These modifications enableone to evaluate accurately some oscillatory infinite integralswith less work. Special emphasis is placed on the evaluationof Fourier and Hankel transforms and some simple algorithmsfor them are given. Convergence properties of these modificationsare analysed in some detail and powerful convergence theoremsare proved for certain cases including those of the Fourierand Hankel transforms treated here. Several numerical examplesare also supplied.     

Computation of an Infinite Integral Using Romberg'S Method

A numerical computation in crystallography involves the integral g(a)=₀ ⁺[(exp ^x +exp^–x ) ^a –exp ^ax –exp^–ax ]dx, 0<a<2. A first approximation value for g(5/3)=4.45 has been given. This result has been obtained by a classical method of numerical integration. It has been followed in an other paper by a second one 4.6262911 obtained from a theoretical formula which seems to lead to a more reliable result. The difficulty when one wants to use a numerical method is the choice of parameters on which the method depends, in this case, the size of the integration interval for instance and the number of steps in Romberg's method. We present a new approach of numerical integration which dynamically allows to take into account both the round-off error and the truncation error and leads to reliable results for every value of a.     

n重积分换元公式的证明

在建立微分线性空间基础上、引入向量外积和有向体积,找到了微分线性变换与有向体积之间的关系,由此给出了n重积分换元公式的一个简单证法.     

一类重积分的新算法

本文利用重积分的中值定理和拟牛顿-莱布尼兹公式,给出了求一类二重积分的新方法,尔后推广到三重积分,最后利用研究的结果给出了三个不等式的猜想与证明.     

关于二阶导数的若干新积分不等式

根据Cauchy-Schwarz不等式,得到了C2([a,b])空间中函数的二阶导数的若干新积分不等式.     

Evaluation of Some Integrals Encountered in Calculations in Many-fermion Systems

The integrals and (l(x)=3j₁(x)/x) and j₁(x) the spherical Bessel function of thefirst order) are evaluated analytically     

一类非负函数无穷积分收敛性的刻画

针对无穷积分教学的需要,在利用随机理论的重要概念——失效函数来刻画出非负函数的递减趋于坐标横轴程度的基础上,运用一般推理方法及无穷积分敛散性的比较判别法,得到了一类非负函数的无穷积分敛散性的较为简便而有效的判别方法.     

Laplace积分的一种新求法

Laplace积分在复变函数、数学分析、Fourier分析中有重要的应用,其求解已有复变函数方法和实方法.在实方法中要用到∫∞0sinxxdx=π2,本文给出另外一种实方法,不需要利用这个积分.     

第一类曲面积分的一种解法

运用第一类曲线积分方法解决一类特殊的第一类曲面积分问题,并举例说明此方法的简便性.     

Comparison of Some Trigonometric Integrals

Mathematical Notes - We show that Preiss–Thomson’s AS-integral is inconsistent with Burkill’s SCPintegral, James’ P2-integral, and Denjoy’s totalization T2s over the...     

Explicit Series Formulae for the Evaluation of Integrals by the Method of Steepest Descents

Explicit series formulae are given for use in the method of steepest descents, where it is required to find the contribution to an integral of the form ∫e^vw(z)φ(z)dz from a well-isolated saddle point z₀. Here v is a large positive parameter, and w^(N)(z0) ≠ 0 while lower-order derivatives at z₀ are equal to zero. Examples where the formulae are helpful are presented.     

A Comparison of Strategies for the Automatic Computation of Two-Dimensional Integrals over Infinite Domains

We present sets of parameterized integral test families over each of the domains [0,∞)² and (?∞,∞)², exhibiting various rates of integrand decay, and use these families to compare the performances of the two general-purpose two-dimensional cubature algorithms r2d2lri and Cubpack++ for integration over such nonfinite domains. The data collected for this comparison is helpful in identifying the respective advantages and disadvantages of the strategies adopted by the two algorithms when dealing with nonfinite domains.     

用待定函数法求解一类不定积分

针对形如∫h（x）/f^2（x）dx的不定积分求解,给出一种待定函数法,并举例验证这种方法的实用性．     

Some New Inverse-type Hilbert-Pachpatte Integral Inequalities

In this paper,some new generalizations of inverse type Hilbert-Pachpatte integral inequalities are proved.The results of this paper reduce to those of Pachpatte(1998,J.Math.Anal.Appl.226,166-179)and Zhan and Debnath(2001,J.Math.Anal.Appl.262,411-418).     

Integral Representations for Harmonic Functions of Infinite Order in a Cone

A harmonic function of infinite order defined in an n-dimensional cone and continuous in the closure can be represented in terms of the modified Poisson integral and an infinite sum of harmonic polynomials vanishing on the boundary.     

计算柱面上对面积的曲面积分的一种新方法

给出了利用对弧长的曲线积分计算柱面上对面积的曲面积分的一种新方法,其计算公式为∫∫_Σf(x,y,z)dS=∫_(L*)ds∫z_1(x,y) z_2(x,y)f(x,y,z)dz,其中积分曲面Σ为垂直于xoy坐标面的柱面片,L*为Σ在xoy坐标面上的投影曲线(平面曲线),z=z1(x,y),z=z2(x,y)分别为过Σ的下边界曲线和上边界曲线的任一不同于Σ的曲面的方程.     

A Boundedness Result for Minimizers of Some Polyconvex Integrals

We consider polyconvex functionals of the Calculus of Variations defined on maps from the three-dimensional Euclidean space into itself. Counterexamples show that minimizers need not to be bounded. We find conditions on the structure of the functional, which force minimizers to be locally bounded.