一道积分题的多种证法

本文利用分析中函数积分的定义和一些基本性质,给出了一道积分题的几何解释和五种证法.其中几种证法是一般教科书上很难见到而在教学中值得引用的一些方法.     

关于积分等式命题的证明

本文研究积分等式问题，给出积分等式命题的若干常用证法。     

Bochner－Martinelli积分的Plemelj公式

关于Ｂｏｃｈｎｅｒ－Ｍａｒｔｉｎｅｌｌｉ积分边界性质的研究，前人已有许多结果，在边界是Ｃ１光滑核密度函数是Ｈｏｌｄｅｒ连续的条件下本文给出了Ｐｌｅｍｅｌｊ公式的一个简单证法，并且极限是任意的而不仅是法向极限或沿角区域的极限．     

函数不等式的积分证法

通过实例介绍了一类函数不等式的积分证法.     

分部积分与Taylor公式

利用定积分的分部积分法由简到繁的推导得到了微分学中的Taylor公式从而给出了Taylor公式的另一种证法,并利用这种方法还可得到复变函数或泛函分析中的Taylor公式及某些函数的渐进级数和更广泛的函数展开.     

一道概率中常用积分的巧证

∫-∞^＋∞e-（x2）/（2）dx=√2π（1）式（1）是概率论中常用的积分,常见的证法是利用了极坐标变换,或利用Γ函数的性质.笔者给出一种利用旋转体体积公式的新证法.     

三分拆的应用——整边三角形

本利用生成函数给出了整边三角形的个数T(n)公式的一个直接证法，然后给出了T（n）的性质，瘼就两种特殊情形给出了T(n)的记数公式。     

一个高等数学命题的证法探析

从探究解题方法方面讨论如何学习高等数学,以求培养学生分析问题的能力.对一条关于周期函数积分的命题给出新的证法,揭示出插入点的含义和周期函数积分的本质,弥补通常证法只侧重凑积分限技巧的不足.     

利用微分方程证明反正弦加法定理

利用微分方程和函数的连续性给出了反正弦加法定理一种新证法,该证法可清晰地显示如何分区域讨论问题.     

巧证一组组合恒等式

一些组合恒等式结构复杂，运算繁琐，方法巧妙，做起来有点爱她不容易的感觉，为此本文介绍一种独特的证明方法——微积分证法．此法是根据等式特点构造二项式函数，再对函数求导或积分，便可得要证的等式．     

On the regularization of a class of integral equations of the first kind whose kernels are discontinuous on the diagonals

For a class of integral equations of the first kind whose kernels are discontinuous on the diagonals, the convergence of the Lavrent??ev regularization method is proved by using methods of the spectral theory of integral operators. These methods lead to a special Dirac system, and finding the asymptotics of fundamental solutions is an important part of the proof.     

The non-integrability of a rotating elliptical billiard

The problem of a point moving inside a rotating ellipse is considered; collisions at the boundary are assumed to be absolutely elastic. It is shown that this discrete dynamical system does not admit of an analytic integral independent of the energy integral. The proof of non-integrability is based on the method of separatrix splitting.     

On existence and asymptotic stability of solutions of a nonlinear integral equation

We prove an existence theorem for a nonlinear integral equation being a Volterra counterpart of an integral equation arising in the traffic theory. The method used in the proof allows us to obtain additional characterization in terms of asymptotic stability of solutions of an equation in question.     

二重积分变量代换公式证明的探讨

在一般教材中二重积分变量代换公式的证明通常采用几何的方法,也有部分数学分析教材给与了严格的分析证明,但证明不便直观的几何说明.本文对二重积分变量代换公式进行了一些探讨,给出了一个较简洁直观的分析证明方法.     

On the existence of a solution to a singular integral equation in electromagnetic reflection

For a singular integral equation arising in a modified approach to boundary integral equations for exterior boundary-value problems from the theory of electromagnetic reflection an existence proof is given.     

Problem with displacement conditions on internal characteristics and the Frankl condition on a segment of the degeneration line for the Gellerstedt equation with a singular coefficient

We study the well-posedness of a problem with displacement conditions on internal characteristics and an analog of the Frankl condition on a segment of the degeneration line for the Gellerstedt equation with a singular coefficient. The uniqueness of a solution is proved with the use of an extremum principle. The proof of the existence uses the method of integral equations.     

Suboptimal Integral Sliding Mode Controller Design for a Class of Affine Systems

This paper proposes an nth-order suboptimal integral sliding mode controller for a class of nonlinear affine systems. First, a general form of integral sliding mode is given. An extended Theta-D method is developed for the optimal control problems characterized by a quadratic cost function with a cross term. Then the extended Theta-D method is employed to determine a suboptimal integral sliding mode. Rigorous proof shows that the controller guarantees semi-global asymptotical stability of affine systems. To verify the accuracy of the extended Theta-D method, a numerical example is provided. To verify the effectiveness of the proposed suboptimal integral sliding mode controller, a numerical example and an application example of an overhead crane system are provided.     

An initial value method for the computation of the characteristic values and functions of an integral operator-II

A convergence proof is given for an initial value method for finding the spectral properties of an integral operator. The algorithm, which is based on a Cauchy system for the Fredholm determinants, is related to the Nyström method and results of Anselone and Atkinson become applicable. The proof is also shown to work for a modification of the procedure due to Kalaba and Scott.     

An elementary sufficiency proof of an absolute minimum for a nonparametric variational problem

For a fixed endpoint, nonparametric simple integral variational problem, there is presented an expansion method proof of a sufficiency theorem for an absolute minimum. In particular, this sufficiency theorem yields readily the proof of a result of the type recently presented by Nehari (Ref. 1), but with an error in formulation and an incorrect proof. The present discussion is in a setting which permits considered arcs to be on the boundary of the set of admissible arcs; thus it contains as particular instances certain types of unilateral variational problems of a control nature.This research was supported by the National Science Foundation under Grant No. GP-36120.     

New proof of the Calabi-Bernstein theorem

By means of a local integral formula on a maximal surface, which involves the hyperbolic angle between the unit normal vector field and a fixed timelike direction, we give a new proof of the parametric Calabi-Bernstein theorem. A suitable modification of the integral formula gives an independent proof to the non-parametric case of this result.