狄拉克δ-函数及有关应用  

Diracδ-function and its related applications

作　　者：郑神州[1] 康秀英[2] ZHENG Shen-zhou;KANG Xiu-ying(College of Science,Beijing Jiaotong University,Beijing 100044,China;Department of Physics,Beijing Normal University,Beijing 100875,China)

机构地区：[1]北京交通大学理学院,北京100044 [2]北京师范大学物理系,北京100875

出　　处：《大学物理》2021年第7期25-29,77,共6页College Physics

基　　金：国家自然科学基金(12071021);北京交通大学研究生课程建设项目(134869522)资助。

摘　　要：狄拉克δ-函数实际上是离散情况下的Kroneckerδ-函数的连续化,它在数学和物理中都有重要的应用.基于广义函数概念引入狄拉克δ-函数的精确定义,证实狄拉克δ-函数不是通常Lebesgue局部可积意义下的普通函数;文中分别以单位矩形脉冲函数、高斯函数、钟形函数和Sinc函数的序列在弱极限意义下来逼近狄拉克δ-函数.另外,验证了狄拉克δ-函数可以作为Heaviside函数的广义导数,以及其高价广义导数,并给出狄拉克δ-函数的卷积性质、伸缩性质、复合变换性质、正交性和狄拉克梳函数,最后引入了狄拉克δ-函数与广义傅里叶变换的关系,以及其在泊松方程Dirichlet边值问题求解中的应用.It is indicated that Diracδ-function is a continuation of the discrete Kroneckerδ-function,which plays an important role in both mathematics and physics.In this paper,the precise definition of Diracδ-function is introduced based on the concept of generalized functions,and it is proved that the Diracδ-function is not a usual function in the Lebesgue sense of local integrable one.To this end,the Diracδ-function is here approximated in the sense of weak limit by making use of the sequences of the unit rectangle impulse functions,Gauss functions,Bell shaped functions and Sinc-functions,respectively.In addition,it is checked that the Diracδ-function is obtained as a generalized derivative of the Heaviside function,and its higher derivative is also shown.Moreover,the convolutions,scales,compound transformations,orthogonality and Comb Dirac functions are recalled,respectively.Finally,the relationship between Diracδ-function and generalized Fourier transform is introduced,and we present an application to solve the Dirichlet boundary value problem of the Poisson equation.

关 键 词：狄拉克 δ-函数 广义函数 弱极限 广义傅里叶变换格林函数 

分 类 号：O4-1[理学—物理]