狄拉克δ -函数及有关应用
收稿日期: 2020-10-10
修回日期: 2020-11-05
网络出版日期: 2021-07-09
基金资助
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
Dirac δ-function and its related applications
Received date: 2020-10-10
Revised date: 2020-11-05
Online published: 2021-07-09
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.
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
郑神州, 康秀英 . 狄拉克δ -函数及有关应用[J]. 大学物理, 2021 , 40(7) : 25 . DOI: 10.16854 /j.cnki.1000-0712.200456
δ-function，which plays an important role in both mathematics and physics. In this paper，the precise
definition of Dirac δ-function is introduced based on the concept of generalized functions，and it
is proved that the Dirac δ-function is not a usual function in the Lebesgue sense of local
integrable one. To this end，the Dirac δ-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 Dirac
δ-function is obtained as a generalized derivative of the Heaviside function，and its higher derivative is also shown.
Moreover，the convo- lutions，scales，compound transformations，orthogonality and Comb Dirac functions are recalled，respectively. Fi- nally，the relationship between Dirac δ-function and generalized Fourier transform is introduced，and we present an application to solve the Dirichlet boundary value problem of the Poisson equation.
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