Abstract: It is indicated that Dirac δ-function is a continuation of the discrete Kronecker

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.

Key words: Dirac δ-function, generalized function, weakly limits, generalized Fourier transform, Green func-tion