Dirac delta function integral representation

This relation may be obtained by the same derivation as that leading to equation (B.28), using the integral representation (C.7) for the three-dimensional Dirac delta function. 

The partition function, Z(4>y), cannot be calculated exactly. It could be rewritten using the integral representation of the functional Dirac delta function and evaluated within the saddle place approximation. The calculations lead to the following expression [36,126,128] ... 

Dirichlet function, which is an approximation of Delta function, S x). Various approximate representations of Dirac delta function are provided in Van der Pol Bremmer (1959) on pp 61-62. This clearly shows that we recover the applied boundary condition at y = 0. Therefore, the delta function is totally supported by the point at infinity in the wave number space (which is nothing but the circular arc of Fig. 2.20 i.e. the essential singularity of the kernel of the contour integral). 

The fourth and fifth integrals in (16) are Fourier representations of the three-dimensional Dirac delta function, whence... 

Substitution of Eq. (18) in Eq. (22) followed by integration over p using the Fourier representation of the Dirac delta function... 

How can such charge distributions be represented mathematically There is no problem in mathematical representation of the electronic parte, they are simply some functions of the position r in space -pei. t ) and —pel.B( ) for each molecule. The integrals of the corresponding electronic distributions yield, of course, —Na and —Nb (in a.u.), or minus the number of the electrons (minus, because the electrons carry negative charge). How do we write the nuclear charge distribution as a function of r There is no way to do it without the Dirac delta function. With the function our task is simple ... 

Equation (3.4d) has led mathematicians frequently to claim that the representation of the Wiener measure in (3.10) is undefined. Their complaint is reminiscent of the disrepute in which Dirac delta functions were held by mathematicians for a number of years. There are mathematically acceptable formulations, or notational transcriptions, of these functional integrals. These formulations may make for good mathematics, but they are physically unnecessary. When in doubt, we just remember that the functional integrals are defined in terms of the limit of an iterated integral. 

By making use of a well-known integral representation of the Dirac delta-function 6 x) appearing in Eq. (31), this equation can be rewritten in the following compact form... 

In view of the integral representation (1.113) of the Dirac delta function this is equivalent to... 

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