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## Properties of the Dirac delta function

1. Sifting property. Given function continuous at ,

When integrated, the product of any (well-behaved) function and the Dirac delta yields the function evaluated where the Dirac delta is singular. The sifting property also applies if the arguments are exchanged: .

2. Integral.

where is the Heaviside unit step function defined as

3. Units. Since the definition of the Dirac delta requires that the product is dimensionless, the units of the Dirac delta are the inverse of those of the argument . That is, has units , and has units .

4. Definition for radial, 2-D, and 3-D geometries. For two- and three- dimensional problems with vector coordinate , the Dirac delta function is defined:

where is differential volume. The units of are given by [], and three important cases are the listed below.

1. 1-D radial cylindrical coordinates: , and units of are [meters].

2. 1-D radial spherical coordinates: and units of are [meters].

3. 2-D Cartesian coordinates: dv = dx dy, and units of are [meters].

Next: Representations of . Up: Dirac delta function Previous: Dirac delta function
2004-01-21