Defined as the limit of a class of Delta Sequences. Sometimes called the Impulse Symbol.
The most commonly used (equivalent) definitions are

(1) 
(the socalled Dirichlet Kernel) and
where is the Fourier Transform. Some identities include

(5) 
for ,

(6) 
where is any Positive number, and

(7) 

(8) 

(9) 
where denotes Convolution,

(10) 

(11) 

(12) 

(13) 
(13) can be established using Integration by Parts as follows:
Additional identities are

(15) 

(16) 

(17) 
where the s are the Roots of . For example, examine

(18) 
Then , so
and
, and we have

(19) 
A Fourier Series expansion of gives

(20) 

(21) 
so
The Fourier Transform of the delta function is

(23) 
Delta functions can also be defined in 2D, so that in 2D Cartesian Coordinates

(24) 
and in 3D, so that in 3D Cartesian Coordinates

(25) 
in Cylindrical Coordinates

(26) 
and in Spherical Coordinates,

(27) 
A series expansion in Cylindrical Coordinates gives





(28) 
The delta function also obeys the socalled Sifting Property

(29) 
See also Delta Sequence, Doublet Function, Fourier TransformDelta Function
References
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 481485, 1985.
Spanier, J. and Oldham, K. B. ``The Dirac Delta Function .''
Ch. 10 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 7982, 1987.
© 19969 Eric W. Weisstein
19990524