## Delta Function

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

 (1)

(the so-called Dirichlet Kernel) and
 (2) (3) (4)

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:
 (14)

 (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
 (22)

The Fourier Transform of the delta function is
 (23)

Delta functions can also be defined in 2-D, so that in 2-D Cartesian Coordinates
 (24)

and in 3-D, so that in 3-D 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 so-called Sifting Property

 (29)

Spanier, J. and Oldham, K. B. The Dirac Delta Function .'' Ch. 10 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 79-82, 1987.