## 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.