“Delta Function”的概念、定义、习题解题方法及翻译-数学辞典

单词

Delta Function

释义

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)

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)


			(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)

See also Delta Sequence, Doublet Function, Fourier Transform--Delta Function

References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 481-485, 1985.

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

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