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Mathematical Techniques, Identities, and Tables
Appendix A
Properties of Dirac Delta Functions
1. δ (x) can be expressed in terms of the limit of some ordinary functions such that
(in the limit of some parameter) the ordinary function satisfies the definition for
δ (x). For example,
d(x) = lim
s:0
a
1 > A2s 2B
12ps e-x2 b or d(x) =
For these two examples, δ(-x) = δ(x), so for these cases δ(x) is said to be an even-sided
delta function. The even-sided delta function is used throughout this text, except when
specifying the PDF of a discrete random variable.
d(x) = e lim
a: q (aeax ), x … 0
0, x 7 0
lim c a ax
asin
a: q p ax bd
This is an example of a single-sided delta function; in particular, this is a left-sided delta
function. This type of delta function is used to specify the PDF of a discrete point of a
random variable. (See Appendix B.)
2. Sifting property:
L
q
-q
w(x) d(x - x 0 ) dx = w(x 0 )
3. For even-sided delta functions
where b 7 a.
4. For left-sided delta functions,
0, x 0 6 a
b
1
2 w(a), x 0 = a
w(x) d(x - x 0 ) dx = e w(x 0 ), a 6 x 0 6 b
La
1
2 w(b), x 0 = b
0, x 0 7 b
where b 7 a.
b
0, x 0 … a
w(x) d(x - x 0 ) dx = c w(x 0 ), a 6 x 0 … b
La
0, x 0 7 b
5.
q
w(x) d (n) (x - x 0 ) dx = (-1) n w (n) (x 0 )
L-q
where the superscript (n) denotes the nth derivative with respect to x.