1549380323-Statistical Mechanics Theory and Molecular Simulation

Diracδ-function 655

Consider, next, aδ-function of the formδ(ax). This can be simplified according to

δ(ax) =

1

|a|

δ(x), (A.14)

which also implies thatδ(−x) =δ(x). Similarly, aδ-function of the formδ(x^2 −a^2 )
can be rewritten equivalently as

δ

(

x^2 −a^2

)

=

1

2 |a|

[δ(x−a) +δ(x+a)]. (A.15)

More generally, given aδ-functionδ(g(x)) whereg(x) is a function withnzeroes at
points, ̄x 1 ,...,x ̄n, such thatg′( ̄xi) 6 = 0, theδ-function can be simplified according to

δ(g(x)) =

∑n

i=1

δ(x−x ̄i)

|g′( ̄xi)|

. (A.16)

It is easy to see that eqns. (A.14) and (A.15) are special cases of eqn. (A.16). Eqn.
(A.16) can also be proved usingδ-sequences by performing a change of variablesy=
g(x) and employing similar arguments to those used in proving eqn. (A.3).