Overview and Fundamental Ideas - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 41

The following derivative property

d {sgn t}/dt = 2 (t) (1.146)

is also applicable, where sgn(t) = +1, t > 0; -1, t < 0. Important integral
properties involving the delta function include

(x-a) (x-b) dx = (a-b) (1.147)

e i

(a-x) d

= 2 (x-a) (1.148)

'(t- ) (t) dt = - (t- ) '(t) dt = - '( ) (1.149)

Also, if x 0 is defined by u(x 0 ) = 0, with u(x) monotonically increasing or
decreasing, it can be shown that

{u(x)} (x) dx = (x 0 ) / |u'(x 0 )| (1.150)

If (t) = 1, t > 0; and 0, t < 0, it is possible to show that

t

( - ) d = (t- ) (1.151)

Finally,

e -st '(t- ) dt = s e -s

, where s > 0 (1.152)

In particular, when = 0 ,

e -st '(t) dt = s, where s > 0 (1.153)

so that we have the inverse

L -1 {s} = '(t) (1.154)

Note that Equation 1.153 alone is not a valid transform, since it does not
vanish as s . The dipole forcing function '(t), though, generally appears on
the right-sides of differential equations; the quotient formed by s and the
Laplace transform of the differential operator, representing the transform of the
quantity of physical interest, does vanish as s increases.