Page 111 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

INTEGRAL TRANSFORMS: THE LAPLACE TRANSFORM 101
and with

G(s ) = L[ f (t)] = sF(s ) − f (0)

we find that

d f
2
2
L = s F(s ) − sf (0) − f (0) (6.25)

dt 2
In general
d f d f
n n−1
n

L = s F(s ) − s n−1 f (0) − s n−2 f (0) − ··· − (0) (6.26)
dt n dt n−1
m
The Laplace transform of t may be found by using the gamma function,
∞

m m −st
L[t ] = t e dt and let x = s t (6.27)
0
∞ ∞
x

m dx 1 (m + 1)
m −x m −x
L[t ] = e = x e dx = (6.28)
s s s m+1 s m+1
x=0 x=0
which is true for all m > −1 even for nonintegers.
6.2.10 Laplace Transforms of Integrals

t
 

L  f (τ)dτ  = L[g(t)] (6.29)
τ=0
where dg/dt = f (t). Thus L[dg/dt] = sL[g(t)]. Hence
t
 
1

L  f (τ)dτ  = F(s ) (6.30)
s
τ=0
6.2.11 Derivatives of Transforms

∞

F(s ) = f (t)e −st dt (6.31)
t=0