Page 110 - Essentials of applied mathematics for scientists and engineers
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book Mobk070 March 22, 2007 11:7
100 ESSENTIALS OF APPLIED MATHEMATICS FOR SCIENTISTS AND ENGINEERS
1
h
t 0-h t 0
FIGURE 6.3: The Dirac delta function
The Laplace transform is found term by term,
1
L[ f (t)] = {1 − 2e −sk [1 − e −sk + e −2sk − e −3sk ··· ]}
s
1 2e −sk 1 1 − e −sk
= 1 − = (6.21)
s 1 + e −sk s 1 + e −sk
6.2.8 The Dirac delta function
Consider a function defined by
− U t 0 −h
U t 0
lim = δ(t 0 ) h → 0 (6.22)
h
L[δ(t 0 )] = e −st 0 (6.23)
The function, without taking limits, is shown in Fig. 6.3.
6.2.9 Transforms of derivatives
∞ ∞
df df
L = e −st dt = e −st d f (6.24)
dt dt
t=0 t=0
and integrating by parts
∞
df
L = f (t)e −st ∞ + s f (t)e −st dt = sF(s ) − f (0)
dt 0
t=0
To find the Laplace transform of the second derivative we let g(t) − f (t). Taking the Laplace
transform
L[g (t)] = sG(s ) − g(0)
