Page 106 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

96 ESSENTIALS OF APPLIED MATHEMATICS FOR SCIENTISTS AND ENGINEERS
in which γ – iL and γ + iL are complex numbers. We will put off using the inversion integral
until we cover complex variables. Meanwhile, there are many tables giving Laplace transforms
and inverses. We will now spend considerable time developing the theory.

6.2 SOME IMPORTANT TRANSFORMS
6.2.1 Exponentials
First consider the exponential function:

∞ ∞
1

e
L[e −at ] = e −at −st dt = e −(s =a)t dt = (6.4)
s + a
t=0 t=0
If a = 0, this reduces to

L[1] = 1/s (6.5)

6.2.2 Shifting in the s-domain

∞

L[e at f (t)] = e −(s −a)t f (t)dt = F(s − a) (6.6)
t=0

6.2.3 Shifting in the time domain
Consider a function defined as

f (t) = 0 t < a f (t) = f (t − a) t > a (6.7)

Then

∞ a ∞

e −s τ f (τ − a)dτ = 0dτ + e −s τ f (τ − a)dτ (6.8)
τ=0 τ=0 τ=a
Let τ − a = t.Then

∞

e −s (t+a) f (t)dt = F(s )e −as = L[ f (t − a)] (6.9)
t=0

the shifted function described above.