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Basic principles

The parameter s belongs to some domain on the real line or in the complex

plane. We will choose s appropriately so as to ensure the convergence of the
Laplace integral (1.1). In a mathematical and technical sense, the domain of s
is quite important. However, in a practical sense, when differential equations are
solved, the domain of s is routinely ignored. When s is complex, we will always

use the notation s = x + iy.
The symbol L is the Laplace transformation, which acts on functions f =
−1
f(t) and generates a new function, F(s) = Lf(t). The notation L will also use
−1
to denote the inverse Laplace transform of F, i.e. f(t) = L F(s) means that
Lf(t) = F(s).

Next we will give examples on computing the Laplace transform of given
functions by definition.
.

Example 1.1, f(t) ≡ 1 for t ≥ 0. Evaluate Lf(t).
. . . . .
.
Solution.

∞
∫ ∞
1 1
F(s) = e −st 1 dt = − e −st =
s 0 s
0
provided of course that s > 0 (if s is real). Thus we have

1
L(1) = (s > 0). (1.2)
s

If s ≤ 0, then the integral would diverge and there would be no resulting
Laplace transform. If we had taken s to be a complex variable, the same
calculation, with Re(s) > 0, would have given L(1) = 1/s.
In fact, let us just verify that in the above calculation the integral can be

treated in the same way even if s is a complex variable. We require the
well-known Euler formula

iθ
e = cos θ + i sin θ, θ — real,

iθ
and the fact that |e | = 1. The claim is that (ignoring the minus sign as
well as the limits of integration to simplify the calculation)

∫ st
e
st
e dt = , (1.3)
s
for s = x + iy is any complex number ̸= 0. To see this observe that

∫ ∫ ∫ ∫
st
xt
xt
e dt = e (x+iy)t dt = e cos ytdt + i e sin ytdt
. . . .
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