The Laplace transform has been introduced in order to simplify several mathematical operations. These operations center upon the solution of linear differential equations. Several basic properties of the Laplace transform are given here.
If the Laplace transforms of f1(t) and f2(t) are F1(s) and F2(s), respectively, then
[f1(t) ± f2(t)] = F1(s) ± F2(s).
If the Laplace transform of f(t) is F(s), the multiplication of the function f(t) by a constant K results in a Laplace transform KF(s).
If the Laplace transform of f(t) is F(s), the transform of the first time derivative (t) of f(t) is given by
where f(0+) is the initial value of f(t), evaluated as t → 0 from the positive region. The transform of the second time derivative (t) of f(t) is given by
where (0+) is the first derivative of f(t) evaluated at t = 0+. The Laplace transform of the nth derivative of a function is given by
The notation f(n−1)(0+) represents the (n − 1)th derivative of f(t) with respect to time evaluated at t = 0+.
If the Laplace transform of f(t) is F(s), the transform of the time integral of f(t) is given by
where [ f(t)dt]t=0+ signifies that the integral is evaluated as t → 0 from the positive region. In general, for nth-order integration,
The Laplace transform of a time function f(t) delayed in time by T equals the Laplace transform of f(t) multiplied by e−sT:
If the Laplace transform of f(t) is F(s), then the Laplace transform of
e−atf(t)
is obtained as follows:
Therefore, multiplying f(t) by e−at is equivalent to replacing s by (s + a) in the Laplace transform. In addition, changing s to (s + a) is equivalent to multiplying f(t) by e−at.
If the Laplace transform of f(t) is F(s), and if lims→∞ sF(s) exists, then the initial value of the time function is given by
If the Laplace transform of f(t) is F(s), and if sF(s) is analytic on the imaginary axis and in the right half-plane, then the final value of the time function is given by