Proofs on Laplace Transform
July 9, 2025
Let f be a function defined on an interval that contains (0,+\infin). The Laplace transform of f is the function F defined by the proper intergral.
The domain of F is the set of all s for which this improper integral converges. The Laplace transform of f is also denoted by \mathcal{L}\{f\}(s).
We will be proving some Laplace transforms of such functions with specific forms, utilizing some integration techniques, and established integrals. For convenience’s sake, such remarks and special functions will be presented before the proofs.
Laplace Transform of f(t) = 1
Proof:
Laplace Transform of f(t) = e^{at}
Proof:
For the proofs of the Laplace transforms of f(t) = t^n \text{ and } f(t) = e^{at}t^n, we will be using some remarks involving the Gamma Function, as shown on the next page.
Laplace Transform of f(t) = t^n
Proof:
To apply the concept of Gamma function, we will use the substitution:
Notice that we are applying the substitution on a definite integral, thus, the bounds have changes. Hence,
This implies that,
For the proofs of the Laplace transforms of f(t) = \sin(bt), we will use some remarks involving standards. Note that these are mainly employed as an alternative to performing multiple iterations of integration by parts, which can be quite tedious.
Laplace Transform of f(t) = \sin(bt)
Proof:
Laplace Transform of f(t) = \cos(bt)
Proof:
Laplace Transform of f(t) = e^{at}t^n
Proof:
To apply the concept of Gamma Function, we will use the substitution:
Notice that we are applying the substitution on a definite integral, thus, the bounds have changed. Hence,
This implies that,
Laplace Transform of f(t) = e^{at}\sin(bt)
Proof:
Laplace Transform of f(t) =e^{at}\cos(bt)
Proof:
Let f be a function whose derivative f' is piecewise continuous on [0, \, \infin) and of exponential order. The Laplace transform of the derivative f'(t) is then defined by the improper integral
where \sigma_0 is the abscissa of convergence for \mathcal{L}\{f\}(s).
where F(s)=\mathcal{L}\{f(t)\}(s)
Proof:
We then apply integration by parts using the following substitutions:
Consequently,
where F(s)=\mathcal{L}\{f(t)\}(s)
Proof: