Fundamental Theorems of the Laplace Transform

Topics: Laplace Transform

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(theorems)

There are several theorems that prove to be fundamental when working with Laplace transforms:

$$\begin{array}{rlr}\mathcal{L}\{f(at)\}& =\frac{1}{a}F\left(\frac{s}{a}\right)& \text{(t. 9)}\\ \mathcal{L}\{f^n(t)\}& =s^{n}F(s)-s^{n-1}f(0)-\cdots -f^{n-1}(0)& \text{(t. 10)}\\ \mathcal{L}\{e^{at}f(t)\}& =F(s-a)& \text{(t. 11)}\\ \mathcal{L}\{f(t-a)U(t-a)\}& =e^{-as}F(s)& \text{(t. 12)}\\ \mathcal{L}\{t^{n}f(t)\}& =(-1{)}^n\frac{d^{n}F(s)}{d{s}^n}& \text{(t. 13)}\\ \mathcal{L}\{f(t)g(t)\}& =F(s)G(s)& \text{(t. 14)}\end{array}$$

…where $U(t-a)$ is the shifted unit step function:

$$U(t-a)=\{\begin{array}{ll}0& \text{if }0\le t\le a\\ 1& \text{if }t>a\end{array}$$

Theorem Numbering

The theorems are numbered according to my Ecuaciones Diferenciales II course.