Laplace Transform of Simple Functions

Topics: Laplace Transform

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(theorems)

The Laplace transform of simple functions like $f(t)=c,f(t)=t$, etc. is as follows:

$$\begin{array}{rlrl}\mathcal{L}\{c\}& =\frac{c}{s}& s>0& \text{(theorem 1)}\\ \mathcal{L}\{t^n\}& =\frac{n!}{s^{n+1}}& s>0& \text{(theorem 2)}\\ \mathcal{L}\{\sin (at)\}& =\frac{a}{s^2+a^2}& s>0& \text{(theorem 3)}\\ \mathcal{L}\{\cos (at)\}& =\frac{s}{s^2+a^2}& s>0& \text{(theorem 4)}\\ \mathcal{L}\{e^{at}\}& =\frac{1}{s-a}& s>0& \text{(theorem 5)}\\ \mathcal{L}\{\sinh (at)\}& =\frac{a}{s^2-a^2}& s>0& \text{(theorem 6)}\\ \mathcal{L}\{\cosh (at)\}& =\frac{s}{s^2-a^2}& s>0& \text{(theorem 7)}\end{array}$$

Theorem Numbering

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