Question 6.9: Computing a Laplace Transform Find the Laplace transform of ...
Computing a Laplace Transform
Find the Laplace transform of f(t)=\cos (\omega t) u(t).
Known Quantities: Function to be Laplace-transformed.
Find: F(s)=\mathcal{L}\{f(t)\}.
Schematics, Diagrams, Circuits, and Given Data: f(t)=\cos (\omega t) u(t).
Assumptions: None.
Analysis: Using equation 6.51
\mathcal{L}[f(t)]=F(s)=\int_{0}^{\infty} f(t) e^{-s t} d t (6.51)
and applying Euler’s identity to \cos (\omega t) gives:
F(s)=\int_{0}^{\infty} \frac{1}{2}\left(e^{j \omega t}+e^{-j \omega t}\right) e^{-s t} d t=\frac{1}{2} \int_{0}^{\infty}\left(e^{(-s+j \omega) t}+e^{(-s-j \omega) t}\right) d t
\begin{aligned}&=\left.\frac{1}{-s+j \omega} e^{-(s+j \omega) t}\right|_{0} ^{\infty}+\left.\frac{1}{-s-j \omega} e^{-(s-j \omega) t}\right|_{0} ^{\infty} \\&=\frac{1}{-s+j \omega}+\frac{1}{-s-j \omega}=\frac{s}{s^{2}+\omega^{2}}\end{aligned}
Comments: Table 6.1 contains a list of common Laplace transform pairs.
Table 6.1 Laplace transform pairs
\begin{array}{ll} \hline {\boldsymbol{f}(\boldsymbol{t})} & \boldsymbol{F}(\boldsymbol{s}) \\\hline\delta(t) (\text{unit impulse}) & 1 \\\\u(t) (\text{unit step}) & \frac{1}{s} \\e^{-a t} u(t) & \frac{1}{s+a}\\ \\\sin \omega t u(t) & \frac{\omega}{s^2+\omega^2}\\ \\\cos \omega t u(t) & \frac{s}{s^2+\omega^2} \\\\e^{-a t} \sin \omega t u(t) & \frac{\omega}{(s+a)^2+\omega^2}\\ \\e^{-a t} \cos \omega t u(t) & \frac{s+a}{(s+a)^2+\omega^2} \\\\t u(t) & \frac{1}{s^2} \\ \hline\end{array}