Chapter 7.1, Problem 49E

Discussion Problems

Use part (c) of Theorem 7.1.1 to show that

$ℒ\left\{e^{(a+ib)t}\right\}=\frac{s-a+ib}{{(s-a)}^2+b^2},$

where a and b are real and i² = −1. Show how Euler’s formula (page 136) can then be used to deduce the results

$\begin{array}{rrr}\hfill ℒ\left\{e^{at}\cos bt\right\}& \hfill =& \hfill \frac{s-a}{{(s-a)}^2+b^2}\\ \hfill ℒ\left\{e^{at}\sin bt\right\}& \hfill =& \hfill \frac{b}{{(s-a)}^2+b^2}.\end{array}$