Use part (c) of Theorem 4.1.1 to show that +5e(aib)t 6 s 2 a ib (s 2 a) 2 b2, where a
Chapter 4, Problem 49(choose chapter or problem)
Use part (c) of Theorem 4.1.1 to show that
\(\mathscr{L}\left\{e^{(a+i b) t}\right\}=\frac{s-a+i b}{(s-a)^{2}+b^{2}}\),
where a and b are real and \(i^{2}=-1\). Show how Euler’s formula (page 121) can then be used to deduce the results
\(\mathscr{L}\left\{e^{a t} \cos b t\right\}=\frac{s-a}{(s-a)^{2}+b^{2}}\)
and \(\mathscr{L}\left\{e^{a t} \sin b t\right\}=\frac{b}{(s-a)^{2}+b^{2}}\).
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