Solution Found!
Use part (c) of Theorem 7.1.1 to show that where a and b
Chapter 7, Problem 50E(choose chapter or problem)
Use part (c) of Theorem 7.1.1 to show that
\(\mathscr{L}\left\{e^{(a+i b) t}\right\}=\frac{s-a+i b}{(s-a)^{2}+b^{2}}\)
and \(i^{2}=-1\). Show how Euler’s formula (page 133) can then be used to deduce the results
\(\mathscr{L}\left\{e^{a t} \cos b t\right\}=\frac{s-a}{(s-a)^{2}+b^{2}}\)
\(\mathscr{L}\left\{e^{a t} \sin b t\right\}=\frac{b}{(s-a)^{2}+b^{2}}\).
Text Transcription:
Le^(a+ib)t=fracs-a+ib(s-a)^2+b^2
i^2=-1
Le^atcosbt=fracs-a(s-a)^2+b^2
Le^atsinbt=fracb(s-a)^2+b^2
Questions & Answers
QUESTION:
Use part (c) of Theorem 7.1.1 to show that
\(\mathscr{L}\left\{e^{(a+i b) t}\right\}=\frac{s-a+i b}{(s-a)^{2}+b^{2}}\)
and \(i^{2}=-1\). Show how Euler’s formula (page 133) can then be used to deduce the results
\(\mathscr{L}\left\{e^{a t} \cos b t\right\}=\frac{s-a}{(s-a)^{2}+b^{2}}\)
\(\mathscr{L}\left\{e^{a t} \sin b t\right\}=\frac{b}{(s-a)^{2}+b^{2}}\).
Text Transcription:
Le^(a+ib)t=fracs-a+ib(s-a)^2+b^2
i^2=-1
Le^atcosbt=fracs-a(s-a)^2+b^2
Le^atsinbt=fracb(s-a)^2+b^2
ANSWER:Step 1 of 3
In this problem we have to prove the basic transformer function.
Given that part c of the given 7.1.1
Part (c) of the 7.1.1 states that