complex roots

Apr 2009
303
37
a) Show that \(\displaystyle e^{(r_{1}+r_{2})t}=e^{r_{1}t}e^{r_{2}t}\) for any complex number \(\displaystyle r_{1}\) and \(\displaystyle r_{2}\).

b) Show that \(\displaystyle \frac{d}{dt}e^{rt}=re^{rt}\).

For (a) I am stuck because \(\displaystyle e^{(r_{1}+r_{2})t}=e^{(\lambda+i\mu+\lambda-i\mu)t}=e^{2\lambda\\t}\). I am not sure where to go from here. Should I just consider to arbitrary positive complex number rather than conjugate roots? I am considering roots because that is all that is talked about in the section.

For (b) do I just differentiate \(\displaystyle e^{\lambda\\t}(cos(\mu\\t)+isin(\mu\\t)\) since \(\displaystyle e^{(\lambda+i\mu)t}=e^{\lambda\\t}(cos(\mu\\t)+isin(\mu\\t)\) or can I just look at the power series expansion of \(\displaystyle e^{(\lambda+i\mu)t}\) and differentiate that?
 

dwsmith

MHF Hall of Honor
Mar 2010
3,093
582
Florida
a) Show that \(\displaystyle e^{(r_{1}+r_{2})t}=e^{r_{1}t}e^{r_{2}t}\) for any complex number \(\displaystyle r_{1}\) and \(\displaystyle r_{2}\).

b) Show that \(\displaystyle \frac{d}{dt}e^{rt}=re^{rt}\).

For (a) I am stuck because \(\displaystyle e^{(r_{1}+r_{2})t}=e^{(\lambda+i\mu+\lambda-i\mu)t}=e^{2\lambda\\t}\). I am not sure where to go from here. Should I just consider to arbitrary positive complex number rather than conjugate roots? I am considering roots because that is all that is talked about in the section.

For (b) do I just differentiate \(\displaystyle e^{\lambda\\t}(cos(\mu\\t)+isin(\mu\\t)\) since \(\displaystyle e^{(\lambda+i\mu)t}=e^{\lambda\\t}(cos(\mu\\t)+isin(\mu\\t)\) or can I just look at the power series expansion of \(\displaystyle e^{(\lambda+i\mu)t}\) and differentiate that?
Part (a)
Where does it say \(\displaystyle r_1\) and \(\displaystyle r_2\) are conjugates?

\(\displaystyle r_1=a\pm b\mathbf{i}\)
\(\displaystyle r_2=c\pm d \mathbf{i}\)

\(\displaystyle e^{((a+b\mathbf{i})+(c+d\mathbf{i}))t}=e^{(a+b\mathbf{i})t+(c+d\mathbf{i})t}=e^{(a+b\mathbf{i})t}e^{(c+d\mathbf{i})t}=e^{r_1t}e^{r_2t}\)
 
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