# complex roots

#### Danneedshelp

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
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?

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

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

• Danneedshelp