Results 1 to 2 of 2

Math Help - complex roots

  1. #1
    Senior Member Danneedshelp's Avatar
    Joined
    Apr 2009
    Posts
    303

    complex roots

    a) Show that e^{(r_{1}+r_{2})t}=e^{r_{1}t}e^{r_{2}t} for any complex number r_{1} and r_{2}.

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

    For (a) I am stuck because 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 e^{\lambda\\t}(cos(\mu\\t)+isin(\mu\\t) since e^{(\lambda+i\mu)t}=e^{\lambda\\t}(cos(\mu\\t)+isi  n(\mu\\t) or can I just look at the power series expansion of e^{(\lambda+i\mu)t} and differentiate that?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Mar 2010
    From
    Florida
    Posts
    3,093
    Thanks
    5
    Quote Originally Posted by Danneedshelp View Post
    a) Show that e^{(r_{1}+r_{2})t}=e^{r_{1}t}e^{r_{2}t} for any complex number r_{1} and r_{2}.

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

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

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

    e^{((a+b\mathbf{i})+(c+d\mathbf{i}))t}=e^{(a+b\mat  hbf{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}
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Complex 5th Roots
    Posted in the Pre-Calculus Forum
    Replies: 2
    Last Post: February 4th 2011, 03:09 PM
  2. Complex roots
    Posted in the Calculus Forum
    Replies: 6
    Last Post: December 23rd 2009, 10:46 AM
  3. Complex Roots
    Posted in the Pre-Calculus Forum
    Replies: 8
    Last Post: August 18th 2009, 05:50 PM
  4. Complex Roots
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: March 12th 2009, 08:38 PM
  5. Complex Roots
    Posted in the Advanced Algebra Forum
    Replies: 7
    Last Post: March 22nd 2008, 07:16 PM

Search Tags


/mathhelpforum @mathhelpforum