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Exponential function

  1. Aug 1, 2009 #1
    1. The problem statement, all variables and given/known data

    This topic is under linear system differential equation.Solve the system by using exponential method. Just want to ask the expansion of exponential function

    2. Relevant equations

    e^x=1+x+(x^2)/2!+(x^3)/3!+........

    3. The attempt at a solution
    then how about the e^(-1)=?
    Besides what is the function of sin x and cos x in continued function (such in e^x)?
    Thanks!
     
  2. jcsd
  3. Aug 1, 2009 #2
    I'm not really for sure what your question even is. What problem are you trying to solve? Are you asking for the power series for sine and cosine?
    [tex]\begin{align*}
    \sin x &= x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \\
    \cos x &= 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots
    \end{align*}[/tex]

    Or are you asking for the following identities?
    [tex]\begin{align*}
    \sin x &= \frac{e^{ix}-e^{-ix}}{2i} \\
    \cos x &= \frac{e^{ix} + e^{-ix}}{2}
    \end{align*}[/tex]
     
  4. Aug 1, 2009 #3

    Mark44

    Staff: Mentor

    e-1 = 1 + (-1) + (-1)2/2! + (-1)3/3! + ... + (-1)n/n! + ...
     
  5. Aug 1, 2009 #4

    Mark44

    Staff: Mentor

    Also, you have posted what appears to be the same question twice, which is frowned upon in this and most other forums.
     
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