Exponential function

1. Aug 1, 2009

Kenji Liew

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
Besides what is the function of sin x and cos x in continued function (such in e^x)?
Thanks!

2. Aug 1, 2009

n!kofeyn

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?
\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*}

Or are you asking for the following identities?
\begin{align*} \sin x &= \frac{e^{ix}-e^{-ix}}{2i} \\ \cos x &= \frac{e^{ix} + e^{-ix}}{2} \end{align*}

3. Aug 1, 2009

Staff: Mentor

e-1 = 1 + (-1) + (-1)2/2! + (-1)3/3! + ... + (-1)n/n! + ...

4. Aug 1, 2009

Staff: Mentor

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