Solve Exponential Function: Expansion of e^x & Sin/Cos x

[Kenji Liew]

Homework Statement

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

Homework Equations

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

The Attempt at a Solution

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

 

[Cyosis]

The power series of the exponential, $e^x$, is $\sum_{n=0}^\infty x^n/n!$.

So if you have $e^{-x}$ you can compute the power series by substituting $x \rightarrow -x$ into the power series. Try it out.

I don't understand your second question. Are you asking for the power series of the sine and cosine? Or for the complex exponential representation?

 

[Kenji Liew]

The power series of the exponential, $e^x$, is $\sum_{n=0}^\infty x^n/n!$.

So if you have $e^{-x}$ you can compute the power series by substituting $x \rightarrow -x$ into the power series. Try it out.

I don't understand your second question. Are you asking for the power series of the sine and cosine? Or for the complex exponential representation?

Thanks for the first part.
I just now find the cosine x can be written in cosine x=1-(x^2)/2!+(x^4)/4!+...
I really no idea what this series call for...
How about the sine x?

 

[Cyosis]

It's called the series expansion of the sine/cosine or the power series of the sine/cosine. I would suggest memorizing/deriving the series expansions for the more common functions.

Check this http://en.wikipedia.org/wiki/Taylor_seriesp out for a list of series expansions.

 

[Kenji Liew]

It's called the series expansion of the sine/cosine or the power series of the sine/cosine. I would suggest memorizing/deriving the series expansions for the more common functions.

Check this http://en.wikipedia.org/wiki/Taylor_seriesp out for a list of series expansions.

Thanks a lot. You really help me up! =)