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Exponential function

  • Thread starter Kenji Liew
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  • #1
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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!
 

Answers and Replies

  • #2
Cyosis
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The power series of the exponential, [itex]e^x[/itex], is [itex]\sum_{n=0}^\infty x^n/n![/itex].

So if you have [itex]e^{-x}[/itex] you can compute the power series by substituting [itex]x \rightarrow -x[/itex] 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?
 
  • #3
25
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The power series of the exponential, [itex]e^x[/itex], is [itex]\sum_{n=0}^\infty x^n/n![/itex].

So if you have [itex]e^{-x}[/itex] you can compute the power series by substituting [itex]x \rightarrow -x[/itex] 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?
 
  • #4
Cyosis
Homework Helper
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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 [Broken] out for a list of series expansions.
 
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  • #5
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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 [Broken] out for a list of series expansions.
Thanks a lot. You really help me up! =)
 
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