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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^(-x)=?
    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

    Cyosis

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    Homework Helper

    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?
     
  4. Aug 1, 2009 #3
    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?
     
  5. Aug 1, 2009 #4

    Cyosis

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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.
     
    Last edited by a moderator: May 4, 2017
  6. Aug 1, 2009 #5
    Thanks a lot. You really help me up! =)
     
    Last edited by a moderator: May 4, 2017
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