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Find function is analytic on R and has Maclaurin expansion

  1. Feb 6, 2008 #1
    Prove the function is analytic on R and find its Maclaurin expansion.

    (a) e[tex]^{x}[/tex]cosx


    Well, I did some work. I can show that

    e[tex]^{x}[/tex]=[tex]\sum[/tex]x[tex]^{k}[/tex]/k!

    cosx=[tex]\sum[/tex]((-1)[tex]^{k}[/tex]x[tex]^{2k}[/tex])/(2k)!

    are analytic and have the above Maclaurin expression.

    My problem is multiplying these two expressions together, finding the Maclaurin
    expression and proving it is analytic.

    I would be grateful for all your help. I think the hardest part will be showing how
    it is analytic.

    Thanks,
    HF08
     
  2. jcsd
  3. Feb 6, 2008 #2

    EnumaElish

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    Science Advisor
    Homework Helper

    If you were to multiply the two series like you would two polynomials, wouldn't that give you a power series representation of exp(x)cos(x)?

    Here is a link that may be useful: http://en.wikipedia.org/wiki/Cauchy_product
     
    Last edited: Feb 6, 2008
  4. Feb 6, 2008 #3
    I know the Maclaurin Series now :-)

    e[tex]^{x}[/tex]cos x = [tex]\sum[/tex][tex]\sum[/tex][(-1)[tex]^{j}[/tex]]/((2j)!(k-2j)1))

    Whew! That was alot of work for me, but I got it. Thanks. The result is correct. So how do I show this is analytic? I'm stuck, please, please help me.

    Thanks,
    HF08
     
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