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Fourier analysis prob.

  1. Oct 12, 2005 #1

    quasar987

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    I'm puzzled and don't know where to begin with this question; it goes like

    "Consider the function f:R²-->R defined by

    [tex]f(x,y) = \sum_{n=1}^{\infty}\frac{(-1)^n}{n^2}sin(nx)sin(ny)[/tex]

    Show that f is continuous."

    Any hint?

    .
     
    Last edited: Oct 12, 2005
  2. jcsd
  3. Oct 13, 2005 #2
    if you can use the fact that a linear combination of continuous functions is continuous, then the problem is greatly simplified--it becomes a matter of showing that sin(nx)sin(xy) is continuous for any n.


    ...maybe?
     
  4. Oct 13, 2005 #3

    quasar987

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    Hey Brad, I appreciate your interest in my problem, however, I don't think an infinite sum qualifies as a linear combination. Nevertheless, your idea made me remember a theorem of 1 variable analysis and I was able to generalize it to an n variable function which solves the problem. :)
     
  5. Oct 13, 2005 #4
    glad i could be of indirect service! :biggrin:
     
  6. Oct 13, 2005 #5

    quasar987

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    Next they say "evaluate [itex]f(3\pi / 4, -5\pi /4)[/itex]". I realize that this is just

    [tex]\sum_{n=1}^{\infty} \frac{(-1)^n}{n^2} sin^2(\frac{3n \pi }{4})[/tex]

    but how do I find the sum? I tried squeezing the sum btw 0 and [itex]\sum (-1)^n/n^2[/itex] but this sum is not 0, so I can't conclude. After this attemp I'm all out of idea. :grumpy:
    Any help welcome.
     
  7. Oct 14, 2005 #6

    quasar987

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    Solved.
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