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Math Help - Odd and Even functions...

  1. #1
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    Question Odd and Even functions...

    Hi!

    my teacher wrote that from symmetry considerations we could say that:

    F(x)=Sin((n/L)*Pi*x) is an even function for odd n's around X=L/2, and odd for even n's.

    could somebody please explain why is that so?

    I tried to test it:
    x=L/2 => f=Sin(2*Pi*n) and this function is zero at X=L/2 for every n I use. odd or even...!

    thanks!
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  2. #2
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    Quote Originally Posted by dudinka View Post
    Hi!

    my teacher wrote that from symmetry considerations we could say that:

    F(x)=Sin((n/L)*Pi*x) is an even function for odd n's around X=L/2, and odd for even n's.

    could somebody please explain why is that so?

    I tried to test it:
    x=L/2 => f=Sin(2*Pi*n) and this function is zero at X=L/2 for every n I use. odd or even...!

    thanks!
    Odd about L/2 means that F(L/2-x) = -F(L/2+x), and even means that F(L/2-x) = F(L/2+x).

    Also sin((2n+1)/L pi (L/2)) = sin( (2n+1)pi /2 ) which is +/-1 for all n, and
    sin((2n)/L pi (L/2)) = sin( (n pi /2 ) which is 0 for all n.

    RonL
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  3. #3
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    I'm not sure I got it yet:

    1.
    sin((2n)/L pi (L/2)) = sin( (n pi /2 ) which is 0 for all n
    sin((2n)/L pi (L/2)) = sin (n pi ) -?


    2. Is there a way to notice these qualities quickly?
    I mean, he says these things in no time...
    Am I missing something? or I should check the qualities with this:
    F(L/2-x) = -F(L/2+x) and so on...

    (it's related to Fourier analysis of strings modes)

    thanks!
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  4. #4
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    Quote Originally Posted by dudinka View Post
    I'm not sure I got it yet:

    1.

    sin((2n)/L pi (L/2)) = sin (n pi ) -?


    2. Is there a way to notice these qualities quickly?
    I mean, he says these things in no time...
    Am I missing something? or I should check the qualities with this:
    F(L/2-x) = -F(L/2+x) and so on...

    (it's related to Fourier analysis of strings modes)

    thanks!
    In this case its the symmetries of the trig functions that you need to know.

    RonL
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  5. #5
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    10x!
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  6. #6
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    Quote Originally Posted by dudinka View Post
    10x!
    The factorial function does not have any symmetries that I am ware of.

    RonL
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  7. #7
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    :-)
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