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Show that the functions sin x, sin 2x, sin 3x, are orthogonal

  1. Oct 30, 2005 #1
    Show that the functions sin x, sin 2x, sin 3x, ... are orthogonal on the interval (0,pi) with respect to p(x) = 1 (where p is supposed to be rho)

    i know i have to use this
    [tex] \int_{0}^{\pi} \phi (x) \ psi (x) \rho (x) dx = 0 [/tex] and i have no trouble doing it for n = 1, and n=2
    but how wouldi go about proving it for hte general case that is

    [tex] \int_{0}^{pi} \sin{nx} \sin{(n+1)x} dx [/tex] for n in positive integers

    would i use this identity :
    [tex] \sin{s) \sin{t} = \frac{ \cos{s-t} - \cos{s+t} }{2} [/tex]
    i proved it using this identity... but isnt this identity a bit too obscure? Isnt there a less 'weird' way of doing this?
     
  2. jcsd
  3. Oct 30, 2005 #2

    Physics Monkey

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    One very easy way of proceeding is to write
    [tex]
    \sin{(n x)} = \frac{e^{i n x} - e^{-inx} }{2i},
    [/tex]
    expand both the sines under the integral as complex exponentials and the result falls out immediately.
     
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