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If f and g are orthogonal, are f* and g orthogonal?

  1. Sep 28, 2013 #1
    I am curious:

    if f and g are (complex) orthogonal functions, are f* and g also orthogonal? (* denotes complex conjugate).

    I would think the answer is no, in general, but I just want to confirm
     
  2. jcsd
  3. Sep 28, 2013 #2

    Office_Shredder

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    What are you integrating over and what functions are you allowed?

    For example if you're integrating over the unit circle and all you ask for are integrable functions, then if f = 1/sqrt(z) and g = 1/sqrt(z) (where z has an argument between 0 and 2pi), then
    [tex] \oint fg dz = 2\pi i [/tex]
    but
    [tex] \oint f^* g dz = \oint 1 dz = 0 [/tex]

    So f and g are orthogonal, but f* and g are not.
     
    Last edited: Sep 29, 2013
  4. Sep 29, 2013 #3
    A simpler example for [itex]f,g\in \mathbb C^2,[/itex] (which we could write as [itex]f,g: \{1,2\}\to \mathbb C[/itex] if you like):

    Let [itex]f:= (1,i), g:=(1,-i).[/itex] Then [tex]\langle f, g \rangle = (1)(\overline{1})+(i)(\overline{-i}) = (1)(1)+ (i)(i) = 0,[/tex] and [tex]\langle f^*, g \rangle = (\overline{1})(\overline{1})+(\overline{i})(\overline{-i}) = (1)(1)+ (-i)(i) = 2\neq 0,[/tex] so that [itex]f,g[/itex] are orthogonal, but [itex]f^*,g[/itex] aren't.
     
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