- #1

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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

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- Thread starter Aziza
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- #1

- 190

- 1

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

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.

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

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- #3

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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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