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

Aziza
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

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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
$$\oint fg dz = 2\pi i$$
but
$$\oint f^* g dz = \oint 1 dz = 0$$

So f and g are orthogonal, but f* and g are not.

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1 person
economicsnerd
A simpler example for $f,g\in \mathbb C^2,$ (which we could write as $f,g: \{1,2\}\to \mathbb C$ if you like):

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