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

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

Office_Shredder
Staff Emeritus
Gold Member
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.

Last edited:
• 1 person
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.