Negative frequencies

[Niles]

Hi

Say I have a real quantity given by


x(t) = \int_{ - \infty }^\infty {\tilde x(\omega )e^{ - i\omega t} d\omega }


Now I complex conjugate it (remember it is real)


x(t) = \int_{ - \infty }^\infty {\tilde x^* (\omega )e^{ + i\omega t} d\omega }


How is it that I from this can conclude that we must have the relation


{\tilde x^* (\omega )} = {\tilde x(-\omega )}

?

Best,
Niles.

[Niles]

I figured it out. Just make a substitution, and it all becomes obvious.