Parseval's Relation

Let $F(\nu)$ and $G(\nu)$ be the Fourier Transforms of $f(t)$ and $g(t)$, respectively. Then

$\int_{-\infty}^\infty f(t)g^*(t)\,dt$
$\quad = \int_{-\infty}^\infty \left[{\int_{-\infty}^\infty F(\nu)e^{-2\pi i\nu t}\,d\nu \int_{-\infty}^\infty G^*(\nu')e^{2\pi i\nu't}\,d\nu'}\right]\,dt$
$\quad = \int_{-\infty}^\infty F(\nu)\int_{-\infty}^\infty G^*(\nu')\delta(\nu'-\nu)\,d\nu'\,d\nu$
$\quad = \int_{-\infty}^\infty F(\nu)G^*(\nu)\,d\nu.$

See also Fourier Transform, Parseval's Theorem

References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, p. 425, 1985.