# What Is the Difference between Parseval's Theorem and Plancherel Theorem?

Wikipedia gives Parseval's theorem as follows. Parseval's Theorem

$\int_{-\infty&space;}^{\infty&space;}\left&space;|&space;x(t))&space;\right&space;|^{2}dt&space;=&space;\frac{1}{2\Pi&space;}&space;\int_{-\infty&space;}^{\infty&space;}&space;\left&space;|&space;X(\omega&space;))&space;\right&space;|^{2}d\omega$ , $X(\omega&space;)$ is the Fourier transform of x(t).

And Plancheral's theorem is given as

$\int_{-\infty&space;}^{\infty&space;}\left&space;|&space;(f(x)))&space;\right&space;|^{2}dx&space;=&space;\int_{-\infty&space;}^{\infty&space;}&space;\left&space;|&space;\hat{f}&space;(\varepsilon&space;)\right&space;|^{2}d\varepsilon$,

where $\hat{f}&space;(\varepsilon&space;)$ is the fourier analog.

According to the links you shared, and using Wikipedia's notation, Parseval assumes $$A(x),B(x)$$ are two square integrable (with respect to the Lebesgue measure), complex-valued functions on $$\mathbb {R}$$ of period $$2\pi$$ with Fourier series
$$A(x)=\sum_{n=-\infty}^\infty a_ne^{inx}$$ and $$B(x)=\sum_{n=-\infty}^\infty b_ne^{inx}$$ Then $$\sum_{n=-\infty}^\infty a_n\overline{b_n} = \frac{1}{2\pi} \int_{-\pi}^\pi A(x)\overline{B(x)} \, \mathrm{d}x$$ Plancherel provides a continuous version of Parsevals theorem (in a sense that $$a_n$$ and $$b_n$$ are now the $$f(x)$$ and $$g(x)$$, respectively and $$A(x),B(x)$$ are replaced by $$\widehat{f}(\xi), \widehat{g}(\xi)$$)