PX275 - H5 - parseval's theorem

I_{t o t a l} = \int_{- \infty}^{\infty} | u (x) |^{2} d x

where, $| u |^{2} = u u^{*}$

u (x) = \frac{1}{2 π} \int_{- \infty}^{\infty} \tilde{u} (k) e^{i k x} d k

\begin{aligned} I_{t o t a l} & = \int_{- \infty}^{\infty} \underset{u (x)}{\underset{⏟}{\frac{1}{2 π} \int_{- \infty}^{\infty} \tilde{u} (k) e^{i k x} d k}} \underset{u^{*} (x)}{\underset{⏟}{\frac{1}{2 π} \int_{- \infty}^{\infty} {\tilde{u}}^{*} (k^{'}) e^{- i k^{'} x} d k^{'}}} d x \\ = ∭_{- \infty}^{\infty} \frac{1}{2 π} \frac{1}{2 π} \tilde{u} (k) {\tilde{u}}^{*} (k^{'}) e^{i (k - k^{'}) x} d k d k^{'} d x \\ = \iint_{- \infty}^{\infty} \frac{1}{2 π} \tilde{u} (k) {\tilde{u}}^{*} (k^{'}) δ (k - k^{'}) d k d k^{'} \\ = \frac{1}{2 π} \int_{- \infty}^{\infty} {\tilde{u}}^{*} (k^{'}) \tilde{u} (k^{'}) d k^{'} \\ ∴ I_{t o t a l} & = \frac{1}{2 π} \int_{- \infty}^{\infty} | \tilde{u} (k) |^{2} d k \end{aligned}

\int_{\infty}^{\infty} | u (x) |^{2} d x = \frac{1}{2 π} \int_{- \infty}^{\infty} | \tilde{u} (k) |^{2} d k

\int_{\infty}^{\infty} f (x) g^{*} (x) d x = \frac{1}{2 π} \int_{- \infty}^{\infty} \tilde{f} (k) {\tilde{g}}^{*} (k) d k