2.3.6 So­lu­tion dot-f

Ques­tion:

Ver­ify that the most gen­eral mul­ti­ple of $\sin(x)$ that is nor­mal­ized on the in­ter­val 0 $\raisebox{-.3pt}{$\leqslant$}$ $x$ $\raisebox{-.3pt}{$\leqslant$}$ $2\pi$ is $e^{{\rm i}\alpha}\sin(x)$$\raisebox{.5pt}{$/$}$​$\sqrt{\pi}$ where $\alpha$ is any ar­bi­trary real num­ber. So, us­ing the Euler for­mula, the fol­low­ing mul­ti­ples of $\sin(x)$ are all nor­mal­ized: $\sin(x)$$\raisebox{.5pt}{$/$}$​$\sqrt{\pi}$, (for $\alpha$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0), $-\sin(x)$$\raisebox{.5pt}{$/$}$​$\sqrt{\pi}$, (for $\alpha$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\pi$), and ${\rm i}\sin(x)$$\raisebox{.5pt}{$/$}$​$\sqrt{\pi}$, (for $\alpha$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\pi$$\raisebox{.5pt}{$/$}$​2).

An­swer:

A mul­ti­ple of $\sin(x)$ means $c\sin(x)$, where $c$ is some com­plex con­stant, so the mag­ni­tude is

$$\vert\vert c\sin(x)\vert\vert = \sqrt{\left\langle\vphantom{... ...gle } = \sqrt{\int_0^{2\pi} (c \sin(x))^*(c\sin(x)){ \rm d}x}$$

You can al­ways write $c$ as $\vert c\vert e^{{\rm i}\alpha}$ where $\alpha$ is some real an­gle, and then you get for the norm:

$$\vert\vert c\sin(x)\vert\vert = \sqrt{\int_0^{2\pi} \left(\v... ... \vert c\vert^2\sin^2(x) { \rm d}x} = \vert c\vert \sqrt{\pi}$$

So for the mul­ti­ple to be nor­mal­ized, the mag­ni­tude of $c$ must be $\vert c\vert$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1/$\sqrt{\pi}$, but the an­gle $\alpha$ can be ar­bi­trary.