/sci/ - Science & Math

>>14638873
The if part is your job.
We break up [math]\exp \left( i \left( p + \dfrac{2 \pi}{a} m \right) x \right) = \exp (ipx) \exp \left( \dfrac{2 i \pi}{a} m x \right) [/math]. From this we notice that [math]span \left\{ \exp \left( i \left( p + \dfrac{2 \pi}{a} m \right) x \right) \right\}[/math] is, actually, basically any complex function with period [math]a[/math] times [math]e^{ipx}[/math].
So if [math]e^{ipx} = f(x) e^{iqx}[/math], where [math]f(x)[/math] is [math]a[/math]-periodic, we can do [math]e^{i(p-q)x} = f(x)[/math], hence [math]p-q = \frac{2 \pi j}{a}[/math]