How do I show that the equation E(k) = 2-4cos(ka) is a parabola when k=0 and when k=pi/a?

Question

It's evident from the graph but I'm not sure how to show this mathematically. This dispersion relation is supposed to be roughly parabolic

Answer 1

Almost any minimum or maximum will be approximately parabolic. You can calculate the Taylor series at $(x_0,y_0)$ and get $y \approx y_0+y'(x_0)(x-x_0)+y''(x_0)(x-x_0)^2\dots$ Since you are at a minimum or maximum, you have $y'(x_0)=0$ so that term goes away. The higher order terms are small when you are near the minimum or maximum. In your case of $y=2-4 \cos (4x)$, one minimum is $(0,-2)$ and the Taylor series is $y \approx -2+32x^2-\frac {128}3x^4+\dots$ For $|x| \ll 1$ you can ignore the quartic and higher terms.