Consider the integral $ f(\alpha,\beta)= \int_0^{2\pi}\,dx \sqrt{1- \cos(\alpha x ) \cos(\beta x)}$ as a function of the two parameters $\alpha,\beta$. I am interested in the asymptotic behavior for $\alpha, \beta \gg 1$.
For $\alpha = \beta$ the integral can be evaluated explicitly with the result $ f(\alpha , \alpha) = \frac{2}{\alpha} \left[ \lfloor 2 \alpha\rfloor + \sin^2 \left(\frac\pi2 \{ 2 \alpha \} \right) \right]$ with $\{ x \} = x - \lfloor x \rfloor$. For large $\alpha$ the function $f(\alpha, \alpha)$ thus approaches $4$.
If we see what happens if we keep $\beta$ large but fixed and vary $\alpha$, we see that $\beta \approx \alpha$ with $f(\alpha,\beta) \approx 4$ looks like a minimum of the function and it quickly approaches $2\pi$ (at least for $\alpha$ large) for $\beta$ sufficiently different from $\alpha$. However, there are oscillations on top of the mean value $2\pi$. In the figure you see a numerical evaluation of the integral for $\beta=20$ and $\alpha$ between 0 and 40.
- Is the value of $f(\alpha,\beta)$ for $\alpha,\beta \gg 1$ and $\alpha \neq \beta$ indeed $2\pi$?
- Why the value $f=4$ for $\alpha \approx \beta$ is lower than the generic value $2\pi$ (for $\alpha$, $\beta$ large)?
- What is the period of the (fast) oscillations as a function of $\alpha$ with $\beta$ fixed which are visible in the plot?
- What is the shape of the envelope? (it is a peaked function -> Lorentzian, or Gaussian, or ...?)
- Does anybody know how to obtain a good asymptotic expression for $f(\alpha, \beta)$?