I encountered an issue when doing some problems in solid state physics, and I spent a whole day trying to clear this up, unsuccessfully. I'm posting it here because my issue is purely mathematical.
Namely, there appears an integral of the Dirac $\delta$-function of this form: $n(E)=\frac{Na}{\pi}\int_\frac{-\pi}{a}^\frac{\pi}{a} \delta(E-E_s+2J\cos ka)dk$
where $N,a,E,E_s,J,k$ are real.
This is supposed to be:
$n(E)=\frac{N}{\pi J\sqrt{1-(\frac{E-E_s}{2J})^2}}(\theta(E-E_b)-\theta(E-E_t))$
where $E_b=E_s-2J, E_t=E_s+2J$, and $\theta$ is the Heaviside step function.
Now, the integral of $\delta$ is either $1$ or $0$, depending whether the argument vanishes inside the integral limits or not. How exactly is this fact transformed into those terms involving $\theta$? I have tried manipulating using definitions of $\theta$, but to no avail.
I suppose this may be very elementary, but I'd be grateful for any help.