I have a problem with a passage in this integral with a Delta function:
\begin{align} I=\frac{A\pi}{h^2}\int_{0}^{+\infty} dt \delta(\epsilon-a^2t^2-b) \end{align} Now i want to use this identity: $\delta(f(t))=\frac{\delta(t-t_0)}{f'(t_0)}$ where $t_0$:$f(t_0)=0$ so that $t_0=\frac{\sqrt{\epsilon-b\sigma}}{a}$
I tried to solve the integral in this way: \begin{align} I=\frac{A\pi}{2a^2t_0h^2}\int_{0}^{+\infty} dt \delta(t-t_0)=\frac{A\pi}{4a^2t_0h^2} \end{align} but is incorrect, the book, after the use of the identity, says that:
$$I=\frac{A\pi}{h^2}\theta(\epsilon-b\sigma)\int_{0}^{+\infty} dt\frac{1}{2a^2t_0}[\delta(t-t_0)+\delta(t+t_0)] $$
I just can't figure out where that theta function and that second delta function came from.
$a$ and $b$ are constants, $\epsilon$ is a variable.