I'm learning about something in Physics and after a few days of searching around I've realized my issue is a mathematical one. The quantity in question is called the "microcanonical partition function", and is written as follows, where $\{p,q\}$ are all independent variables and called the "phase space", and $E\equiv E(\{p,q\})$,
$$\Omega(E)=\int...\int_{\{p,q\}}\delta(E-H(p_1,q_1,...,p_N,q_N))dp_1dq_1...dp_Ndq_N \tag{1}$$
The paper I was reading in particular manipulated this by taking the Laplace transform of it (this part I understand): $$Z(\beta)=\int^{\infty}_{0} e^{-\beta E}\Omega(E)dE \tag{2}$$
And due to the properties of the dirac-delta function, we evaluate our exponential function at $E=H$, $$Z(\beta)=\int...\int_{\{p,q\}}e^{-\beta H}dp_1dq_1...dp_Ndq_N \tag{3}$$
This itself is a physically important quantity (canonical ensemble partition function) and therefore (1) is related to (3) via the L.T., even though physically they can be derived entirely separately. Quite an interesting revelation.
Upon reflection, I realize that I have no idea how to evaluate (1), given that the dirac-delta function is a function E, which is shifted by a function in and of itself (H). I would know how to evaluate this for example,
$$\int_{a}^{b}f(E)\delta(E-H)dE=f(H)\tag{4}$$
Where $a < H<b$. But in (1) the differential components are written as the components that H depends on.
Is there a term for this type of dirac-delta integral? Is there any theory that can be leveraged to expand it?