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Is there a general approach for handling the integration (and more specifically, addressing the limits) of general multivariate functions including a (multivariate) delta function? I suppose the most general form looks like $$ \iiint_Af(x,y,z)\delta(g(x,y,z))dxdydz $$

Unlike in the case of Multivariate integrals involving Dirac delta functions, it seems possible that depending on the form of $g(x,y,z)$ and $A$, one might have to change the limits of integration on the remaining integrals. (For example, in that case, what happens to the limits of the first integral if the delta function looks like $\delta(y-3x)$?)

In my specific case, I have a set of $n$ integrals that look like $$ \int_0^\Delta dt^\prime\int_0^{t^\prime}dt^{\prime\prime}\int_0^{t^{\prime\prime}}dt^{\prime\prime\prime}\ldots\int_0^{t^{\prime^{n-1}}}dt^{\prime^n} \;f(g(\{t\})\delta(c - g(\{t\})) $$ with $$ g(\{t\})=t^\prime-t^{\prime\prime}+t^{\prime\prime\prime}-\ldots $$ and $$ f(g(\{t\})) = e^{-mg(\{t\})}e^{-n(\Delta-g(\{t\}))}=e^{-m(t^\prime-t^{\prime\prime}+\ldots)}e^{-n(\Delta - t^\prime + t^{\prime\prime} - \ldots)} $$ and its not clear to me how to approach this.

Thanks!

Edit: After working through the first handful of $n$'s, it's apparent that the answer is going to look like $$ e^{-mc}e^{-n(\Delta-c)}\frac{(c(\Delta-c))^{(n-1)/2}}{((n-1)/2)!)^2}\text{ if }n\text{ odd} $$

and $$ e^{-mc}e^{-n(\Delta-c)}\frac{c^{n/2}(\Delta-c))^{n/2-1}}{(n/2-1)!(n/2)!}\text{ if }n\text{ even} $$ but how one would get from there to here, even mildly rigorously, is beyond me.

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