I am trying to evaluate an integral, which includes a delta function, as a power series in a small parameter $\beta$. The integral is of the form
$$ \int d^2\mathbf{k'} \, f(\mathbf{k},\mathbf{k'}) \delta \left(g(\mathbf{k},\mathbf{k'}) \right). $$
Here, $\mathbf{k}=(k_1,k_2)$ is a vector parameter. It happens that $$ g(\mathbf{k},\mathbf{k'}) = k_1 + \beta g_1(\mathbf{k},\mathbf{k}')+O(\beta^2), $$ so the zeroth-order term is simply $$ \delta (k_1) \int d^2\mathbf{k'} \, f(\mathbf{k},\mathbf{k'}) $$ (the result of the integral should itself be thought of a distribution in $\mathbf{k}$). I am having trouble, however, conceptualizing what the first-order (in $\beta$) correction to this should be. In general, we can try something like the following
$$ \int dx \, \delta(g_0(x) + \beta g_1(x))f(x) = \int \left(\frac{dg_0}{dx}\right)^{-1} du \, \left[\delta(u) + \beta\ g_1(g_0^{-1}(u)) \delta'(u) \right] f(g_0^{-1}(u))$$
and then integrate the second term by parts. However, this doesn't work in the above case, because here $g_0$ does not depend on the integration variable $\mathbf{k'}$.
If it's helpful, I can provide explicit forms of $f$ and $g$. But I do have other, similar integrals to evaluate, so I am interested in general features of the problem of finding a power series expansion for a delta function of a complicated argument.