Simple Dirac-Delta Integral

Question

What is $\iint \delta (ax^2+by-c) \, dx \, dy$?

I am aware of the translational and scaling rules, as well as $\int\delta (x) \,dx=1$.

Answer 1

It's ill-defined. The best you can do is take a continuous compactly supported $\varphi:\mathbb R^2 \to \mathbb R$ and compute : \begin{align} \iint \varphi(x,y) \delta( ax^2 + by + c) \text{d}x \text{d}y &= \frac{1}{b} \int \varphi\left(x,-\frac{c+ax^2}{b}\right)\text{d}x \end{align}

Edit : We can actually do better : the distribution $\delta(ax^2 + by +c)$ is positive and of order $0$, so it corresponds to a positive Radon measure $\mu$. In this sens $\int \varphi \text d\mu$ is well-defined for any measurable positive function (but it might be infinite). In particular, for $\varphi = 1$ the constant function, we get : $$\iint \delta(ax^2 + by +c)\text dx \text dy = +\infty$$