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I need help evaluating the following integral:

$$\int \delta(x + uxy - a)\delta(y + vxy - b)p(x,y)dxdy$$

where $\delta(x)$ is Dirac-delta function, and $p(x,y)$ is some sufficiently well behaved function. The parameters $a,b,u,v$ are all real.

I'd know how to do this if the $x$ (or the $y$) was in only one of the delta functions. The problem is that they are "coupled" and I'm not sure how to proceed. I do know that the result is some number times $f$ evaluated at the values of $x,y$ that make the arguments to the delta functions zero. I just don't know what the front factor is.

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1 Answers1

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There is a property of the Dirac Delta function one can use:

$$\delta\left(f\left(x,y\right)\right)\delta\left(g\left(x,y\right)\right)=\frac{\delta\left(x-x_{0}\right)\delta\left(y-y_{0}\right)}{\left|\frac{\partial f}{\partial x}\frac{\partial g}{\partial y}-\frac{\partial g}{\partial x}\frac{\partial f}{\partial y}\right|} $$

where $(x_0,y_0)$ is the (unique) common zero of $f$ and $g$ (if the zero is not unique the formula is more complicated). I don't have a reference now, but I got this formula from https://math.stackexchange.com/a/619471/10063. (By the way, if know where I can find a proof of the general $n$-dimensional generalization of this formula, please drop it in a comment).

It follows that:

$$\delta\left(x+uxy-a\right)\delta\left(y+vxy-b\right)=\frac{\delta\left(x-x_{0}\right)\delta\left(y-y_{0}\right)}{1+vx+uy}$$

and immediately, the value of the integral is:

$$\frac{p(x_0,y_0)}{1+vx_0 + uy_0}$$

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