# Dirac delta function composition

I've seen following identity before $$\int f(x)\delta(g(x))dx=\sum_i\frac{f(x_i)}{|g'(x_i)|} \tag{1}$$ Where $x_i$ are the roots to $g(x)$, but I've also seen cases where it has been written as $$\int f(x)\delta(g(x))dx=\frac{f(x)}{|g'(x)|} \tag{2}$$ Where you don't take the roots into account. (e.g check out equation (23) here)

Now to my question: is (2) some kind of special case of (1)?

Example: Suppose I have the following equation $$\int x \delta(x+\sqrt{x^2+y^2+m^2-2xy\cos\theta}-y-m)dx$$ Where both $y,m$ are assumed to be real and positive.

Hence $f(x)=x$ and $g(x)=x+\sqrt{x^2+y^2+a^2-2xy\cos\theta}-y-a$ which has a root at $$x_0=\frac{my}{m+y-y\cos\theta}=\frac{y}{1+\dfrac{y}{m}(1-\cos\theta)}$$ The derivative of $g(x)$ is $$g'(x)=1+\dfrac{x-y\cos\theta}{\sqrt{x^2+y^2+m^2-2xy\cos\theta}}$$ Is it equally valid to just write that \begin{align} \int x \delta(x+\sqrt{x^2+y^2+m^2-2xy\cos\theta}-y-m)dx=\frac{x}{1+\dfrac{x-y\cos\theta}{\sqrt{x^2+y^2+m^2-2xy\cos\theta}}} \end{align} which would correspond to equation (2), as it is to write \begin{align} \int x \delta(x+\sqrt{x^2+y^2+m^2-2xy\cos\theta}-y-m)dx= \frac{y}{1+\dfrac{y}{m}(1-\cos\theta)} \frac{1}{1+\dfrac{\dfrac{y}{1+\dfrac{y}{m}(1-\cos\theta)}-y\cos\theta}{\sqrt{\Big(\dfrac{y}{1+\dfrac{y}{m}(1-\cos\theta)}\Big)^2+y^2+m^2-2\Big(\dfrac{y}{1+\dfrac{y}{m}(1-\cos\theta)}\Big)y\cos\theta}}} \end{align} Which would correspond to using equation (1)?

• Where did you find the identity (2)? I may be wrong, but sifting property of delta makes the integrals into discrete values, leading me to think that (1) might be the right identity. Commented Jan 12, 2018 at 16:12
• I imagine the second author is being sloppy and either assuming $g$ only has one root, or assuming the unspecified domain of integration only includes one root, and calling it $x.$ Under those assumptions and notational abuses, yes, (2) is a special case of (1). Commented Jan 12, 2018 at 16:13
• Yes in the cases I've seen it $g(x)$ only has one root. Commented Jan 12, 2018 at 16:18

You are integrating over $x$, so by definition your result cannot depend on $x$ anymore, it can only depend on some constant, say $x_0$.
This, your equation $1$ is correct, and $2$ is incorrect - it should be $x_0$ instead of $x$, and there is an implicit assumption that there is only one root in the domain of integration.
Your results from your example, are both incorrect for the same reason - they should not include $x$, just $x_0$. If you only have a single root, the corrected equation $2$ and equation $1$ are of course identical.
• It's no more wrong to write $\int \delta(g(x))\,dx=1/g'(x)$ than it is to write $\int^x\sin x\,dx=-\cos x.$ Which is to say, they're both bad notations, and you might consider them wrong, but some people definitely write that way. Commented Jan 13, 2018 at 3:47
• It certainly led OP to think it should be a function of $x$, which is not so good Commented Jan 13, 2018 at 4:45