Is there a physically motivated "trick" to evaluate this convolution?

Question

I'm working on adapting some of the formalism in this paper to a system I'm working with. The part I'm interested in amounts to convolving a density profile $\rho(r)$ with a smoothing kernel $S(r,\epsilon)$. The smoothing kernel used in this particular case is Plummer smoothing:

$$S(r;\epsilon) = \frac{3}{4\pi}\frac{\epsilon^2}{(r^2+\epsilon^2)^{5/2}}$$

and the density profile is left as a general (but spherically symmetric) function. The convolution (incidentally, I'm pretty sure this is a correct use of the term convolution, but would appreciate a confirmation/correction if I'm using it incorrectly) I want to evaluate is:

$$\rho(\vec{r};\epsilon) = \int d\vec{r}' \rho(\vec{r}') S(|\vec{r}-\vec{r}'|;\epsilon)$$

Following along with the paper, this is going to be a lot easier to evaluate in cylindrical coordinates with the vector $\vec{r}$ lying along the $z$-axis, and without too much difficulty I arrive at the first part of his equation (8), but get stuck trying to get the second part. Equation (8) reads:

$$\rho(r,\epsilon) = \frac{3\epsilon^2}{2}\int_{-\infty}^{\infty}dz\int_0^{\infty}dRR\frac{\rho\left(\sqrt{R^2+z^2}\right)}{(R^2+(z-r)^2+\epsilon^2)^{5/2}} \\= \frac{3\epsilon^2}{2}\int_{-\infty}^{\infty}dz\int_0^{\infty}dRR\frac{\rho\left(\sqrt{R^2+(z-r)^2}\right)}{(R^2+z^2+\epsilon^2)^{5/2}}.\tag{8}$$

He points out that "the equality holds because the outer integral is taken over the entire $z$ axis", but I don't see how this follows.

A couple details on the geometry. The cylindrical coordinates are $R$ and $z$ (and $\phi$ I guess, but that just gives a $2\pi$). $r$ is the distance from the origin of the spherically symmetric $\rho(r)$. It follows that:

$$|\vec{r}'| = \sqrt{R^2+z^2}$$

and

$$|\vec{r}'-\vec{r}|^2 = R^2 + (z-r)^2.$$

I thought the solution might be as simple as translating everything along the $z$-axis a bit, something like $\tilde{z}=z+r$, but that doesn't seem to work out. My next hunch is that the "trick" that I need to see has something to do with the behavior of $\lim_{r\to\infty}\rho(r)$: this had better be zero to impose a finite total mass (of course this does not guarantee finite mass, and indeed there are several commonly used density profiles that have unbounded total mass, in practice they get truncated at some $r$, and they always still limit to 0 as $r\to\infty$), but I'm still stuck. So, anyone see what makes that equality true?

Answer 1

Hint: It seems the substitution $z \longrightarrow z^{\prime}=r-z$ will work to prove the second equality of eq. (8). (Geometrically, this corresponds to a reflection of the $z$-axis around $z=\frac{r}{2}$.)

Answer 2

As @Qmechanic points out in his answer, making the substitution $z \rightarrow z - r$ (going from the first version to the second in the identity) works. That's the math answer.

The physical intuition is that, in order for the system to have perfect cylindrical symmetry, it must extend to $z = \pm \infty$. That corresponds to integrating your $z$ variable from $-\infty$ to $+\infty$, as you have. If you do that, then all values of $r$ must behave identically; $\rho(r,\epsilon)$ is independent of $r$. If there were any $r$ dependence left over, not all points along the $z$-axis would be the same, and you would not have cylindrical symmetry.