$n$-dimensional integral of delta function

Question

As part of an exercise for quantum mechanics I have to solve the following integral:

$$ \int \delta\left( a- \frac{x^2}{b} \right)\; \mathrm{d}^nx = a^{\frac{n}{2}-1} \cdot b^{\frac{n}{2}} .\tag{1}$$

I do have the solution to the integral, but I can't figure out how to get to it. I know I have to use the following identity:

$$\delta\left( x^2 - a^2 \right) = \frac{1}{2|a|}\left[ \delta \left(x-a \right) + \delta \left( x+a \right) \right] \tag{2}$$

It would be great if someone could give me some hints.

edit 1:

By using the identity from above I get: $$\delta\left(a-\frac{x^2}{b}\right) = \frac{1}{2 \sqrt{a}} \left[\delta\left(\sqrt{a}-\frac{x}{\sqrt{b}}\right) + \delta\left(\sqrt{a}+\frac{x}{\sqrt{b}}\right)\right]\tag{3}$$

I wasn't sure as the delta function is very common in physics. — kaos, Dec 01 '16 at 11:23
Yes, the Delta function is seen in physics, however, your question is about doing an integral, which makes it not about physics but doing maths. Hence my suggestion. — Kyle Kanos, Dec 01 '16 at 12:08
Hint: Rewrite in spherical coordinates. There is a factor ${\rm Vol}(S^{n-1})$ missing in eq. (1). — Qmechanic, Dec 01 '16 at 12:37

score 1 · Answer 1 · answered Dec 01 '16 at 11:42

1

Expand the argument of the $\delta$-function to the first order in $x$ around the point where it vanishes.

answered Dec 01 '16 at 11:42

Andrey Feldman

111

I just used the identity stated above on my problem. Is this what you ment by expanding? – kaos Dec 01 '16 at 11:58
@kaos I meant that it is the way to derive (2) and the case of arbitrary function $f(x)$ under the $\delta$. – Dec 01 '16 at 13:07

score 1 · Accepted Answer · edited Apr 13 '17 at 12:21

Assume that $a,b>0$. Define for later convenience $$f(r)~:=~\frac{r^2}{b}-a.$$ Then OP's integral becomes $$I~:=~\int_{\mathbb{R}^n} \!\mathrm{d}^nx~\delta\left(\frac{x^2}{b}-a\right) ~\stackrel{\text{Spher. coord.}}{=}~{\rm Vol}(S^{n-1})\int_{\mathbb{R}_+} \!\mathrm{d}r~r^{n-1}~\delta(f(r))$$ $$~=~{\rm Vol}(S^{n-1})\int_{\mathbb{R}_+} \!\mathrm{d}r~r^{n-1}~\sum_{r_0>0}^{f(r_0)=0}\frac{1}{|f^{\prime}(r)|} \delta(r\!-\!r_0)~\stackrel{r_0=\sqrt{ab}}{=}~{\rm Vol}(S^{n-1})\int_{\mathbb{R}_+} \!\mathrm{d}r~r^{n-1}~\frac{b}{2r}\delta(r\!-\!r_0)$$ $$~=~{\rm Vol}(S^{n-1}) \frac{r_0^{n-2}b}{2} ~=~\frac{\pi^{\frac{n}{2}}}{\Gamma\left(\frac{n}{2}\right)} a^{\frac{n}{2}-1}b^{\frac{n}{2}},$$ where we have used the formula for the volume of an $n$-sphere, and the substitution rule for Dirac delta distributions, cf. e.g. this & this math.SE posts.

$n$-dimensional integral of delta function

edit 1:

2 Answers2

Linked