Several posts discuss the representation of the delta function in polar coordinates in 2D or 3D, e.g. Dirac delta in polar coordinates or Delta function at the origin in polar coordinates
Does anyone have a reference for a representation of $\delta(|\mathbf x|)$ or $\delta(r)$ in general dimension $n \geq 2$? I am guessing the following:
$$ \delta(r)/r^{n-1} = s_{n-1} \delta(\mathbf x), \qquad s_{n-1} := {2\pi^{n/2} \over \Gamma(n/2)} \text{ (area of unit sphere)} $$
or equivalently (?)
$$
\delta(|\mathbf x|) = s_{n-1} |\mathbf x|^{n-1} \delta(\mathbf x)
$$
My proof: For $r_0 \neq 0$ the general curvilinear correspondence is
$$ \delta(\mathbf {x – x}_0) \equiv \delta(r – r_0) \delta(\mathbf u – \mathbf u_0) / r^{n-1}, \qquad \mathbf {x, x}_0\in\mathbb R^n\quad r,r_0>0\quad \mathbf{u, u}_0\in\mathbb S^{n-1} \text{ (unit sphere)} $$
The case $\mathbf x_0 = \mathbf 0 \Leftrightarrow (r_0 = 0, \mathbf u_0\in\mathbb S^{n-1})$ corresponds to a singular Jacobian of the transformation and we may "integrate out" all ignorable spherical coordinates $\mathbf u$:
$$ \begin{aligned}\delta(\mathbf x) \int_{\mathbb S^{n-1}} d\mathbf u
& = \delta(r) / r^{n-1} \int_{\mathbb S^{n-1}} \delta(\mathbf u – \mathbf u_0) d\mathbf u, \qquad\text{i.e.}
\\
s_{n-1} \delta(\mathbf x) & = \delta(r) / r^{n-1}
\end{aligned} $$
as required.
Best Answer
I will write $\omega_d := s_{d-1} = \frac{2\,\pi^{d/2}}{\Gamma(d/2)}$.
If $f(x) = \tilde{f}(|x|)$ with $\tilde{f}\in L^1_{\mathrm{loc}}$, then for any $\varphi\in C^\infty(\mathbb{R})$, $$ \langle \tilde{f},\varphi\rangle = \int_{\mathbb{R}_+} \tilde{f}(r)\,\varphi(r)\,\mathrm{d}r = \frac{1}{\omega_d}\int_{\mathbb{R}^d} f(x)\frac{\varphi(|x|)}{|x|^{d-1}}\,\mathrm{d}x = \frac{1}{\omega_d}\left\langle f, \frac{\varphi(|x|)}{|x|^{d-1}}\right\rangle $$ so if we want to define a generalization of the notion of radial change of variable, we might want to define $\tilde{f}(r)$ by setting $$ \langle \tilde{f},\varphi\rangle := \frac{1}{\omega_d}\left\langle f, \frac{\varphi(|x|)}{|x|^{d-1}}\right\rangle $$ in the general case. In particular, we will have the relation $\omega_d\,|x|^{d-1} \tilde{f}(|x|) = f(x)$. Remark however that the above definition does not make sense if we replace $f$ by $\delta_0$. $$ \frac{\delta_0(x)}{|x|^{d-1}} \text{ is not a priori a well defined distribution} $$ Defining $g(r) = \tilde{f}(|r|)$ for $r\in\mathbb{R}$, we can however try to solve the equation $\omega_d\,r^{d-1} g(r) = \delta_0$. Taking the Fourier transform ($\hat{g}(y) = \int_{\mathbb{R}} g(x)\,e^{-2i\pi x\cdot y}\,\mathrm{d}x$) we get $$ \frac{\omega_d}{(-2i\pi)^{d-1}} \hat{g}^{(d-1)} = 1 $$ so by integrating $d-1$ times, we get $$ \hat{g}(y) = \frac{(-2i\pi)^{d-1}}{\omega_d} \left(\frac{y^{d-1}}{(d-1)!} + \sum_{k=1}^{d-1} a_k\,x^{k-1}\right) $$ so that (if I did not messed up with the power on the $-1$) $$ g(r) = \frac{(-1)^{d-1}}{\omega_d} \left(\frac{\delta_0^{(d-1)}}{(d-1)!} + \sum_{k=1}^{d-1} C_k\,\delta_0^{(k-1)}\right) $$ (where $\delta_0^{(n)}$ is the $n$-th derivative of the Dirac delta centered in $0$) and we can see that there is no uniqueness of $\delta_0(r)/r^{d-1}$ (which is natural since $x^n \delta_0^{(n-1)} = 0$). However, the Dirac is homogeneous, and thus so is its radial representation. Therefore $$ g = \frac{(-1)^{d-1}}{\omega_d\,(d-1)!} \,\delta_0^{(d-1)} $$ which can be written in a more informal way with $r=|x|$ $$\boxed{ \delta_0(x) = \frac{(-1)^{d-1}}{\omega_d\,(d-1)!} \,\delta_0^{(d-1)}(r) }$$
Remark: Of course, we can now choose to define $$ \frac{\delta_0(r)}{r^{d-1}} := \frac{(-1)^{d-1}}{\,(d-1)!} \,\delta_0^{(d-1)}(r) $$ as the homogeneous solution of the equation $r^{d-1} g(r) = \delta_0$, and so we find as you was writing $\omega_d\,\delta_0(x) = \frac{\delta_0(r)}{r^{d-1}}$, but now we know the true meaning of this notation.