= Solution
For the real decaying Bessel form used here, take $k>0$. The original PDF has $K_m(r)$ as the last factor in its auxiliary differential equation, although both differentiated factors have argument $kr$; that last argument must also be $kr$ for the stated <Modified Bessel differential equation>.
Put $B(r)=K_m(kr)$. The corrected identity is $B''=-B'/r+(k^2+m^2/r^2)B$. Since the field is radial, the <radial Laplacian> gives
$$
\begin{aligned}
(\Delta-k^2)(r^{-p}B)
&=r^{-p}\left[B''+\frac{2-2p}{r}B'+\left(\frac{p(p-1)}{r^2}-k^2\right)B\right]\\
&=r^{-p}\left[\frac{1-2p}{r}B'+\frac{p(p-1)+m^2}{r^2}B\right].
\end{aligned}
$$
Away from the point source this must vanish. For $k>0$, a constant ratio of $rB'$ to $B$ would make $B$ a pure power, incompatible with the $k^2r^2$ term of its equation. Therefore both displayed coefficients must vanish:
$$
\boxed{p=\frac12,\qquad m^2=\frac14,\quad\text{so }m=\pm\frac12.}
$$
The two orders give the same <Modified Bessel function of the second kind>: $K_{-1/2}(z)=K_{1/2}(z)=\sqrt{\pi/(2z)}e^{-z}$.<NIST Digital Library of Mathematical Functions>[§10.39, equation 10.39.2, equality of the two half-integer orders and their elementary form]{ref}
The source describes $k$ merely as real, but the following displayed real square-root formula implicitly needs $k>0$. For the operator with arbitrary real $k$, its decaying fundamental solution instead uses $\kappa=|k|$; $k=0$ is obtained by a limit. The unmodified real $A K_m(kr)$ ansatz is not a real Bessel formula for all negative $k$.
Back to article page
