Consider the "Euclidean propagator",
\begin{align} \Delta_E(x_1-x_2)&\equiv \frac{m}{(2\pi)^2|x_1-x_2|} K_1[m|x_1-x_2|] \tag{*}\label{*}\\ &=\int_{\mathbb{R}^4} \frac{d^4p}{(2\pi)^4}\frac{e^{i p\cdot (x_1-x_2)}}{p^2+m^2}, \end{align} where $K_1$ is a modified Bessel function of the second kind; $m\in \mathbb{R}$; $p,x_1,x_2$ are (real) 4-vectors in Cartesian coordinates; and "$\cdot$" denotes the dot product.
Now consider switching to 4-dimensional spherical coordinates,
\begin{align}
p_1&=|p|\cos(\phi_1)\\
p_2&=|p|\sin(\phi_1)\cos(\phi_2)\\
p_3&=|p|\sin(\phi_1)\sin(\phi_2)\cos(\phi_3)\\
p_4&=|p|\sin(\phi_1)\sin(\phi_2)\sin(\phi_3),\\
\end{align}
where $\phi_1,\phi_2$ range over $[0,\pi]$ while $\phi_3$ ranges over $[0,2\pi]$.
Choosing $\phi_3$ to coincide with the angle between the 4-vectors $p$ and $x_1-x_2$, then yields
\begin{align} \Delta_E (x_1-x_2)&=\frac{\pi}{(2\pi)^4} \int^\infty_0 d|p|\frac{|p|^3}{|p|^2+m^2} \int_0^{2\pi}d\phi_3 e^{i |p||x_1-x_2|\cos{\phi_3}}\\ &= \frac{2\pi^2}{(2\pi)^4} \int^\infty_0 d|p| \frac{|p|^3}{|p|^2+m^2}J_0[|p||x_1-x_2|], \tag{**}\label{**} \end{align} where $J_0$ is a Bessel function of the first kind.
If my manipulations are correct, then $\eqref{**}$ should evaluate to a functional of the modified Bessel function \eqref{*}--which is bounded everywhere except the coincidence limit $x_1\to x_2$. However, Mathematica seems to indicate that \eqref{**} does not converge on $|p|:[0,\infty)$. Have I made an unspotted error in my manipulations or is there some subtlety to evaluating the integral $\eqref{**}$ that I'm missing?