= Solution
Use the same specified homogeneous Newtonian background and <Jeans swindle>, but describe <stars> by a mass-normalized <galactic distribution function>. The <Collisionless Boltzmann equation> is $\partial_tf+\mathbf v\cdot\nabla_xf-\nabla\phi\cdot\nabla_vf=0$. Let $f=f_0(\mathbf v)+f_1e^{i(\mathbf k\cdot\mathbf x-\omega t)}$ and similarly perturb the <gravitational potential>. To first order,
$$
(\mathbf k\cdot\mathbf v-\omega)f_1-\phi_1\mathbf k\cdot\nabla_vf_0=0,\qquad-k^2\phi_1=4\pi G\int f_1\,d^3v.
$$
Eliminating $f_1$ and the nonzero <gravitational potential> amplitude gives the <collisionless Jeans dispersion relation>
$$
\boxed{1+\frac{4\pi G}{k^2}\int\frac{\mathbf k\cdot\nabla_vf_0}{\mathbf k\cdot\mathbf v-\omega}\,d^3v=0.}
$$
For growing modes $\operatorname{Im}\omega>0$ there is no real-velocity pole. Real and damped frequencies require a causal contour prescription or analytic continuation; one must not simply integrate through a pole without specifying it.
Take $b>0$ in the <Cauchy velocity distribution>. Its normalization is $\int f_0d^3v=\rho_0$. Orient $\mathbf k$ along the $u=v_z$ axis and integrate over the perpendicular <velocities>:
$$
F(u)=\int f_0\,dv_xdv_y=\frac{\rho_0b}{\pi(u^2+b^2)}.
$$
At the marginal mode $\omega=0$, $F'(u)/u=-2\rho_0b/[\pi(u^2+b^2)^2]$, with its finite limiting value at zero. The provided <Gamma function> <integral> at power two gives
$$
\int_{-\infty}^{\infty}\frac{F'(u)}u\,du=-\frac{2\rho_0b}{\pi}\frac{\pi}{2b^3}=-\frac{\rho_0}{b^2}.
$$
Consequently
$$
\boxed{k_J=\frac{\sqrt{4\pi G\rho_0}}b.}
$$
To check that this really separates growing modes, put $c=\omega/k$ in the upper half-plane. Residue <integration>, or <integration> by parts followed by the Cauchy resolvent, gives $\int F'(u)/(u-c)\,du=\rho_0/(c+ib)^2$. The upper-half-plane solution has $\omega=i(\sqrt{4\pi G\rho_0}-kb)$, which grows exactly for $0<k<k_J$. At the threshold it approaches the marginal mode continuously. Although $b$ plays the threshold role of a <velocity> scale, this distribution's <second moment> diverges: the radial integrand for $\int v^2 f_0d^3v$ tends to a nonzero constant at large <speed>. It is therefore incorrect to identify $b^2$ with a finite <velocity dispersion> or to infer this result by substituting an rms <speed> into a fluid formula.
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