= Collisionless Jeans dispersion relation
{title2=$1+\frac{4\pi G}{k^2}\int\frac{\mathbf k\cdot\nabla_v f_0}{\mathbf k\cdot\mathbf v-\omega}\,d^3v=0$}
Linearize the <Collisionless Boltzmann equation> and the perturbation Poisson equation about a homogeneous <velocity> distribution $f_0$. A <plane wave> gives $f_1=\phi_1\mathbf k\cdot\nabla_vf_0/(\mathbf k\cdot\mathbf v-\omega)$. Integrating this density and using $-k^2\phi_1=4\pi G\int f_1d^3v$ yields $1+(4\pi G/k^2)\int[\mathbf k\cdot\nabla_vf_0/(\mathbf k\cdot\mathbf v-\omega)]d^3v=0$. Growing modes have $\operatorname{Im}\omega>0$ and no real-velocity pole; real and damped frequencies require the causal continuation prescription. The Newtonian homogeneous background requires the <Jeans swindle>.
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