= Two-dimensional Landau effective-mass relation
{title2=$m^*/m=1+F_1/2$}
For an isotropic spinless continuum <Fermi liquid> with the <Principle of Galilean relativity>, let $D^*=Am^*/(2\pi\hbar^2)$ be the total <density of states> and use a <Landau quasiparticle energy functional> whose double sum has no explicit $1/A$. Then
$$
\frac{m^*}{m}=1+D^*\int_0^{2\pi}\frac{d\phi}{2\pi}\,f(\phi)\cos\phi.
$$
Indeed, a small boost has radial displacement $p_0\cos\theta$. The one-<quasiparticle> term costs $Np_0^2/(2m^*)$, and the quadratic interaction multiplies that by $D^*\int f\cos\phi\,d\phi/(2\pi)$, since $\int\cos(\theta+\phi)\cos\theta\,d\theta=\pi\cos\phi$. Matching the <Galilean boost energy of an interacting gas> proves the formula. With <Fourier series> $D^*f=F_0+\sum_{\ell\geq1}F_\ell\cos(\ell\phi)$, the angular average is $F_1/2$. If the functional instead contains $1/A$, use its intensive kernel $Af$ and the <density of states> per area.
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