The energy spectrum \(\varepsilon(p)\) of a gas composed of mutually interacting, spin-half fermions is given by (Galitskii, 1958; Mohling, 1961)

\[
\begin{aligned}
\frac{\varepsilon(p)}{p_{F}^{2} / 2 m} \simeq x^{2}+\frac{4}{3 \pi}\left(k_{F} aight)+ & \frac{4}{15 \pi^{2}}\left(k_{F} aight)^{2} \\
\times & {\left[11+2 x^{4} \ln \frac{x^{2}}{\left|x^{2}-1ight|}-10\left(x-\frac{1}{x}ight) \ln \left|\frac{x+1}{x-1}ight|ight.} \\
& \left.\quad-\frac{\left(2-x^{2}ight)^{5 / 2}}{x} \ln \left(\frac{1+x \sqrt{ }\left(2-x^{2}ight)}{1-x \sqrt{ }\left(2-x^{2}ight)}ight)ight],
\end{aligned}
\]

where \(x=p / p_{F} \leq \sqrt{ } 2\) and \(k=p / \hbar\). Show that, for \(k\) close to \(k_{F}\), this spectrum reduces to

\[
\begin{aligned}
\frac{\varepsilon(p)}{p_{F}^{2} / 2 m} \simeq x^{2} & +\frac{4}{3 \pi}\left(k_{F} aight) \\
& +\frac{4}{15 \pi^{2}}\left(k_{F} aight)^{2}\left[(11-2 \ln 2)-4(7 \ln 2-1)\left(\frac{k}{k_{F}}-1ight)ight] .
\end{aligned}
\]

Using equations (11.8.10) and (11.8.11), check that this expression leads to the result

\[
\frac{m^{*}}{m} \simeq 1+\frac{8}{15 \pi^{2}}(7 \ln 2-1)\left(k_{F} aight)^{2}
\]