= Finite-temperature electron equation of state
{title2=$P_e(\rho,T)$}
Neglecting interactions and thermal pairs, the <Fermi-Dirac distribution> $f(p)=[\exp((\varepsilon(p)-\mu_{\rm kin})/(k_BT))+1]^{-1}$ with kinetic energy $\varepsilon(p)=\sqrt{m_e^2c^4+p^2c^2}-m_ec^2$ determines the <Electron> contribution to a stellar <equation of state>. The kinetic <chemical potential> is fixed by the electron <number density>, not an independently chosen pressure fraction:
$$
n_e=\frac1{\pi^2\hbar^3}\int_0^\infty p^2f(p)\,dp,\qquad
P_e=\frac1{3\pi^2\hbar^3}\int_0^\infty\frac{p^4c^2}{\sqrt{m_e^2c^4+p^2c^2}}f(p)\,dp.
$$
The electron kinetic <internal energy> per volume is $(\pi^2\hbar^3)^{-1}\int_0^\infty p^2\varepsilon(p)f(p)\,dp$. In the nondegenerate limit $P_e=n_ek_BT$; at zero <temperature> the integrals become the <equation of state of a cold electron gas>. These are limits of one component and must not be added together as independent electron pressures. When <electron-positron thermal pair abundance> matters, both charge species must be included, with charge neutrality fixing their net density.
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