Table 1: Summary of analytic solutions for the polarisation as discussed in Sect. 2.
Polarisation $\chi$ for General expression for $\chi$ Approximation for $\chi$

T=0 $\left(2\sum_{\nu=1}^{\frac{\mu_{\rm e}^2-m^2}{2eB}}\sqrt{1-\frac{2\nu eB}{\mu_{\rm e}^2-m^2}}+1\right)^{-1}$ ${3eB}/(2\mu_{\rm e}^2)$ for $\mu_{\rm e}\gg\sqrt{2eB},m$

$T \gg 2eB \gg \mu, m$ $\left(3\frac{T^2}{eB}\frac{\zeta(3)}{\ln{2}}-1\right)^{-1}$ ${eB\ln{2}}/(3\zeta(3)T^2)$ as $T^2 \gg 2eB$

$m \gg T \gg 2eB, \mu_{\rm e}=0$ $\left(\frac{2T}{eB}\frac{mK_0[m/T]+T K_1[m/T]}{K_1[m/T]}-1\right)^{-1}$ eB/(2mT) for $m \gg T$

$2eB \gg T \gg m, \mu$ $\left(1+\frac{4}{\ln(2)}\sqrt{\frac{\pi T}{4eB}}\exp(-\sqrt{2eB}/T) \right)^{-1}$ ...

**Table 1:** Summary of analytic solutions for the polarisation as discussed in Sect. 2.
Polarisation $\chi$ for	General expression for $\chi$	Approximation for $\chi$
T=0	$\left(2\sum_{\nu=1}^{\frac{\mu_{\rm e}^2-m^2}{2eB}}\sqrt{1-\frac{2\nu eB}{\mu_{\rm e}^2-m^2}}+1\right)^{-1}$	${3eB}/(2\mu_{\rm e}^2)$ for $\mu_{\rm e}\gg\sqrt{2eB},m$
$T \gg 2eB \gg \mu, m$	$\left(3\frac{T^2}{eB}\frac{\zeta(3)}{\ln{2}}-1\right)^{-1}$	${eB\ln{2}}/(3\zeta(3)T^2)$ as $T^2 \gg 2eB$
$m \gg T \gg 2eB, \mu_{\rm e}=0$	$\left(\frac{2T}{eB}\frac{mK_0[m/T]+T K_1[m/T]}{K_1[m/T]}-1\right)^{-1}$	eB/(2mT) for $m \gg T$
$2eB \gg T \gg m, \mu$	$\left(1+\frac{4}{\ln(2)}\sqrt{\frac{\pi T}{4eB}}\exp(-\sqrt{2eB}/T) \right)^{-1}$	...

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