We then have \[\begin{split} n&=\!\int\limits_{-\infty}^\infty\!\!\!d\ve\>g\ns_\uar(\ve)\,f(\ve-\mu) \> + \int\limits_{-\infty}^\infty\!\!\!d\ve\>g\ns_\dar(\ve)\,f(\ve-\mu) \\ &=\half\!\int\limits_{-\...We then have n=∞∫−∞d\veg\ns\uar(\ve)f(\ve−μ)+∞∫−∞d\veg\ns\dar(\ve)f(\ve−μ)=\half∞∫−∞d\ve{g(\ve−\mutBH)+g(\ve+\mutBH)}f(\ve−μ)=∞∫−∞d\ve{g(\ve)+(\mutBH)2g″(\ve)+…}f(\ve−μ). We now invoke the Sommerfeld expension to find the temperature dependence: \[\begin{split…
