ゾンマーフェルト展開
, {\displaystyle \int _{-\infty }^{\infty }{\frac {d\epsilon }{2\pi }}{\frac {1}{2!}}\left({\frac {\epsilon }{2\pi }}\right)^{2}\left\{{\frac {1}{1+e^{\beta (\epsilon -\mu )}}}-\theta (-\epsilon )\right\}={\frac {1}{3!}}\left({\frac {\mu }{2\pi }}\right)^{3}+\left({\frac {\mu }{2\pi }}\right){\frac {T^{2}}{4!}},}
∫ − ∞ ∞ d ε 2 π 1 3 ! ( ε 2 π ) 3 { 1 1 + e β ( ε − μ ) − θ ( − ε ) } = 1 4 ! ( μ 2 π ) 4 + 1 2 ! ( μ 2 π ) 2 T 2 4 ! + 7 8 T 4 6 ! , {\displaystyle \int _{-\infty }^{\infty }{\frac {d\epsilon }{2\pi }}{\frac {1}{3!}}\left({\frac {\epsilon }{2\pi }}\right)^{3}\left\{{\frac {1}{1+e^{\beta (\epsilon -\mu )}}}-\theta (-\epsilon )\right\}={\frac {1}{4!}}\left({\frac {\mu }{2\pi }}\right)^{4}+{\frac {1}{2!}}\left({\frac {\mu }{2\pi }}\right)^{2}{\frac {T^{2}}{4!}}+{\frac {7}{8}}{\frac {T^{4}}{6!}},}
∫ − ∞ ∞ d ε 2 π 1 4 !