ゾンマーフェルト展開
( ε 2 π ) 4 { 1 1 + e β ( ε − μ ) − θ ( − ε ) } = 1 5 ! ( μ 2 π ) 5 + 1 3 ! ( μ 2 π ) 3 T 2 4 ! + ( μ 2 π ) 7 8 T 4 6 ! , {\displaystyle \int _{-\infty }^{\infty }{\frac {d\epsilon }{2\pi }}{\frac {1}{4!}}\left({\frac {\epsilon }{2\pi }}\right)^{4}\left\{{\frac {1}{1+e^{\beta (\epsilon -\mu )}}}-\theta (-\epsilon )\right\}={\frac {1}{5!}}\left({\frac {\mu }{2\pi }}\right)^{5}+{\frac {1}{3!}}\left({\frac {\mu }{2\pi }}\right)^{3}{\frac {T^{2}}{4!}}+\left({\frac {\mu }{2\pi }}\right){\frac {7}{8}}{\frac {T^{4}}{6!}},}
∫ − ∞ ∞ d ε 2 π 1 5 ! ( ε 2 π ) 5 { 1 1 + e β ( ε − μ ) − θ ( − ε ) } = 1 6 ! ( μ 2 π ) 6 + 1 4 ! ( μ 2 π ) 4 T 2 4 ! + 1 2 ! ( μ 2 π ) 2 7 8 T 4 6 ! + 31 24 T 6 8 ! .