松原振動数
n B ′ ( ξ ) = − β 4 c s c h 2 β ξ 2 , {\displaystyle n_{B}^{\prime }(\xi )=-{\frac {\beta }{4}}\mathrm {csch} ^{2}{\frac {\beta \xi }{2}},}
n F ′ ( ξ ) = − β 4 s e c h 2 β ξ 2 . {\displaystyle n_{F}^{\prime }(\xi )=-{\frac {\beta }{4}}\mathrm {sech} ^{2}{\frac {\beta \xi }{2}}.}
積で表すと、
n η ′ ( ξ ) = − β n η ( ξ ) ( 1 + η n η ( ξ ) ) . {\displaystyle n_{\eta }^{\prime }(\xi )=-\beta n_{\eta }(\xi )(1+\eta n_{\eta }(\xi )).}
温度ゼロの極限では、
n η ′ ( ξ ) = η δ ( ξ ) as β → ∞ . {\displaystyle n_{\eta }^{\prime }(\xi )=\eta \delta (\xi ){\text{ as }}\beta \rightarrow \infty .}