Appendix A

The integrals in (4.61) are of the form

$$I_1 = \int_\Delta^{\infty} (\epsilon-\Delta)^{1/2} \exp\left(\dis... ...+\Delta)^{1/2} \exp\left(\displaystyle\frac{-\epsilon}{k_{B}T}\right) d\epsilon$$ (6.1)

Setting $t = \displaystyle \frac{\epsilon-\Delta}{k_BT}$ in $I_1$ and $t = \displaystyle \frac{\epsilon+\Delta}{k_BT}$ in $I_2$, the integrals reduce to

$$I_1 = (k_BT)^{3/2} \exp\left(\displaystyle\frac{-\Delta}{k_{B}T}\right) \int_{0}^{\infty} t^{1/2} \exp{(-t)} dt,$$ (6.2)

$$I_2 = (k_BT)^{3/2} \exp\left(\displaystyle\frac{\Delta}{k_{B}T}\right) \int_{\Delta/k_BT}^{\infty} t^{1/2} \exp{(-t)} dt$$ (6.3)

which gives

$$I_1 = (k_BT)^{3/2} \cdot\exp\left(\displaystyle\frac{-\Delta}{k_{... ...ft(\displaystyle\frac{\Delta}{k_{B}T}\right) \Gamma\left(\frac{3}{2},-z\right).$$ (6.4)

Similarly, the integrals in (4.66) are of the form

$$I_3 = \int_\Delta^{\infty} \sqrt{\epsilon (\epsilon-\Delta)} \exp... ...silon+\Delta)} \exp\left(\displaystyle\frac{-\epsilon}{k_{B}T}\right) d\epsilon$$ (6.5)

Setting $t = \displaystyle \frac{\epsilon-\Delta}{k_BT}$ in $I_1$ and $t = \displaystyle \frac{\epsilon+\Delta}{k_BT}$ in $I_2$, the integrals reduces to

$$I_3 = (k_BT)^{2} \exp\left(\displaystyle\frac{-\Delta}{k_{B}T}\right) \int_{0}^{\infty} \sqrt{t(t+\Delta/k_BT)} \exp{(-t)} dt,$$ (6.6)

$$I_4 = (k_BT)^{2} \exp\left(\displaystyle\frac{\Delta}{k_{B}T}\right) \int_{\Delta/k_BT}^{\infty} \sqrt{t(t-\Delta/k_BT)} \exp{(-t)} dt$$ (6.7)

which gives

$$I_3 = (k_BT)^{2}\exp{\left(\frac{-\Delta}{k_BT}\right)}\left[ \fr... ...\left(\frac{\Delta}{2k_BT}\right)} K_1\left(\frac{\Delta}{2k_BT}\right)\right],$$ (6.8)

$$I_4 = (k_BT)^{2}\exp{\left(\frac{\Delta}{k_BT}\right)}\left[ \fra... ...{\left(\frac{\Delta}{2k_BT}\right)} K_1\left(\frac{\Delta}{2k_BT}\right)\right]$$ (6.9)

Here, $K_1$ denotes the modified Bessel function of the second kind.