Properties of $C(t)$

One can write

$$C(t) = \int_0^{\infty} d\omega \rho(\omega) \left[\coth\left(\beta\omega/2\right) \cos(\omega t) - i \sin(\omega t)\right],$$ (57)

where we used the useful identity

$$\coth \left({\beta\omega}/{2}\right) = 1+2n_B(\omega).$$ (58)

Calculation of the integral with $\rho (\omega )$ given by Eq.(7.54) yields

$$C(t) = {2}\alpha \omega_c^{1-s} \beta^{-(s+1)}\times$$ (59)

$$\Gamma(s+1)\left[ \zeta\left(s+1,\frac{1+\beta\omega_c-i\omega_ct... ...ega_c}\right) +\zeta\left(s+1,\frac{1+i\omega_ct}{\beta\omega_c}\right)\right],$$

where $\Gamma$ is the Gamma function and

$$\zeta(z,u)\equiv \sum_{n=0}^{\infty} \frac{1}{(n+u)^z},\quad u\ne 0,-1,-2,...$$ (60)

is the generalised Zeta function (cf. W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorem for the Special Functions of Mathematical Physics, Springer, Berlin 1966). The zero temperature limit is obtained either from the $\beta\to \infty$ limit of Eq.(7.59) or directly by calculating the integral,

$$C(t) = {2}\alpha \omega_c^{s+1} \Gamma(s+1) \left(1+i\omega_c t\right)^{-(s+1)}.$$ (61)