content_image/366183.jpg content_image/366184.jpg content_image/366180.jpg content_image/366181.jpg content_image/366178.jpg content_image/366179.jpg content_image/366177.jpg
where Q denotes an arbitrary operator and C the Hankel path in the complex plane. We obtain [13, 48] 

<img src='content_image/366183.jpg'>

<img src='content_image/366184.jpg'>

where 

<img src='content_image/366180.jpg'>

<img src='content_image/366181.jpg'>

and ν$_{q}$ is given by Eq. (42) and (66) for ideal gases and harmonic oscillators, respectively. 

## 2. The normal average 

Next we discuss evaluations by the normal average given by 

By using Eqs. (B3) and (B4), we obtain 

<img src='content_image/366178.jpg'>

<img src='content_image/366179.jpg'>

<img src='content_image/366177.jpg'>

