La fonction de partition dans l'ensemble canonique d'un gaz parfait relativiste à 1 particule est :
où
est l'énergie cinétique de la particule. En séparant les variables, on obtient
![{\displaystyle Z(T,V,1)={\frac {4\pi V}{h^{3}}}\exp u\int _{0}^{+\infty }\exp \left[-u{\sqrt {1+\left({\frac {p}{mc}}\right)^{2}}}\right]p^{2}\,{\text{d}}p}](https://wikimedia.org/api/rest_v1/media/math/render/svg/153b6ff1519cc48b3455ee056d05f5b9a6921665)
,
en posant 
. Le changement de variable
donne
.
Par parties[3]
![{\displaystyle I=\underbrace {{\frac {2}{3}}{\Big [}(\zeta ^{2}-1)^{3/2}\exp(-u\zeta ){\Big ]}_{1}^{+\infty }} _{=0}+{\frac {2}{3}}\underbrace {\int _{1}^{+\infty }\left(\zeta ^{2}-1\right)^{3/2}\exp(-u\zeta )\,{\text{d}}\zeta } _{={\frac {3}{2}}K_{2}(u)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/63e07a7b6ae24997004dcb6406c7579239bea300)
.
Le résultat se trouve en utilisant l'absence d'interaction entre particules :
![{\displaystyle Nmc^{2}\left[{\frac {K_{1}(u)}{K_{2}(u)}}+{\frac {3}{u}}-1\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ef8dac2eceae86f62d9b18373c93b0677de24b3)
![{\displaystyle -NkT\left\{\log \left[{\frac {4\pi V}{N}}\left({\frac {mc}{h}}\right)^{3}{\frac {K_{2}(u)}{u}}\right]+1\right\}-Nmc^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/44455003f7c157b3fe1f7ea1ee5e70c51150ab08)
![{\displaystyle Nmc^{2}\left[{\frac {K_{1}(u)}{K_{2}(u)}}+{\frac {4}{u}}-1\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/554792184e830ef79ea50ffcb109b61e206d2b54)
![{\displaystyle -NkT\log \left[{\frac {4\pi V}{N}}\left({\frac {mc}{h}}\right)^{3}{\frac {K_{2}(u)}{u}}\right]-Nmc^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a2896f57b3890f1fea386803f4e452186a4108a7)
![{\displaystyle Nku\left\{u+{\frac {3}{u}}-{\frac {K_{1}(u)}{K_{2}(u)}}\left[3+u{\frac {K_{1}(u)}{K_{2}(u)}}\right]\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c6442b27e94490ebb18a83b924a12ab061eea89)
![{\displaystyle Nku\left\{u+{\frac {4}{u}}-{\frac {K_{1}(u)}{K_{2}(u)}}\left[3+u{\frac {K_{1}(u)}{K_{2}(u)}}\right]\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c3acdf3571b25a664e9e1631ad700b588fd7e1e)
![{\displaystyle -kT\log \left[{\frac {4\pi V}{N}}\left({\frac {mc}{h}}\right)^{3}{\frac {K_{2}(u)}{u}}\right]-mc^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/17de38ad8631d1857186e28659d20365dc906c41)
![{\displaystyle Nk\left\{\log \left[{\frac {4\pi V}{N}}\left({\frac {mc}{h}}\right)^{3}{\frac {K_{2}(u)}{u}}\right]+u{\frac {K_{3}(u)}{K_{2}(u)}}\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c45ec3162e2f3e8e81d3c4d8809ac0309f62db88)
