Transforming a sum in an integral I have a second question about the article "Imperfect Bose Gas with Hard-Sphere Interaction". The authors begins with the sum:
$$\frac{{E_2 }}{{E_0 }} = \frac{{16\pi ^2 a^2 \lambda ^2 }}{{V^2 }}\sum\limits_{} {'\frac{{\langle n_\alpha  \rangle \langle n_\gamma  \rangle \langle n_\lambda  \rangle }}{{\frac{1}{2}\left( {k_\alpha ^2  + k_\beta ^2  - k_\gamma ^2  - k_\lambda ^2 } \right)}}\delta \left( {\vec k{}_\alpha   + \vec k{}_\beta   - \vec k{}_\gamma   - \vec k{}_\lambda  } \right)} 
$$
which represents the second order perturbation term of the energy. ${\langle n_\alpha  \rangle }$ is:
$$\langle n_\alpha  \rangle  = \sum\limits_{n = 0}^\infty  {n\left( {ze^{ - \beta \varepsilon _\alpha  } } \right)} ^n /\sum\limits_{n = 0}^\infty  {\left( {ze^{ - \beta \varepsilon _\alpha  } } \right)} ^n  = \frac{{ze^{ - \beta \varepsilon _\alpha  } }}{{1 - ze^{ - \beta \varepsilon _\alpha  } }}
$$
In the sum the terms with a vanishing denominator are omitted. There is also the
restriction ${\vec k{}_\alpha   \ne \vec k{}_\beta  }\ \ $  and ${\vec k{}_\gamma   \ne \vec k{}_\lambda  }\ \ $. Now the passage that is unclear to me is the passage to the integral:
$$\frac{{E_2 }}{{E_0 }} = \frac{{16\pi  a^2 \lambda ^2 }}{{V^2 }}\left( {\frac{{4V^3 }}{{\pi ^3 \lambda ^5 }}} \right)\sum\limits_{i,j,k}^\infty  {\frac{{z^{j + k + l} }}{{\left( {j + k + l} \right)^{1/2} \left( {j - k} \right)l}}} \frac{{\partial J}}{{\partial u}}
$$
where:
$$J = \int\limits_0^\infty  {\int\limits_0^\infty  {dqdq\frac{{\cosh \left( {upq} \right)}}{{q^2  - p^2 }}} } e^{ - vq^2  - wp^2 } 
$$
and:
$$u = \frac{{\hbar ^2 \beta }}{{2m}}\left( {\frac{{2\left( {j - k} \right)l}}{{j + k + l}}} \right)
$$
$$v = \frac{{\hbar ^2 \beta }}{{2m}}\left( {\frac{{\left( {j + k} \right)l}}{{j + k + l}}} \right)
$$
$$w = \frac{{\hbar ^2 \beta }}{{2m}}\left( {\frac{{\left( {j + k} \right)l + 4jk}}{{j + k + l}}} \right)
$$
$\frac{{\partial J}}{{\partial u}}$ is calculated here: Evaluating the integral $I(u,v,w)=\iint_{(0,\infty)^2}\sinh(upq) e^{-vq^2 - wp^2}pq(q^2-p^2)^{-1}dpdq$
Do you have any suggestion?
 A: Maybe we don't have to do thermal average of n and we need :
$$n = \frac{1}{{z^{ - 1} e^{\varepsilon /kT}  - 1}} = \sum\limits_{n = 1}^\infty  {\left( {ze^{ - \varepsilon /kT} } \right)^l } 
$$
so:
$$\sum\limits_{} {'... = \sum\limits_{i,j,k = 1}^\infty  {\sum\limits_{} ' } } z^{ijk} \frac{{e^{ - \frac{{\hbar ^2 k_a^2j }}{{2m}}} e^{ - \frac{{\hbar ^2 k_\gamma ^2k }}{{2m}}} e^{ - \frac{{\hbar ^2 k_\lambda ^2l }}{{2m}}} }}{{\frac{1}{2}\left( {k_a^2  + k_b^2  - k_\gamma ^2  - k_\lambda ^2 } \right)}}
$$
This explain the factor $z^{ijk}$ and the sum. How can we continue?Now we have to transform the sum' in a integral. The density of state depends of $kdk$ so i think we have to evaluate:
$$\int\limits_0^\infty  {\frac{{e^{ - \frac{{\hbar ^2 k_a^2j }}{{2m}}} e^{ - \frac{{\hbar ^2 k_\gamma ^2k }}{{2m}}} e^{ - \frac{{\hbar ^2 k_\lambda ^2l }}{{2m}}} }}{{\frac{1}{2}\left( {k_a^2  + k_b^2  - k_\gamma ^2  - k_\lambda ^2 } \right)}}k_a k_b k_\gamma  k_\lambda  } dk_a dk_b dk_\gamma  dk_\lambda  
$$
We have to evaluate two integrals, i think.
A: I have tried to solve this integral:
$$F\left( {a,b,c,d} \right) = \int {\frac{{e^{ - \frac{{\hbar ^2 }}{{2m}}\left( {ak_a^2  + bk_b^2  + ck_\gamma ^2  + dk_\lambda ^2 } \right)} }}{{\left( {k_a^2  + k_b^2  - k_\gamma ^2  - k_\lambda ^2 } \right)}}k_a k_b k_\gamma  k_\lambda  dk_a dk_b dk_\gamma  dk_\lambda  } 
$$
from which we can calculate our integral putting $b=0$. We have:
$$\frac{{\partial F}}{{\partial a}} + \frac{{\partial F}}{{\partial b}} - \frac{{\partial F}}{{\partial c}} - \frac{{\partial F}}{{\partial d}} \propto \frac{1}{{abcd}}
$$
Now we put:
$$i=a+b+c+d$$
$$l=a+b-c-d$$
$$m=a-b$$
$$n=c-d$$
So:
$$\frac{\partial }{{\partial a}} = \frac{\partial }{{\partial i}} + \frac{\partial }{{\partial l}} + \frac{\partial }{{\partial m}}
$$
$$\frac{\partial }{{\partial b}} = \frac{\partial }{{\partial i}} + \frac{\partial }{{\partial l}} - \frac{\partial }{{\partial m}}
$$
$$\frac{\partial }{{\partial c}} = \frac{\partial }{{\partial i}} - \frac{\partial }{{\partial l}} + \frac{\partial }{{\partial n}}
$$
$$\frac{\partial }{{\partial d}} = \frac{\partial }{{\partial i}} - \frac{\partial }{{\partial l}} - \frac{\partial }{{\partial n}}
$$
$$\frac{\partial }{{\partial a}} + \frac{\partial }{{\partial b}} - \frac{\partial }{{\partial c}} - \frac{\partial }{{\partial d}} = 4\frac{\partial }{{\partial l}}
$$
and:
$$a=(i+l+2m)/4$$
$$b=(i+l-2m)/4$$
$$c=(i-l+2n)/4$$
$$d=(i-l-2n)/4$$
and the equation becomes:
$$\frac{{\partial F'}}{{\partial l}} \propto \frac{1}{{\left( {\frac{{i + l}}{2} + m} \right)\left( {\frac{{i + l}}{2} - m} \right)\left( {\frac{{i - l}}{2} + n} \right)\left( {\frac{{i - l}}{2} - n} \right)}}
$$
The integration can be evaluated with derive, and the substitutions gives an expression divergent if we put $b=0$. What is wrong?
