Virial expansion: Resolving these integrals

[Korbid]

Hi! From this text: http://arxiv.org/pdf/nucl-th/0004061v1.pdf
I need to resolve these integrals.

1) Equation (5), $\int e^{-\omega_1/T}d{\vec k}_1 =?$ where $\omega_1=\sqrt{m^2+{\vec k}^2_1}$

What function is $K_2(m/T)$?
​

2) Equation (26), $\rho_s(T)$

Thanks!

 

[vanhees71]

$K_2$ is a modified Bessel function.

https://en.wikipedia.org/wiki/Bessel_function#Modified_Bessel_functions:_I.CE.B1.2C_K.CE.B1

$\rho_s$ is the scalar density of the gas, defined in the unnumbered formula directly following Eq. (26).

 

[Korbid]

Thanks, vanhees71!
But, how I can solve the integral of the scalar density. With Matlab, for example?

 

[Korbid]

Is there anyone who can tell me the solution of $\rho_s$? Please.

 

[vanhees71]

I think, it's not so easy to get this integral analytically. I'd evaluate it numerically, which is not big deal.

 

[Korbid]

Well, i think the same thing...Thank you vanhees71!

 

[Korbid]

Hi guys! I didn't give up!
Finally, I found the solution of $\rho_s$

$\rho_s=\frac{g}{(2\pi)^3}\int \frac{m}{\omega}e^{-\omega/T}d\vec{k}=\frac{g}{(2\pi)^3}\int \frac{m}{\sqrt{m^2+k^2}}e^{-\frac{\sqrt{m^2+k^2}}{T}}d\vec{k}=\frac{mg}{(2\pi)^3}\int^{\infty}_0 4\pi\frac{k^2}{\sqrt{m^2+k^2}}e^{-\frac{\sqrt{m^2+k^2}}{T}}dk=\frac{g}{2\pi^2}m^2TK_1(m/T)$

where $\int^{\infty}_0 \frac{x^{2n}}{\sqrt{x^2+b^2}}e^{-a\sqrt{x^2+b^2}}dx=\frac{2^n}{\sqrt{\pi}}\Gamma(n+1/2)(b/a)K_1(ab)$

 

[vanhees71]

Wow, great! Where did you find this integral? I checked it numerically with Mathematica, and it's correct.

 

[Korbid]

vanhees71,

Yeah, it's correct, I checked too with Matlab!

I found it in this text, http://129.81.170.14/~vhm/papers_html/final22.pdf, equation (4.2)

Thanks for your help!

 

[vanhees71]

I see, the good old integral tables are not yet completely superfluous. :-)