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Virial expansion: Resolving these integrals

  1. Sep 26, 2015 #1
    Hi! From this text: http://arxiv.org/pdf/nucl-th/0004061v1.pdf
    I need to resolve these integrals.

    1) Equation (5), [itex] \int e^{-\omega_1/T}d{\vec k}_1 =?[/itex] where [itex]\omega_1=\sqrt{m^2+{\vec k}^2_1}[/itex]
    What function is [itex] K_2(m/T)[/itex]?
    2) Equation (26), [itex]\rho_s(T)[/itex]

    Thanks!
     
  2. jcsd
  3. Sep 26, 2015 #2

    vanhees71

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  4. Sep 26, 2015 #3
    Thanks, vanhees71!
    But, how I can solve the integral of the scalar density. With Matlab, for example?
     
  5. Sep 28, 2015 #4
    Is there anyone who can tell me the solution of [itex]\rho_s[/itex]? Please.
     
  6. Sep 28, 2015 #5

    vanhees71

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    I think, it's not so easy to get this integral analytically. I'd evaluate it numerically, which is not big deal.
     
  7. Sep 28, 2015 #6
    Well, i think the same thing...Thank you vanhees71!
     
  8. Sep 30, 2015 #7
    Hi guys! I didn't give up!
    Finally, I found the solution of [itex]\rho_s[/itex]

    [itex]\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)[/itex]

    where [itex]\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)[/itex]
     
  9. Sep 30, 2015 #8

    vanhees71

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    Wow, great! Where did you find this integral? I checked it numerically with Mathematica, and it's correct.
     
  10. Sep 30, 2015 #9
  11. Sep 30, 2015 #10

    vanhees71

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