Solving Stat Mech Integral with Wolfram Alpha

[ergospherical]

Looking to evaluate an integral of the form $$\int_0^{\infty} \frac{p^2 dp}{\mathrm{exp}(a\sqrt{p^2+b^2}) \pm 1} $$Changing to ##x(p) = a\sqrt{p^2 + b^2}## gives $$\frac{1}{a^3} \int_0^{\infty} \frac{\sqrt{x^2-(b/a)^2}}{e^x \pm 1} dx$$Wolfram alpha doesn't tell me anything useful, sadly.

 

[jedishrfu]

Looking at my Schaum's Outline of Mathematical Handbook of Formulas and Tables (aka old school) there are some similar integrals.

The solution involves the ##\Gamma(n)## See formulas 18.81 and 18,82

https://archive.org/details/murray-...hematical-handbook-of-formu/page/113/mode/2up

Sadly, you'll still have to do some work to get it into the right form though.

 

[Couchyam]

Looking to evaluate an integral of the form $$\int_0^{\infty} \frac{p^2 dp}{\mathrm{exp}(a\sqrt{p^2+b^2}) \pm 1} $$Changing to ##x(p) = a\sqrt{p^2 + b^2}## gives $$\frac{1}{a^3} \int_0^{\infty} \frac{\sqrt{x^2-(b/a)^2}}{e^x \pm 1} dx$$Wolfram alpha doesn't tell me anything useful, sadly.

You might want to double check your algebra for changing variables (e.g. do the bounds of the integral over ##x## make sense?)