Hansel, Gretyl, Riemann and Zeta

When solving problem 2e) in class today (integrate blackbody flux over all frequencies to obtain bolometric flux) we arrived at a strange integral after some substitutions:

Simplified, we had: Integrate[x^3/(e^x-1)),dx]

To solve such integrals ew can use the Riemann zeta function ζ(s) defined as

ζ(s)=Sum(1/n^s) for n=1 to infinity

 You can use this to solve integrals in statistical mechanics ζ(s) =1/Tau(s)*Integral[x^(s-1)/(e^x-1),dx]
where Tau[s+1]=s!




so Integral[x^3/(e^x-1)),dx]=ζ(s) *Tau(s), where s=4


 ζ(4)=Pi^4/90 using the summation above, and Tau(4)=3!=6 so we estimate the integral as Pi^4/15.

I used these equations extensively over the summer calculating pressures and energy densities for nonideal adiabatic (reversible reactions with chemical potential 0) fluids in the early universe. For example, you can use this to show that the energy density of photons in the early universe is Pi^2*T^4/15 (for a purely relativistic case). Since energy density in a radiation dominated universe scales as the scale factor a^-4, we see that the temperature of a radiation dominated background (the cosmic microwave background) must scale as a^-1 .