Further Fun With the Planck Function, or, When All Else Fails Use Mathematica

In my last post I showed an integration trick by using the Planck function:

To do so, I took the limit where h*nu >> k*T. This condition is valid for high frequencies (or probably less common, low temperatures). Then I could approximate it as this:

I forgot to mention that this is known as the Wien approximation. And after integrating I got the result:

Remember, my approximation was for high frequencies. Does it make sense to integrate over all frequencies? Maybe as an approximation, but it is important to keep valid limits in mind.

Using Mathematica, I can integrate the exact Planck function:
Mathematica> Integrate[2 h \[Nu]^3/c^2/(Exp[h \[Nu]/k/T] - 1), {\[Nu], 0, Infinity}]
This gives the solution:

Notice that my factor was 12, while the correct factor should be 2*pi^4/15=12.9879.
My approximation was close! But it was a little lower than the exact answer. I'm just trying to illustrate the importance of always thinking about the validity of your answers.

I'm curious how one would analytically integrate the complete Planck function. I'm not sure myself.

Now, what about the other possible limit, h*nu << k*T? Then we can do a Taylor expansion:

 Around a=0 using the exponential term:

We can discard everything on the order of x^2 or higher since x is small. The whole equation becomes:

This is known as the Rayleigh–Jeans law. Notice how my integral over all frequencies would now be infinite? This was known as the 'ultraviolet catastrophe.' These high-frequency and low-frequency approximations were what Planck used to ultimately formulate his function, valid at all frequencies.