Stirling’s like approximation for log factorial
Posted by peeterjoot on December 24, 2012
Back in my undergrad days, I rarely took notes, since most of what we covered was in the text books. The only notes I’d taken from the course where we did some thermodynamics was half an scrap paper with a simple Stirling like approximation for (sitting in my old text ). Here it is
First write the factorial in product form
so that it’s logarithm is a sum
We can now derivatives of both sides with respect to , ignoring the fact that is a discrete variable. For the left hand side, writing , we have
Now for the right hand side
Merging 1.2.3 and 1.2.4 into two differentials, and integrating we have
Even though we are interested in large , we note that with we have
but this is small compared to for large , so we have
Expontiating, this is
I think that our Professor’s point at the time was that when we only care about the logarithm of we can get away with 1.2.9, and can avoid the complexity and care required to do a proper Stirling’s approximation.
 C. Kittel and H. Kroemer. Thermal physics. WH Freeman, 1980.