Reducing logarithms of factorials

1. Sep 5, 2009

IHateMayonnaise

1. The problem statement, all variables and given/known data

Part of a much bigger problem, but I am hung up on solving the following:

$$ln\left [ \left(\frac{N+n}{2}\right ) ! \right ] = \left ( \frac{N+n+1}{2}\right) \frac{ln(N+n)}{2}\right )$$

I am trying to follow a proof in http://books.google.com/books?id=CD...on random walk&pg=PA205#v=onepage&q=&f=false" My confusion comes from eqn. 10.34. Clearly this is following the Stirling Approximation, where the associated substituted approximation is eqn. 10.33.

2. Relevant equations

The stirling approximation (Eqn. 10.33), as well as Eqns. 10.31 and 10.32.

3. The attempt at a solution

In trying to reduce the following:

$$ln\left [ \left(\frac{N+n}{2}\right ) ! \right ]$$

I do not understand how I am to use the stirling approximation since it specifies the definition of $$ln(n!)$$, and not what I have above. Is there some identity that I am not remembering? I know that:

$$ln\left [ \left(\frac{N+n}{2}\right ) ! \right ]$$

is NOT the same as

$$ln\left [ \frac{(N+n)!}{2!} \right ]$$

But other than that I cannot remember a relevant identity. Thoughts?

IHateMayonnaise

EDIT: To solve use the Stirling's approximation twice. Nevermind :)

Last edited by a moderator: Apr 24, 2017