TITLE: These functions are very similar. Is there an explanation?
QUESTION [5 upvotes]: Blue: $x \ln x-x$
Brown: $\ln \frac{\Gamma(x+1/2)}{\sqrt{\pi}}$

REPLY [2 votes]: Well, I post my own answer. The real reason is that (up to some constant terms and factors), the first function is $D^{-1}D^{-1}[\frac1x]$ and the second function is $D^{-1}\Delta^{-1}[\frac1x]$. So, in one case we replace one normal integration operator with discrete integration. There is no surprise, the functions should be similar.