When I first learned this technique in undergraduate Quantum Mechanics at the University of Minnesota — Minneapolis I was truly blown away.  Richard Feynman references this technique in his memoirs.  I am pretty sure it was in Surely You’re Joking Mr. Feynman!  When I have taught this technique to my physics students at SUNY Geneseo nd The College at Brockport, well they have shown impressive enthusiasm for the technique as well.  It is a very cool technique, and it applies not only to Gamma and related functions that appear in Quantum Mechanics.  It is also very useful for computing otherwise nasty integrals in four-dimensional space-time that appear in loop-level Feynman integrals in Quantum Field Theory.  Whether or not there is an immediate application, it is an extremely cool technique to explore!  Read more in the full article.