October 2010 UTC I ve updated the motivation again The recurrence relation needs to be satisfied for positive real inputs as mentioned in Bohr Mollerup theorem If I wrote complex last night not worth looking in the history oops my bad I ve changed the variable to x since z tends to imply complex I d be happy to hear a motivation for logarithmic convexity or for an entirely different true uniqueness condition to be used The one you wrote below if it s true would be fine so long as you motivated the final condition Of the two conditions under consideration I prefer logarithmic convexity but I almost always prefer mental pictures to algebra The main point of my edit was that the motivation section didn t motivate the gamma function it only motivated one of infinitely many alternativeswhere the alternatives are nontrivial Any text which accomplishes the goal of giving a brief uniqueness argument is exactly what I had in mind 08 23 27