Functions

Apart than computer algorithms, what's so special about log-convex functions? Is it so important that it's better than being "correct"?

I know that when Euler's gamma was made, the function wasn't meant to be extended into negative numbers, but people ended up doing so anyways. Now I'm wondering why people didn't consider defining an interpolation function for (-n)!.

I would like to avoid discussing whether it is better or not - or more "correct" for that matter - but it is worth noting that it is the first function of its kind, it is very simple compared to other functions (and thus is easier to compute, too), and have this log-convex property. I also think, but I am not sure, that gamma applications for negative values are sparse, and we have stayed with this function because it works for most applications.

As Euler's Gamma function is log-convex, that also means it is more "smooth": It does not make those bumps and jumps as Hadamard's Gamma does. It is quite important that it works like that for e.g. the gamma distribution, whenever you use it with non-integer alpha values and want the correct probability. Whether that has anything to do with the usage of Euler's Gamma as the "standard" gamma function, I do not know, but I am quite sure it does not.

Noah