Is the function uniquely determined (as a meromorphic function on ) by the functional equation and a single value, say ?
And/or is there any other characterization of the Gamma function that uniquely determines it on ?
Here are the criterion that uniquely determine in :
1.)
2.)
3.)
Ask if you'd like to see why this is true.
This is a nice theorem, because if you want to show something is equal to the Gamma function, all you need to do is show it satisfies these three properties!
Last edited by chiph588@; Mar 27th 2010 at 02:57 PM.
Another characterization (on ) : is the only function on that is log-convex (i.e. is convex) and satisfies and for all . This is called Artin's theorem I think. (and the proof is really nice; probably you can find it somewhere online)
Another characterization (on ) : is the only function on that is log-convex (i.e. is convex) and satisfies and for all . This is called Artin's theorem I think. (and the proof is really nice; probably you can find it somewhere online)