05/23/2008, 12:07 PM

Ok, now the promised proof for the uniqueness of the gamma function, which was point 1. here. I nearly only translate it from the mentioned book:

We consider the function on the right half plane, it also satisfies . Hence has a meromorphic continuation to . Poles can be at most at 0, -1, -2 , ....

As we have , hence has a holomorphic continuation to 0 and also to to each , by .

restricted to is bounded because is bounded there. Then is also restricted on given by . defined by is bounded on because and have the same values on . Now , hence is bounded on whole and by Liouville . Hence and .

We consider the function on the right half plane, it also satisfies . Hence has a meromorphic continuation to . Poles can be at most at 0, -1, -2 , ....

As we have , hence has a holomorphic continuation to 0 and also to to each , by .

restricted to is bounded because is bounded there. Then is also restricted on given by . defined by is bounded on because and have the same values on . Now , hence is bounded on whole and by Liouville . Hence and .