Re: Incomplete Gamma function
- To: mathgroup at smc.vnet.net
- Subject: [mg67630] Re: Incomplete Gamma function
- From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
- Date: Sun, 2 Jul 2006 06:28:38 -0400 (EDT)
- References: <F1CC734F-5AD5-446A-ACF9-14487B8D6637@mimuw.edu.pl>
- Sender: owner-wri-mathgroup at wolfram.com
I have found the answer to the question myself, less than 5 minutes after posting it... Looking at http://mathworld.wolfram.com/IncompleteGammaFunction.html I see that, for an integer n we have: Gamma[n-1,n] = (n-1)! Sum[(n-1)^k/k!,{k,0,n-1}] It is completely obvious that this is always an integer. So now the question is: why doesn't Mathematica (or FunctionExpand) make use of this formula to compute Gamma[n,x] for integer n? Andrzej Kozlowski On 2 Jul 2006, at 11:16, Andrzej Kozlowski wrote: > Here is a question I have come across concerning special > functions, which unfortunately, is an area of mathematics I know > next to nothing about. However, as there are many experts in this > field who read this list I hope someone will find this an > interesting (or trivial?) question. > > Consider the following sequence: > > > Table[FunctionExpand[Exp[n-1]*Gamma[n,n-1]],{n,2,14}] > > {2,10,78,824,10970,176112,3309110,71219584,1727242866,46602156800, > 1384438376222,44902138752000,1578690429731402} > > As you see, we get only integers. What happens if n is larger than > 14? Mathematica seems not to be able to answer this: > > > FunctionExpand[Exp[n - 1]* > Gamma[n, n - 1]] /. > n -> 15 > > > E^14*Gamma[15, 14] > > Numerical methods also do not seem to be able to determine if this > is an integer or not. I have looked at Abramowitz & Stegun but I > can't see anything that obviously helps to resolve the issue. Can > anyone help? > > Andrzej Kozlowski > > Tokyo, Japan