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Re: Incomplete Gamma function

  • To: mathgroup at smc.vnet.net
  • Subject: [mg67654] Re: Incomplete Gamma function
  • From: "James Gilmore" <james.gilmore at yale.edu>
  • Date: Mon, 3 Jul 2006 06:38:39 -0400 (EDT)
  • Organization: Yale University
  • References: <F1CC734F-5AD5-446A-ACF9-14487B8D6637@mimuw.edu.pl> <e888ar$930$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Hi Andrzej,

This issue has been reported previously... in 2003 when I came across it 
working on high order hydrogenic wave/greens functions with Mathematica 4.2
and 5.0. 
Lets hope for a amendment in 6??

In[234]:=
FunctionExpand[Gamma[14, z]]
FunctionExpand[Gamma[15, z]]
$Version
Out[234]=
z^13/E^z +
13*(z^12/E^z +
12*(z^11/E^z +
11*(z^10/E^z + 10*(z^9/E^z + 9*(z^8/E^z + 8*(z^7/E^z + 7*(z^6/E^z + 
6*(z^5/E^z + 5*(z^4/E^z + 4*(z^3/E^z +
3*(z^2/E^z + 2*(E^(-z) + z/E^z))))))))))))
Out[235]=
Gamma[15, z]
Out[236]=
"5.2 for Microsoft Windows (June 20, 2005)"


Cheers
James Gilmore

"Andrzej Kozlowski" <akoz at mimuw.edu.pl> wrote in message 
news:e888ar$930$1 at smc.vnet.net...
>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
> 



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