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Re: Bug in definite integral over Gamma function?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg116746] Re: Bug in definite integral over Gamma function?
  • From: Bob Hanlon <hanlonr at cox.net>
  • Date: Sat, 26 Feb 2011 06:05:51 -0500 (EST)

$Version

"8.0 for Mac OS X x86 (64-bit) (November 6, 2010)"

Integrate[x^4*Exp[-x]*Gamma[2, x], {x, 0, Infinity}]

21/8



$Version

"7.0 for Mac OS X x86 (64-bit) (February 19, 2009)"

Integrate[x^4*Exp[-x]*Gamma[2, x], {x, 0, Infinity}]

-(363/8)

Limit[Integrate[x^4*Exp[-x]*Gamma[2, x], {x, 0, a}], a -> Infinity]

21/8

Limit[Integrate[x^4*Exp[-x]*Gamma[2, x], {x, a, Infinity}], a -> 0]

21/8

Integrate[x^4*Exp[-x]*Gamma[2, x], {x, a, Infinity}] /. a -> 0

21/8

Integrate[x^n*Exp[-x]*Gamma[2, x], {x, 0, Infinity}] /. n -> 4

21/8

Integrate[x^4*Exp[-x]*Gamma[a, x], {x, 0, Infinity}] /. a -> 2

21/8


Bob Hanlon

---- H Hogreve <hogreve at ifisr.org> wrote: 

=============
When getting weired results after a large chain of symbolic calculations,
I spotted the reason to something that appears to be a bug in an 
integration, i.e., in

   Integrate[x^4*Exp[-x]*Gamma[2, x], {x, 0, Infinity}]

The two 7.0 versions of Mathematica (32-bit Windows and 64-bit Linux) 
available for checking this integral yield the result

      -363/8

which is obviously incorrect; a correct results can be obtained by
computing the indefinite integral and taking the limits x->0 and
x->Infinity. Moreover, the 6.0 version of Mathematica also gives the
correct result for the definite integral.

Now I am wondering how Mathematica 8 is handling this integral, and
whether there are possibilities in Mathematica 7 to get the correct
definite integral (other than via the indefinite one and boundary
values) ?

Many thanks in advance for any hints,
H. Hogreve




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