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MathGroup Archive 2003

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Yet another incorrect integral

  • To: mathgroup at smc.vnet.net
  • Subject: [mg39338] Yet another incorrect integral
  • From: Bob Stagat <stagat at mrcsb.com>
  • Date: Tue, 11 Feb 2003 04:47:34 -0500 (EST)
  • Sender: owner-wri-mathgroup at wolfram.com

Consider the following integral...

Integrate[x^p*E^(-x^2), {x, z, Infinity}]

Using the substitution x = t^2 it is easy to show that the answer is 
half the incomplete gamma function, (1/2)*Gamma[(p + 1)/2, z^2].

However, if I ask Mathematica to do this integral, here's what I get...

Integrate[x^p E^(-x^2), {x, z, Infinity}]
PowerExpand[%]
% /. z -> 0
%% /. z -> Infinity

Out[146]=
(1/2)*(z^(p + 1)*Gamma[(p + 1)/2, z^2]*
     (z^2)^((1/2)*(-p - 1)) + Gamma[(p + 1)/2])

Out[147]=
(1/2)*(Gamma[(p + 1)/2] + Gamma[(p + 1)/2, z^2])

Out[148]=
Gamma[(p + 1)/2]

Out[149]=
(1/2)*Gamma[(p + 1)/2]

This is incorrect. The correct result should be:

Out[147]=
(1/2)*Gamma[(p + 1)/2, z^2]


Out[148]=
(1/2)*Gamma[(p + 1)/2]

Out[149]=
0

Does anyone understand why Mathematica screws up on such a simple integral?

-Bob Stagat-
-- 


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