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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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