Re: Integration error

*To*: mathgroup at smc.vnet.net*Subject*: [mg108930] Re: Integration error*From*: psycho_dad <s.nesseris at gmail.com>*Date*: Wed, 7 Apr 2010 07:25:08 -0400 (EDT)*References*: <hpf5ij$or8$1@smc.vnet.net>

On Apr 6, 1:22 pm, Jason Alexander <ja... at lse.ac.uk> wrote: > Hello all, > > I'm getting a strange result when calculating what I thought was a relatively straightforward integral. If I evaluate the following expression, I receive an imaginary result when it should, in fact, equal 1. (This happens in Mathematica 7.0.1 on Mac OS X): > > In[1]:= > Integrate[( > 4 (8/(2320 + 3 x (-96 + 3 x)) + 8/(1460 + 3 x (-76 + 3 x)) + 6/( > 800 + 3 x (-56 + 3 x)) + 4/(340 + 3 x (-36 + 3 x)) + 1/( > 80 + 3 x (-16 + 3 x))))/(9 \[Pi]), {x, -Infinity, Infinity}] > > Out[1]:= > 1/3 + (52 I)/9 > > However, if I break the integral into two parts, as below, then I get an answer of 1. > > In[2]:= > Integrate[( > 4 (8/(2320 + 3 x (-96 + 3 x)) + 8/(1460 + 3 x (-76 + 3 x)) + 6/( > 800 + 3 x (-56 + 3 x)) + 4/(340 + 3 x (-36 + 3 x)) + 1/( > 80 + 3 x (-16 + 3 x))))/(9 \[Pi]), {x, -Infinity, 0}] + > Integrate[( > 4 (8/(2320 + 3 x (-96 + 3 x)) + 8/(1460 + 3 x (-76 + 3 x)) + 6/( > 800 + 3 x (-56 + 3 x)) + 4/(340 + 3 x (-36 + 3 x)) + 1/( > 80 + 3 x (-16 + 3 x))))/(9 \[Pi]), {x, 0, Infinity}] // Simplify > > Out[2]:= > 1 > > Furthermore, if I compute the antiderivative, I get the following: > > (1/(9 \[Pi]))4 (-(1/24) ArcTan[1/4 (8 - 3 x)] - > 1/4 ArcTan[1/4 (28 - 3 x)] - 1/3 ArcTan[1/4 (38 - 3 x)] + > 2/3 ArcTan[3/4 (-16 + x)] + 1/3 ArcTan[3/4 (-6 + x)] + > 1/3 ArcTan[1/4 (-38 + 3 x)] + 1/4 ArcTan[1/4 (-28 + 3 x)] + > 1/24 ArcTan[1/4 (-8 + 3 x)]) > > And if I define f[x] to be the above, and then ask Mathematica to calculate > > Limit[f[n] - f[-n], n -> Infinity] Well, NIntegrate seem to be working OK (the result is 1.), so as MajorBob said, always try to check your results! > > I get 1, as expected. > > Is this a bug, or am I overlooking some subtle point in how Mathematica computes improper integrals? > > Cheers, > > Jason > > -- > Dr. J. McKenzie Alexander > Department of Philosophy, Logic and Scientific Method > London School of Economics > Houghton Street, London WC2A 2AE