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MathGroup Archive 2007

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RootSum

  • To: mathgroup at smc.vnet.net
  • Subject: [mg74886] RootSum
  • From: "dimitris" <dimmechan at yahoo.com>
  • Date: Tue, 10 Apr 2007 05:14:18 -0400 (EDT)

Working with integration of rational functions I came to the
conclusion
that when RootSum is generated in the course of the computation
strange behavior of Mathematica (at least for me) takes place.

In a recent thread I show one such example.

As another example consider the integral of (x^3 + 6*x + 7)/(x^6 +
4*x^4 + 3*x^3 + 6)
in [0,Infinity).

The indefinite integral obtained by Mathematica is

F=Integrate[(x^3 + 6*x + 7)/(x^6 + 4*x^4 + 3*x^3 + 6), x]

RootSum[6 + 3*#1^3 + 4*#1^4 + #1^6 & , (7*Log[x - #1] + 6*Log[x -
#1]*#1 + Log[x - #1]*#1^3)/(9*#1^2 + 16*#1^3 + 6*#1^5) & ]

which is of course correct.

Together[D[F, x]]

(7 + 6*x + x^3)/(6 + 3*x^3 + 4*x^4 + x^6)

Application of the Newton-Leibniz formula to get from the
andiderivative the requested
definite integral fails.

Limit[RootSum[6 + 3*#1^3 + 4*#1^4 + #1^6 & , (7*Log[x - #1] + 6*Log[x
- #1]*#1 + Log[x - #1]*#1^3)/
      (9*#1^2 + 16*#1^3 + 6*#1^5) & ], x -> Infinity] -
  Limit[RootSum[6 + 3*#1^3 + 4*#1^4 + #1^6 & , (7*Log[x - #1] +
6*Log[x - #1]*#1 + Log[x - #1]*#1^3)/
      (9*#1^2 + 16*#1^3 + 6*#1^5) & ], x -> 0]

Limit[RootSum[6 + 3*#1^3 + 4*#1^4 + #1^6 & , (7*Log[x - #1] + 6*Log[x
- #1]*#1 + Log[x - #1]*#1^3)/
      (9*#1^2 + 16*#1^3 + 6*#1^5) & ], x -> Infinity] - RootSum[6 +
3*#1^3 + 4*#1^4 + #1^6 & ,
   (7*Log[-#1] + 6*Log[-#1]*#1 + Log[-#1]*#1^3)/(9*#1^2 + 16*#1^3 +
6*#1^5) & ]

My question comes now...

Since for integrals like this Mathematica uses (as it appears in a
recent forum) the Newton-Leibniz formula
how it can evaluate the definite integral?

Integrate[(x^3 + 6*x + 7)/(x^6 + 4*x^4 + 3*x^3 + 6), {x, 0, Infinity}]

-((Log[-Root[6 + 3*#1^3 + 4*#1^4 + #1^6 & , 1]]*(7 + 6*Root[6 + 3*#1^3
+ 4*#1^4 + #1^6 & , 1] +
      Root[6 + 3*#1^3 + 4*#1^4 + #1^6 & , 1]^3))/(Root[6 + 3*#1^3 +
4*#1^4 + #1^6 & , 1]^2*
     (9 + 16*Root[6 + 3*#1^3 + 4*#1^4 + #1^6 & , 1] + 6*Root[6 +
3*#1^3 + 4*#1^4 + #1^6 & , 1]^3))) -
  (Log[-Root[6 + 3*#1^3 + 4*#1^4 + #1^6 & , 2]]*(7 + 6*Root[6 + 3*#1^3
+ 4*#1^4 + #1^6 & , 2] +
     Root[6 + 3*#1^3 + 4*#1^4 + #1^6 & , 2]^3))/(Root[6 + 3*#1^3 +
4*#1^4 + #1^6 & , 2]^2*
    (9 + 16*Root[6 + 3*#1^3 + 4*#1^4 + #1^6 & , 2] + 6*Root[6 + 3*#1^3
+ 4*#1^4 + #1^6 & , 2]^3)) -
  (Log[-Root[6 + 3*#1^3 + 4*#1^4 + #1^6 & , 3]]*(7 + 6*Root[6 + 3*#1^3
+ 4*#1^4 + #1^6 & , 3] +
     Root[6 + 3*#1^3 + 4*#1^4 + #1^6 & , 3]^3))/(Root[6 + 3*#1^3 +
4*#1^4 + #1^6 & , 3]^2*
    (9 + 16*Root[6 + 3*#1^3 + 4*#1^4 + #1^6 & , 3] + 6*Root[6 + 3*#1^3
+ 4*#1^4 + #1^6 & , 3]^3)) -
  (Log[-Root[6 + 3*#1^3 + 4*#1^4 + #1^6 & , 4]]*(7 + 6*Root[6 + 3*#1^3
+ 4*#1^4 + #1^6 & , 4] +
     Root[6 + 3*#1^3 + 4*#1^4 + #1^6 & , 4]^3))/(Root[6 + 3*#1^3 +
4*#1^4 + #1^6 & , 4]^2*
    (9 + 16*Root[6 + 3*#1^3 + 4*#1^4 + #1^6 & , 4] + 6*Root[6 + 3*#1^3
+ 4*#1^4 + #1^6 & , 4]^3)) -
  (Log[-Root[6 + 3*#1^3 + 4*#1^4 + #1^6 & , 5]]*(7 + 6*Root[6 + 3*#1^3
+ 4*#1^4 + #1^6 & , 5] +
     Root[6 + 3*#1^3 + 4*#1^4 + #1^6 & , 5]^3))/(Root[6 + 3*#1^3 +
4*#1^4 + #1^6 & , 5]^2*
    (9 + 16*Root[6 + 3*#1^3 + 4*#1^4 + #1^6 & , 5] + 6*Root[6 + 3*#1^3
+ 4*#1^4 + #1^6 & , 5]^3)) -
  (Log[-Root[6 + 3*#1^3 + 4*#1^4 + #1^6 & , 6]]*(7 + 6*Root[6 + 3*#1^3
+ 4*#1^4 + #1^6 & , 6] +
     Root[6 + 3*#1^3 + 4*#1^4 + #1^6 & , 6]^3))/(Root[6 + 3*#1^3 +
4*#1^4 + #1^6 & , 6]^2*
    (9 + 16*Root[6 + 3*#1^3 + 4*#1^4 + #1^6 & , 6] + 6*Root[6 + 3*#1^3
+ 4*#1^4 + #1^6 & , 6]^3))


What am I missing here?

Dimitris



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