Services & Resources / Wolfram Forums
-----
 /
MathGroup Archive
2007
*January
*February
*March
*April
*May
*June
*July
*August
*September
*October
*Archive Index
*Ask about this page
*Print this page
*Give us feedback
*Sign up for the Wolfram Insider

MathGroup Archive 2007

[Date Index] [Thread Index] [Author Index]

Search the Archive

Infinity appears as a factor in Integrate result!

  • To: mathgroup at smc.vnet.net
  • Subject: [mg74862] Infinity appears as a factor in Integrate result!
  • From: "dimitris" <dimmechan at yahoo.com>
  • Date: Mon, 9 Apr 2007 06:09:42 -0400 (EDT)

Consider the integral

In[13]:=
f = HoldForm[Integrate[1/(x^3 + a*x^2 + b*x + 1), {x, 0, Infinity}]]

Then

In[17]:=
Integrate[1/(x^3 + a*x^2 + b*x + 1), {x, 0, Infinity}]

Out[17]=
If[(Re[Root[1 + b*#1 + a*#1^2 + #1^3 & , 1]] <= 0 || Im[Root[1 + b*#1
+ a*#1^2 + #1^3 & , 1]] != 0) &&
   (Re[Root[1 + b*#1 + a*#1^2 + #1^3 & , 2]] <= 0 || Im[Root[1 + b*#1
+ a*#1^2 + #1^3 & , 2]] != 0) &&
   (Re[Root[1 + b*#1 + a*#1^2 + #1^3 & , 3]] <= 0 || Im[Root[1 + b*#1
+ a*#1^2 + #1^3 & , 3]] != 0),
  -(Log[-Root[1 + b*#1 + a*#1^2 + #1^3 & , 1]]/(b + 2*a*Root[1 + b*#1
+ a*#1^2 + #1^3 & , 1] +
      3*Root[1 + b*#1 + a*#1^2 + #1^3 & , 1]^2)) - Log[-Root[1 + b*#1
+ a*#1^2 + #1^3 & , 2]]/
    (b + 2*a*Root[1 + b*#1 + a*#1^2 + #1^3 & , 2] + 3*Root[1 + b*#1 +
a*#1^2 + #1^3 & , 2]^2) -
   Log[-Root[1 + b*#1 + a*#1^2 + #1^3 & , 3]]/(b + 2*a*Root[1 + b*#1 +
a*#1^2 + #1^3 & , 3] +
     3*Root[1 + b*#1 + a*#1^2 + #1^3 & , 3]^2) +
   Infinity*(1/(b + 2*a*Root[1 + b*#1 + a*#1^2 + #1^3 & , 1] +
3*Root[1 + b*#1 + a*#1^2 + #1^3 & , 1]^2) +
     1/(b + 2*a*Root[1 + b*#1 + a*#1^2 + #1^3 & , 2] + 3*Root[1 + b*#1
+ a*#1^2 + #1^3 & , 2]^2) +
     1/(b + 2*a*Root[1 + b*#1 + a*#1^2 + #1^3 & , 3] + 3*Root[1 + b*#1
+ a*#1^2 + #1^3 & , 3]^2)),
  Integrate[1/(1 + b*x + a*x^2 + x^3), {x, 0, Infinity},
   Assumptions ->  !((Re[Root[1 + b*#1 + a*#1^2 + #1^3 & , 1]] <= 0 ||
Im[Root[1 + b*#1 + a*#1^2 + #1^3 & , 1]] != 0) &&
      (Re[Root[1 + b*#1 + a*#1^2 + #1^3 & , 2]] <= 0 || Im[Root[1 +
b*#1 + a*#1^2 + #1^3 & , 2]] != 0) &&
      (Re[Root[1 + b*#1 + a*#1^2 + #1^3 & , 3]] <= 0 || Im[Root[1 +
b*#1 + a*#1^2 + #1^3 & , 3]] != 0))]]

In the results it appears Infinity!. For anyone who dosn't believe me
try:

In[18]:=
%[[2]]

Out[18]=
-(Log[-Root[1 + b*#1 + a*#1^2 + #1^3 & , 1]]/(b + 2*a*Root[1 + b*#1 +
a*#1^2 + #1^3 & , 1] +
     3*Root[1 + b*#1 + a*#1^2 + #1^3 & , 1]^2)) - Log[-Root[1 + b*#1 +
a*#1^2 + #1^3 & , 2]]/
   (b + 2*a*Root[1 + b*#1 + a*#1^2 + #1^3 & , 2] + 3*Root[1 + b*#1 +
a*#1^2 + #1^3 & , 2]^2) -
  Log[-Root[1 + b*#1 + a*#1^2 + #1^3 & , 3]]/(b + 2*a*Root[1 + b*#1 +
a*#1^2 + #1^3 & , 3] +
    3*Root[1 + b*#1 + a*#1^2 + #1^3 & , 3]^2) +
  Infinity*(1/(b + 2*a*Root[1 + b*#1 + a*#1^2 + #1^3 & , 1] + 3*Root[1
+ b*#1 + a*#1^2 + #1^3 & , 1]^2) +
    1/(b + 2*a*Root[1 + b*#1 + a*#1^2 + #1^3 & , 2] + 3*Root[1 + b*#1
+ a*#1^2 + #1^3 & , 2]^2) +
    1/(b + 2*a*Root[1 + b*#1 + a*#1^2 + #1^3 & , 3] + 3*Root[1 + b*#1
+ a*#1^2 + #1^3 & , 3]^2))

The following term is multiplied with Infinity and the result of this
product
appeared in the results!

In[19]:=
Cases[%, (a_)*Infinity -> a]

Out[19]=
{1/(b + 2*a*Root[1 + b*#1 + a*#1^2 + #1^3 & , 1] + 3*Root[1 + b*#1 +
a*#1^2 + #1^3 & , 1]^2) +
   1/(b + 2*a*Root[1 + b*#1 + a*#1^2 + #1^3 & , 2] + 3*Root[1 + b*#1 +
a*#1^2 + #1^3 & , 2]^2) +
   1/(b + 2*a*Root[1 + b*#1 + a*#1^2 + #1^3 & , 3] + 3*Root[1 + b*#1 +
a*#1^2 + #1^3 & , 3]^2)}

Obviolusly Integarate algorithm failes for this integral.

What is funnier is that

In[20]:=
Integrate[1/(x^3 + a*x^2 + b*x + 1), {x, 0, Infinity}]

Out[20]=
If[(Re[Root[1 + b*#1 + a*#1^2 + #1^3 & , 1]] <= 0 || Im[Root[1 + b*#1
+ a*#1^2 + #1^3 & , 1]] != 0) &&
   (Re[Root[1 + b*#1 + a*#1^2 + #1^3 & , 2]] <= 0 || Im[Root[1 + b*#1
+ a*#1^2 + #1^3 & , 2]] != 0) &&
   (Re[Root[1 + b*#1 + a*#1^2 + #1^3 & , 3]] <= 0 || Im[Root[1 + b*#1
+ a*#1^2 + #1^3 & , 3]] != 0), ComplexInfinity,
  Integrate[1/(1 + x*(b + x*(a + x))), {x, 0, Infinity},
   Assumptions -> (Im[Root[1 + b*#1 + a*#1^2 + #1^3 & , 1]] == 0 &&
Re[Root[1 + b*#1 + a*#1^2 + #1^3 & , 1]] > 0) ||
     (Im[Root[1 + b*#1 + a*#1^2 + #1^3 & , 2]] == 0 && Re[Root[1 +
b*#1 + a*#1^2 + #1^3 & , 2]] > 0) ||
     (Im[Root[1 + b*#1 + a*#1^2 + #1^3 & , 3]] == 0 && Re[Root[1 +
b*#1 + a*#1^2 + #1^3 & , 3]] > 0)]]

I.e. if you try again to obtain the integral Mathematica returns
ComplexInfinity!

Dimitris



  • Prev by Date: Re: Showing the points on a surface about a circle in the plane
  • Next by Date: Re: Enquirey
  • Previous by thread: Re: NDSolve can't solve a complex ODE correctly?
  • Next by thread: Re: Infinity appears as a factor in Integrate result!