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MathGroup Archive 1993

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Integrals Crash MMA

  • To: mathgroup <mathgroup at yoda.physics.unc.edu>
  • Subject: Integrals Crash MMA
  • From: Roberto Sierra <73557.2101 at compuserve.com>
  • Date: 17 Jul 93 02:17:46 EDT

The following double integrals crash MMA 2.03 for the Mac.  It takes
a while, and you get some warnings about ComplexInfinities first, but
they will crash eventually.  I wonder if this problem has been fixed
in MMA 2.2.  The variables x, y, z, and sigma are constants within the
integral.

    Integrate[ (Sqrt[r^2] r Cos[theta]) /
      (r^2 - 2 r x Cos[theta] - 2 r y Sin[theta])^(3/2),
      {theta,0,2Pi}, {r,0,R}
    ]

    Integrate[ (Sqrt[r^2] r Sin[theta]) /
      (r^2 - 2 r x Cos[theta] - 2 r y Sin[theta])^(3/2),
      {theta,0,2Pi}, {r,0,R}
    ]

    Integrate[ (Sqrt[r^2] sigma (x - r Cos[theta])) /
      (r^2 + x^2 + y^2 + z^2 - 2 r x Cos[theta] - 2 r y Sin[theta])^(3/2),
      {theta,0,2Pi}, {r,0,R}
    ]

    Integrate[ (Sqrt[r^2] sigma (y - r Sin[theta])) /
      (r^2 + x^2 + y^2 + z^2 - 2 r x Cos[theta] - 2 r y Sin[theta])^(3/2),
      {theta,0,2Pi}, {r,0,R}
    ]


I tried to evaluate a related 1-D integral, and it took MMA a real
long time before leaving the integral unevaluated.  Why does it take
MMA so long (hours) before rejecting this integral, whereas most
unevaluable integrals are rejected within minutes or seconds?  [Note
that the reported timing requirements are considerably lower than
what is actually required.]  I want to avoid having to wait hours
in the future -- I thought that MMA had crashed at first until I
let it run overnight...

    In[1] :=
        eqn = (lambda (x - R Cos[theta])
            Sqrt[R^2 Cos[theta]^2 + R^2 Sin[theta]^2]) /
            (z^2 + (x - R Cos[theta])^2 + (y - R Sin[theta])^2)^(3/2)
    Out[1] = 
                                        2           2    2           2
        lambda (x - R Cos[theta]) Sqrt[R  Cos[theta]  + R  Sin[theta] ]
        ---------------------------------------------------------------
                2                     2                     2 3/2
              (z  + (x - R Cos[theta])  + (y - R Sin[theta]) )

    In[2] :=
        Timing[eqn1 = Simplify[eqn]]
    Out[2] =
        {38.15 Second, 

                                     2
                        lambda Sqrt[R ] (x - R Cos[theta])
           ------------------------------------------------------------}
             2    2    2    2                                       3/2
           (R  + x  + y  + z  - 2 R x Cos[theta] - 2 R y Sin[theta])

    In[3] :=
        Timing[int1 = Integrate[eqn1, {theta,0,2Pi}]]
    Out[3] =
                                     2
        {4044.6 Second, lambda Sqrt[R ] 
 
                                       x - R Cos[theta]
     Integrate[------------------------------------------------------------, 
                 2    2    2    2                                       3/2
               (R  + x  + y  + z  - 2 R x Cos[theta] - 2 R y Sin[theta])
 
            {theta, 0, 2 Pi}]}

    In[4] :=
        Timing[int = Integrate[eqn, {theta,0,2Pi}]]
    Out[4] =
        {3856.17 Second, lambda Integrate[
 
                                      2           2    2           2
             (x - R Cos[theta]) Sqrt[R  Cos[theta]  + R  Sin[theta] ]
             --------------------------------------------------------, 
                 2                     2                     2 3/2
               (z  + (x - R Cos[theta])  + (y - R Sin[theta]) )
 
            {theta, 0, 2 Pi}]}


-- Roberto Sierra
   Tempered MicroDesigns






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