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MathGroup Archive 1998

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Rational function integration

  • To: mathgroup at smc.vnet.net
  • Subject: [mg14409] Rational function integration
  • From: lebigot at ens.fr (Eric Le Bigot)
  • Date: Sun, 18 Oct 1998 15:10:13 -0400
  • Organization: Ecole Normale Superieure, Paris
  • Sender: owner-wri-mathgroup at wolfram.com

  Hello,

  To my immense surprise, I've not been able to obtain from Mathematica
the result of *very simple integrations* of rational functions.
Furthermore, there seem to be some inconsistencies between different
results given by Mathematica...

  Does anybody know how to make Mathematica compute those integrals of
rational functions ? (the result is not that much interesting in
itself, but the method is: the expressions I actually want to compute
are simply some more complicated rational functions).

  EOL, who loves Mathematica, even though he is a bit disappointed this
time...


  Here is a notebook that gives further details:

(***********************************************************************

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This notebook can be used on any computer system with Mathematica 3.0,
MathReader 3.0, or any compatible application. The data for the
notebook  starts with the line of stars above.

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the following:

* Save the data starting with the line of stars above into a file
  with a name ending in .nb, then open the file inside the application;

* Copy the data starting with the line of stars above to the
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Data for notebooks contains only printable 7-bit ASCII and can be sent
directly in email or through ftp in text mode.  Newlines can be CR, LF
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word CacheID, otherwise Mathematica-compatible applications may try to 
use invalid cache data.

For more information on notebooks and Mathematica-compatible 
applications, contact Wolfram Research:
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Research.
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CellTagsIndexPosition[      5590,        175]*) (*WindowFrame->Normal*)


Notebook[{

Cell[CellGroupData[{
Cell["Some problems with integrals of rational functions",
"Subsection"],

Cell["Easy integral:", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    RowBox[{"Integrate", "[", 
      RowBox[{\(1\/\(\[Epsilon]\^2 + \((0 - x)\)\^2\)\), ",", 
        RowBox[{"{", 
          RowBox[{"x", ",", 
            InterpretationBox[\(-\[Infinity]\),
              DirectedInfinity[ -1]], ",", 
            InterpretationBox["\[Infinity]",
              DirectedInfinity[ 1]]}], "}"}], ",", 
        \(Assumptions \[Rule] {Arg[\[Epsilon]\^2] == 0}\)}], "]"}]],
"Input"],

Cell[BoxData[
    \(\[Pi]\ \ at \(1\/\[Epsilon]\^2\)\)], "Output"] }, Open  ]],

Cell["\<\
The following integral is simply a translation of the previous one, \
and should give the same result, which is not the case:\ \>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    RowBox[{"Integrate", "[", 
      RowBox[{\(1\/\(\[Epsilon]\^2 + \((1 - x)\)\^2\)\), ",", 
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            InterpretationBox[\(-\[Infinity]\),
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            InterpretationBox["\[Infinity]",
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      RowBox[{\(1\/\(\((1 - x)\)\^2 + \[Epsilon]\^2\)\), ",", 
        RowBox[{"{", 
          RowBox[{"x", ",", 
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            InterpretationBox["\[Infinity]",
              DirectedInfinity[ 1]]}], "}"}], ",", 
        \(Assumptions \[Rule] {Arg[\[Epsilon]\^2] == 0}\)}], "]"}]],
"Output"] }, Open  ]],

Cell["\<\
As strange as it is, removing the constraint gives a general \ result:\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    RowBox[{"Integrate", "[", 
      RowBox[{\(1\/\(\[Epsilon]\^2 + \((1 - x)\)\^2\)\), ",", 
        RowBox[{"{", 
          RowBox[{"x", ",", 
            InterpretationBox[\(-\[Infinity]\),
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            InterpretationBox["\[Infinity]",
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Cell[BoxData[
    \(\[Pi]\/\ at \[Epsilon]\^2\)], "Output"] }, Open  ]],

Cell["\<\
However, the following translated integral generates a condition, \
although it is the \"same\" as the previous one:\ \>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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      RowBox[{\(1\/\(\[Epsilon]\^2 + \((0 - x)\)\^2\)\), ",", 
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            InterpretationBox[\(-\[Infinity]\),
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            InterpretationBox["\[Infinity]",
              DirectedInfinity[ 1]]}], "}"}]}], "]"}]], "Input"],

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          SubsuperscriptBox["\[Integral]", 
            InterpretationBox[\(-\[Infinity]\),
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            InterpretationBox["\[Infinity]",
              DirectedInfinity[ 1]]], 
          \(\(1\/\(x\^2 + \[Epsilon]\^2\)\) \[DifferentialD]x\)}]}],
"]"}]], 
  "Output"]
}, Open  ]]
}, Open  ]]
},
FrontEndVersion->"X 3.0",
ScreenRectangle->{{0, 1280}, {0, 1024}}, WindowSize->{520, 600},
WindowMargins->{{244, Automatic}, {160, Automatic}} ]


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Cached data follows.  If you edit this Notebook file directly, not
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Cell[2367, 72, 149, 3, 57, "Text"],

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(***********************************************************************
End of Mathematica Notebook file.
***********************************************************************)


-- 
-------------------------------------------------------------------------------

Eric-Olivier Le Bigot (EOL)
lebigot at lpan.jussieu.fr


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