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: (*********************************************************************** Mathematica-Compatible Notebook 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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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info at wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 4996, 156]*) (*NotebookOutlinePosition[ 5634, 179]*) (* 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\)\), ",", RowBox[{"{", RowBox[{"x", ",", InterpretationBox[\(-\[Infinity]\), DirectedInfinity[ -1]], ",", InterpretationBox["\[Infinity]", DirectedInfinity[ 1]]}], "}"}], ",", \(Assumptions \[Rule] {Arg[\[Epsilon]\^2] == 0}\)}], "]"}]], "Input"], Cell[BoxData[ RowBox[{"Integrate", "[", RowBox[{\(1\/\(\((1 - x)\)\^2 + \[Epsilon]\^2\)\), ",", RowBox[{"{", RowBox[{"x", ",", InterpretationBox[\(-\[Infinity]\), DirectedInfinity[ -1]], ",", 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]\), DirectedInfinity[ -1]], ",", InterpretationBox["\[Infinity]", DirectedInfinity[ 1]]}], "}"}]}], "]"}]], "Input"], 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[ RowBox[{"Integrate", "[", RowBox[{\(1\/\(\[Epsilon]\^2 + \((0 - x)\)\^2\)\), ",", RowBox[{"{", RowBox[{"x", ",", InterpretationBox[\(-\[Infinity]\), DirectedInfinity[ -1]], ",", InterpretationBox["\[Infinity]", DirectedInfinity[ 1]]}], "}"}]}], "]"}]], "Input"], Cell[BoxData[ RowBox[{"If", "[", RowBox[{ \(Arg[\[Epsilon]\^2] \[NotEqual] \[Pi]\), ",", \(\[Pi]\ \ at \(1\/\[Epsilon]\^2\)\), ",", RowBox[{ SubsuperscriptBox["\[Integral]", InterpretationBox[\(-\[Infinity]\), DirectedInfinity[ -1]], 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}} ] (*********************************************************************** Cached data follows. 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