Interesting integral problem
- To: mathgroup at smc.vnet.net
- Subject: [mg47716] Interesting integral problem
- From: Arturas Acus <acus at itpa.lt>
- Date: Fri, 23 Apr 2004 02:30:44 -0400 (EDT)
- Sender: owner-wri-mathgroup at wolfram.com
Dear Group, The following integral http://www.itpa.lt/mathematica/IntegralProblem.nb was my headache for a week, still I cannot decide if it can be expressed using elementary or special functions. The problem is if one has two quite similar integrals, one of them can be calculated explicitly, other seems not. What is so special about the first? If somebody could look at it I would be very gratefull. (************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. 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The task is to integrate integral bellow in spherical coordinates \ (do not forget to multiply it by Sin[\[Theta]]).\ \>", "Text", CellAutoOverwrite->False], Cell[BoxData[ \(\(initialProblem = \(3\ Cos[\[Theta]]\^2\ Sin[\[Theta]]\^4\ \ Sin[2\ \[CurlyPhi]]\^2\)\/\((1 - Sin[\[Theta]]\^2 + Sin[\[Theta]]\^4\ \ \((1 - Sin[\[CurlyPhi]]\^2 + Sin[\[CurlyPhi]]\^4)\))\)\^2;\)\)], \ "Input", CellLabel->"In[1]:=", CellAutoOverwrite->False], Cell["\<\ If we first integrate by \[Theta] (integratation of \ \[CurlyPhi] first gives less promising rezult and looks like a dead \ end) , after a lot of simplification hints we can arrive to the \ following integral over \[CurlyPhi]:\ \>", "Text", CellAutoOverwrite->False], Cell[BoxData[ \(\(\(intEquivalent = \(-\(\(6\ \((\(-1\) + a)\)\ \((4\ \@\(7 + a\) + \@2\ \((5 + a)\)\ \[Pi])\)\)\/\(\((5 + a)\)\ \((7 + a)\)\^\(3/2\)\)\)\) - \(24\ \ \((\(-1\) + a)\)\ \((6 + a)\)\ Log[\(-\@\(5 + a\)\) + \@\(7 + \ a\)]\)\/\((\((5 + a)\)\ \((7 + a)\))\)\^\(3/2\) + \(24\ \((\(-1\) + \ a)\)\ \((6 + a)\)\ Log[\@\(5 + a\) + \@\(7 + a\)]\)\/\((\((5 + a)\)\ \ \((7 + a)\))\)\^\(3/2\);\)\(\[IndentingNewLine]\) \)\)], "Input", CellLabel->"In[2]:=", CellAutoOverwrite->False], Cell["\<\ with a=Cos[4 \[CurlyPhi]]. The following numerical check \ ensures, that no mistake was made.\ \>", "Text", CellAutoOverwrite->False], Cell[CellGroupData[{ Cell[BoxData[{ \(\[Theta]Rand = Random[Real, {0, \[Pi]}]; \[CurlyPhi]Rand = Random[Real, {0, 2\ \[Pi]}]; aRand = Cos[4\ \[CurlyPhi]Rand];\), "\[IndentingNewLine]", \({NIntegrate[ initialProblem*Sin[\[Theta]], {\[CurlyPhi], 0, 2*\[Pi]}, {\[Theta], 0, \[Pi]}], NIntegrate[ Evaluate[ initialProblem* Sin[\[Theta]] /. \[CurlyPhi] \[Rule] \[CurlyPhi]Rand], \ {\[Theta], 0, \[Pi]}]}\)}], "Input", CellLabel->"In[3]:=", CellAutoOverwrite->False], Cell[BoxData[ RowBox[{\(General::"spell1"\), \(\(:\)\(\ \)\), "\<\"Possible \ spelling error: new symbol name \\\"\\!\\(\[CurlyPhi]Rand\\)\\\" is \ similar to existing symbol \\\"\\!\\(\[Theta]Rand\\)\\\". \ \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", \ ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \ ButtonData:>\\\"General::spell1\\\"]\\)\"\>"}]], "Message", CellLabel->"From In[3]:="], Cell[BoxData[ \({2.750736494334113`, 0.5649250647272591`}\)], "Output", CellLabel->"Out[4]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(intEquivalent /. a -> aRand // Chop\)], "Input", CellLabel->"In[5]:=", CellAutoOverwrite->False], Cell[BoxData[ \(0.5649250647272592`\)], "Output", CellLabel->"Out[5]="] }, Open ]], Cell["And we hope for", "Text", CellAutoOverwrite->False], Cell[CellGroupData[{ Cell[BoxData[ \(NIntegrate[ intEquivalent /. a \[Rule] Cos[4\ \[CurlyPhi]], {\[CurlyPhi], 0, 2*\[Pi]}]\)], "Input", CellLabel->"In[6]:=", CellAutoOverwrite->False], Cell[BoxData[ \(2.7507365480157735`\)], "Output", CellLabel->"Out[6]="] }, Open ]], Cell["or better", "Text", CellAutoOverwrite->False], Cell[CellGroupData[{ Cell[BoxData[ \(4\ NIntegrate[ intEquivalent /. a \[Rule] Cos[4\ \[CurlyPhi]], {\[CurlyPhi], 0, \[Pi]/2}]\)], "Input", CellLabel->"In[7]:=", CellAutoOverwrite->False], Cell[BoxData[ \(2.7507365480127253`\)], "Output", CellLabel->"Out[7]="] }, Open ]], Cell["\<\ The problematic part is with the Log[ ]. Can it be \ integrated?\ \>", "Text", CellAutoOverwrite->False], Cell[BoxData[ \(\(problematicPart = \(24\ \((\(-1\) + a)\)\ \((6 + a)\)\ \ Log[\(-\@\(5 + a\)\) + \@\(7 + a\)]\)\/\((\((5 + a)\)\ \((7 + a)\))\)\ \^\(3/2\) + \(24\ \((\(-1\) + a)\)\ \((6 + a)\)\ Log[\@\(5 + a\) + \@\ \(7 + a\)]\)\/\((\((5 + a)\)\ \((7 + a)\))\)\^\(3/2\);\)\)], "Input", CellLabel->"In[8]:=", CellAutoOverwrite->False], Cell["\<\ If not, then what so special (from algebraic point of view) \ has the following quite similar integral, which do can be integrated \ explicitly (even undefined integration actually can be \ performed)\ \>", "Text", CellAutoOverwrite->False], Cell[BoxData[ \(\(simpleProblem = \(12\ Sin[\[Theta]]\^2\ \((1 - \ Sin[\[Theta]]\^2 + Cos[\[CurlyPhi]]\^2\ Sin[\[Theta]]\^4\ Sin[\ \[CurlyPhi]]\^2)\)\)\/\((1 - Sin[\[Theta]]\^2 + Sin[\[Theta]]\^4\ \ \((1 - Sin[\[CurlyPhi]]\^2 + Sin[\[CurlyPhi]]\^4)\))\)\^2;\)\)], \ "Input", CellLabel->"In[10]:=", CellAutoOverwrite->False], Cell["We hope for 16\[Pi]", "Text", CellAutoOverwrite->False], Cell[CellGroupData[{ Cell[BoxData[ \({NIntegrate[ simpleProblem*Sin[\[Theta]], {\[CurlyPhi], 0, 2*\[Pi]}, {\[Theta], 0, \[Pi]}], NIntegrate[ Evaluate[ simpleProblem* Sin[\[Theta]] /. \[CurlyPhi] \[Rule] \[CurlyPhi]Rand], \ {\[Theta], 0, \[Pi]}]}\)], "Input", CellLabel->"In[11]:=", CellAutoOverwrite->False], Cell[BoxData[ \({50.26548187126606`, 8.824897793155591`}\)], "Output", CellLabel->"Out[11]="] }, Open ]], Cell["\<\ First integrating by \[Theta] and simplifying we get\ \>", \ "Text", CellAutoOverwrite->False], Cell[BoxData[ \(\(simpleProblemEquivalent = \(96\ \((3 + a)\)\)\/\(\((5 + a)\)\ \ \((7 + a)\)\) + \(48\ \((1 + a\ \((6 + a)\))\)\ Log[\(-\@\(5 + a\)\) \ + \@\(7 + a\)]\)\/\((\((5 + a)\)\ \((7 + a)\))\)\^\(3/2\) - \(48\ \ \((1 + a\ \((6 + a)\))\)\ Log[\@\(5 + a\) + \@\(7 + a\)]\)\/\((\((5 + \ a)\)\ \((7 + a)\))\)\^\(3/2\);\)\)], "Input", CellLabel->"In[12]:=", CellAutoOverwrite->False], Cell["The numerical check ensures that all is correct", "Text", CellAutoOverwrite->False], Cell[CellGroupData[{ Cell[BoxData[ \(simpleProblemEquivalent /. a \[Rule] aRand\)], "Input", CellLabel->"In[13]:=", CellAutoOverwrite->False], Cell[BoxData[ \(8.824897793177056`\)], "Output", CellLabel->"Out[13]="] }, Open ]], Cell["The problematic part", "Text", CellAutoOverwrite->False], Cell[BoxData[ \(\(simpleProblemEquivalentProblematic = \((\(48\ \((1 + a\ \((6 \ + a)\))\)\ Log[\(-\@\(5 + a\)\) + \@\(7 + a\)]\)\/\((\((5 + a)\)\ \ \((7 + a)\))\)\^\(3/2\) - \(48\ \((1 + a\ \((6 + a)\))\)\ Log[\@\(5 + \ a\) + \@\(7 + a\)]\)\/\((\((5 + a)\)\ \((7 + \ a)\))\)\^\(3/2\))\);\)\)], "Input", CellLabel->"In[14]:=", CellAutoOverwrite->False], Cell["\<\ is of the same complexity as previous integral, but can be \ explicitly integrated:\ \>", "Text", CellAutoOverwrite->False], Cell[CellGroupData[{ Cell[BoxData[ \(4\ Integrate[ simpleProblemEquivalentProblematic /. a \[Rule] Cos[4\ \[CurlyPhi]], {\[CurlyPhi], 0, \[Pi]/2}]\)], "Input", CellLabel->"In[15]:=", CellAutoOverwrite->False], Cell[BoxData[ \(16\ \((1 - 2\ \@3 + \@6)\)\ \[Pi]\)], "Output", CellLabel->"Out[15]="] }, Open ]], Cell["The rezult is correct:", "Text", CellAutoOverwrite->False], Cell[CellGroupData[{ Cell[BoxData[ \({16\ \((1 - 2\ \@3 + \@6)\)\ \[Pi] // N, 4\ NIntegrate[ simpleProblemEquivalentProblematic /. a \[Rule] Cos[4\ \[CurlyPhi]], {\[CurlyPhi], 0, \[Pi]/2}]}\)], "Input", CellLabel->"In[16]:=", CellAutoOverwrite->False], Cell[BoxData[ \({\(-0.7344728135092694`\), \(-0.7344728135092773`\)}\)], \ "Output", CellLabel->"Out[16]="] }, Open ]], Cell["\<\ However if we alternatively rewrite the problematic part, \ the integration fails with $RecursionLimit and $IterationLimit \ errors. It also fails after Apart[], etc... etc....\ \>", "Text", CellAutoOverwrite->False], Cell[BoxData[ \(\(simpleProblemEquivalentProblematicBad = \(48\ \@\(\((5 + a)\)\ \ \((7 + a)\)\)\ \((1 + 6\ a + a\^2)\)\ Log[6 + a - \@\(5 + a\)\ \ \@\(7 + a\)]\)\/\(\(\((5 + a)\)\^2\)\(\ \)\(\((7 + a)\)\^2\)\(\ \)\);\ \)\)], "Input", CellLabel->"In[31]:=", CellAutoOverwrite->False], Cell[CellGroupData[{ Cell[BoxData[ \(4\ Integrate[ simpleProblemEquivalentProblematicBad /. a \[Rule] Cos[4\ \[CurlyPhi]], {\[CurlyPhi], 0, \[Pi]/2}]\)], "Input", CellLabel->"In[32]:=", CellAutoOverwrite->False], Cell[BoxData[ RowBox[{\($RecursionLimit::"reclim"\), \(\(:\)\(\ \)\), \ "\<\"Recursion depth of \\!\\(256\\) exceeded. \ \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", \ ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \ ButtonData:>\\\"$RecursionLimit::reclim\\\"]\\)\"\>"}]], "Message", CellLabel->"From In[32]:="], Cell[BoxData[ RowBox[{\($RecursionLimit::"reclim"\), \(\(:\)\(\ \)\), \ "\<\"Recursion depth of \\!\\(256\\) exceeded. \ \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", \ ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \ ButtonData:>\\\"$RecursionLimit::reclim\\\"]\\)\"\>"}]], "Message", CellLabel->"From In[32]:="], Cell[BoxData[ RowBox[{\($RecursionLimit::"reclim"\), \(\(:\)\(\ \)\), \ "\<\"Recursion depth of \\!\\(256\\) exceeded. \ \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", \ ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \ ButtonData:>\\\"$RecursionLimit::reclim\\\"]\\)\"\>"}]], "Message", CellLabel->"From In[32]:="], Cell[BoxData[ RowBox[{\(General::"stop"\), \(\(:\)\(\ \)\), "\<\"Further output \ of \\!\\($RecursionLimit :: \\\"reclim\\\"\\) will be suppressed \ during this calculation. \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", \ ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \ ButtonData:>\\\"General::stop\\\"]\\)\"\>"}]], "Message", CellLabel->"From In[32]:="], Cell[BoxData[ RowBox[{\($IterationLimit::"itlim"\), \(\(:\)\(\ \)\), \ "\<\"Iteration limit of \\!\\(4096\\) exceeded. \\!\\(\\*ButtonBox[\\\ \"More\[Ellipsis]\\\", ButtonStyle->\\\"RefGuideLinkText\\\", \ ButtonFrame->None, \ ButtonData:>\\\"$IterationLimit::itlim\\\"]\\)\"\>"}]], "Message", CellLabel->"From In[32]:="], Cell[BoxData[ RowBox[{\($IterationLimit::"itlim"\), \(\(:\)\(\ \)\), \ "\<\"Iteration limit of \\!\\(4096\\) exceeded. \\!\\(\\*ButtonBox[\\\ \"More\[Ellipsis]\\\", ButtonStyle->\\\"RefGuideLinkText\\\", \ ButtonFrame->None, \ ButtonData:>\\\"$IterationLimit::itlim\\\"]\\)\"\>"}]], "Message", CellLabel->"From In[32]:="], Cell[BoxData[ RowBox[{\($IterationLimit::"itlim"\), \(\(:\)\(\ \)\), \ "\<\"Iteration limit of \\!\\(4096\\) exceeded. \\!\\(\\*ButtonBox[\\\ \"More\[Ellipsis]\\\", ButtonStyle->\\\"RefGuideLinkText\\\", \ ButtonFrame->None, \ ButtonData:>\\\"$IterationLimit::itlim\\\"]\\)\"\>"}]], "Message", CellLabel->"From In[32]:="], Cell[BoxData[ RowBox[{\(General::"stop"\), \(\(:\)\(\ \)\), "\<\"Further output \ of \\!\\($IterationLimit :: \\\"itlim\\\"\\) will be suppressed \ during this calculation. \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", \ ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \ ButtonData:>\\\"General::stop\\\"]\\)\"\>"}]], "Message", CellLabel->"From In[32]:="], Cell[BoxData[ \($Aborted\)], "Output", CellLabel->"Out[32]="] }, Open ]], Cell["\<\ I rewrited the integral using various substitutions, in \ many forms. All in vain. Thought both these integrals seems quite \ similar, one can be integrated, whereas other no. This looks quite \ interesting and somehow unusual for me. For example if you look in \ Grandstein Rhyzik integral tables, you will find quite general \ formulas, integrability of which do not depend on, say what \ polynomial coefficients are. I home that somebody can shield a light \ on my problem. THANKS.\ \>", "Text", CellAutoOverwrite->False] }, FrontEndVersion->"5.0 for X", ScreenRectangle->{{0, 3200}, {0, 2048}}, WindowToolbars->"EditBar", Evaluator->"Local", CellGrouping->Manual, WindowSize->{1592, 1156}, WindowMargins->{{0, Automatic}, {Automatic, 0}}, PrintingPageRange->{1, 5}, PrintingOptions->{"PaperSize"->{597.562, 842.375}, "PaperOrientation"->"Portrait", "PostScriptOutputFile":>FrontEnd`FileName[{$RootDirectory, "home", \ "acus", "TMW", "Work", "Helis"}, "Helis.nb.ps", CharacterEncoding -> \ "iso8859-1"], "Magnification"->1}, ShowSelection->True, Magnification->1, StyleDefinitions -> "KnygosStilius.nb" ] (******************************************************************* Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. 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