For general discussion
- To: mathgroup at smc.vnet.net
- Subject: [mg45429] For general discussion
- From: akhmel at hotmail.com (Alex)
- Date: Wed, 7 Jan 2004 01:09:28 -0500 (EST)
- Sender: owner-wri-mathgroup at wolfram.com
Here is the subject for general discussion Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ "It is time to have a serious talk about what ", StyleBox["Mathematica", FontSlant->"Italic"], " claims to be able to do and what it really does. I reproduce below some \ quotations for ", StyleBox["Mathematica", FontSlant->"Italic"], " book and then I give some examples and my comments. ", StyleBox["Mathematica", FontSlant->"Italic"], " quotes are given in small font, my comments - in big font." }], "Subtitle"], Cell[BoxData[ RowBox[{ RowBox[{ StyleBox[ RowBox[{"\[FilledSmallSquare]", StyleBox["Integrate", "MR"]}]], StyleBox[" ", "MR"], "can", " ", "evaluate", " ", "integrals", " ", "of", " ", "rational", " ", \(functions . It\), " ", "can", " ", "also", " ", "evaluate", " ", "integrals", " ", "that", " ", "involve", " ", "exponential"}], ",", "logarithmic", ",", \(trigonometric\ and\ inverse\ trigonometric\ functions\), ",", \(so\ long\ as\ the\ result\ comes\ out\ in\ terms\ of\ the\ same\ \ set\ of\ \(\(functions\)\(.\)\)\)}]], "Input"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Here is an example I presented in previous posting. ", StyleBox["Mathematica", FontSlant->"Italic"], " 4.2 was unable to compute it." }], "Subtitle"], Cell[BoxData[ \(Integrate[\(\(2 \[Pi]\ b\)\/a\) \((\(\(3 e\^2\)\/\(\[Lambda] \((\[Lambda]\^2 - e\^2)\) \(\@\(\(a\^2\) \[Lambda]\^2 - l1\^2\)\) \@\(\(a\^2\) \[Lambda]\^2 - l2\^2\)\)\) \ \((x\/\[Lambda]\^2 + \(\[ImaginaryI] y\)\/\(\[Lambda]\^2 - e\^2\))\) - \ \(\((x\/\[Lambda]\^2 + \(\[ImaginaryI] y\)\/\(\[Lambda]\^2 - e\^2\))\)\^3\) \ \((\(a\^2\) \(\[Lambda]\^3\) \((\[Lambda]\^2 - e\^2)\))\)\/\((\(\@\(\(a\^2\) \ \[Lambda]\^2 - l1\^2\)\) \@\(\(a\^2\) \[Lambda]\^2 - l2\^2\))\)\^3)\), {\ \[Lambda], \[Xi], \[Infinity]}]\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "I got an E-mail from Wolfram employee Lichtblau, who was able to do \ indefinite integration using ", StyleBox["Mathematica", FontSlant->"Italic"], " 5, but when he tried to substitute the upper limit \[Infinity], he got a \ result 255K long. This is nonsense, something is fundamentally wrong here." }], "Subtitle"], Cell[BoxData[""], "Input"], Cell[BoxData[ RowBox[{\(In\ fact\), ",", \(if\ you\ give\ an\ integrand\ that\ involves\ only\ such\ \ elementary\ functions\), ",", RowBox[{ "then", " ", "one", " ", "of", " ", "the", " ", "important", " ", "capabilities", " ", "of", " ", StyleBox["Integrate", "MR"], StyleBox[" ", "MR"], "is", " ", "that", " ", "if", " ", "the", " ", "corresponding", " ", "integral", " ", "can", " ", "be", " ", "expressed", " ", "in", " ", "terms", " ", "of", " ", "elementary", " ", "functions"}], ",", RowBox[{"then", " ", StyleBox["Integrate", "MR"], StyleBox[" ", "MR"], "will", " ", "essentially", " ", "always", " ", "succeed", " ", "in", " ", "finding", " ", \(\(it\)\(.\)\)}]}]], "Input"], Cell[BoxData[ StyleBox[\(REALLY?\ \ Here\ is\ an\ integral\ computable\ in\ elementary\ \ \(\(functions\)\(:\)\)\), "Subtitle", FontFamily->"Century Gothic"]], "Input"], Cell[BoxData[ \(Integrate[\(1 - x\^4\/\(\(r\^2\) \[Rho]\^2\)\)\/\(1 + x\^4\/\(\(r\^2\) \ \[Rho]\^2\) - \(\(2 x\^2\)\/\(r\ \[Rho]\)\) Cos[\[Phi]]\)*1\/\(\(\@\(r\^2 - \ x\^2\)\) \@\(\[Rho]\^2 - x\^2\)\), {x, 0, r}]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(FullSimplify[%]\)], "Input"], Cell[BoxData[ \(\((\[ImaginaryI]\ \@\(1 - x\^2\/r\^2\)\ \@\(1 - x\^2\/\[Rho]\^2\)\ \ \((EllipticF[\[ImaginaryI]\ ArcSinh[\@\(-\(1\/r\^2\)\)\ x], r\^2\/\[Rho]\^2] - EllipticPi[ r\/\(\[Rho]\ Cos[\[Phi]] - \[Rho]\ \@\(\(-1\) + \ Cos[\[Phi]]\^2\)\), \[ImaginaryI]\ ArcSinh[\@\(-\(1\/r\^2\)\)\ x], r\^2\/\[Rho]\^2] - EllipticPi[ r\/\(\[Rho]\ \((Cos[\[Phi]] + \@\(-Sin[\[Phi]]\^2\))\)\), \ \[ImaginaryI]\ ArcSinh[\@\(-\(1\/r\^2\)\)\ x], r\^2\/\[Rho]\^2])\))\)/\((\@\(-\(1\/r\^2\)\)\ \@\(\((r - x)\)\ \ \((r + x)\)\)\ \@\(\(-x\^2\) + \[Rho]\^2\))\)\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ We got instead a combination of ridiculous elliptic integrals.\ \>", "Subtitle"], Cell[BoxData[ RowBox[{ RowBox[{ "A", " ", "large", " ", "book", " ", "of", " ", "integral", " ", "tables", " ", "will", " ", "list", " ", "perhaps", " ", "a", " ", "few", " ", "thousand", " ", "indefinite", " ", RowBox[{"integrals", ".", StyleBox["Mathematica", "TI"]}], StyleBox[" ", "TI"], "can", " ", "do", " ", "essentially", " ", "all", " ", "of", " ", "these", " ", \(integrals . And\), " ", "because", " ", "it", " ", "contains", " ", "general", " ", "algorithms", " ", "rather", " ", "than", " ", "just", " ", "specific", " ", "cases"}], ",", RowBox[{ StyleBox["Mathematica", "TI"], StyleBox[" ", "TI"], "can", " ", "actually", " ", "do", " ", "a", " ", "vastly", " ", "wider", " ", "range", " ", "of", " ", \(\(integrals\)\(.\)\)}]}]], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Let us see, if this is really so:", "Subtitle"], Cell[BoxData[{ \(\(R0 = \@\(\((x - x0)\)\^2 + \((\ y - y0)\)\^2 + z\^2\);\)\), "\ \[IndentingNewLine]", \(\(R1 = \@\(\((x - x1)\)\^2 + \((y - y1)\)\^2 + h\^2\);\)\), "\ \[IndentingNewLine]", \(\(RR = \@\(\((x1 - x0)\)\^2 + \((y1 - y0)\)\^2 + \((z + h)\)^2\);\)\), \ "\[IndentingNewLine]", \(\(f1 = z\/R0\^3;\)\), "\[IndentingNewLine]", \(\(f2 = h\/R1\^3;\)\[IndentingNewLine]\), "\[IndentingNewLine]", \(Integrate[ Integrate[ f1\ f2, {x, \(-\[Infinity]\), \[Infinity]}], {y, \(-\[Infinity]\), \ \[Infinity]}]\), "\[IndentingNewLine]", \(\)}], "Input"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "No result. The exact result is ", Cell[BoxData[ \(\(2 \[Pi]\ \((z + h)\)\)\/RR\^3\)]], ". ", StyleBox["Mathematica", FontSlant->"Italic"], " was unable to find it." }], "Subtitle"], Cell[BoxData[ RowBox[{ StyleBox[ RowBox[{"\[FilledSmallSquare]", StyleBox["Integrate", "MR"]}]], StyleBox[" ", "MR"], "can", " ", "give", " ", "results", " ", "in", " ", "terms", " ", "of", " ", "many", " ", "special", " ", \(\(functions\)\(.\)\)}]], "Input"], Cell[BoxData[ RowBox[{ StyleBox["Integrate", "MR"], StyleBox[" ", "MR"], "can", " ", "evaluate", " ", "essentially", " ", "all", " ", "indefinite", " ", "integrals", " ", "and", " ", "most", " ", "definite", " ", "integrals", " ", "listed", " ", "in", " ", "standard", " ", "books", " ", "of", " ", \(\(tables\)\(.\)\)}]], "Input"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Let us see if ", StyleBox["Mathematica", FontSlant->"Italic"], " knows about Elliptic integrals" }], "Subtitle"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(\(Integrate[ 1\/\@\(\((\[Lambda]\^2 - 1)\) \((\[Lambda]\^2 - e\^2)\)\), {\[Lambda], \ \[Xi], \[Infinity]}]\)\(\[IndentingNewLine]\)\(\[IndentingNewLine]\) \)\)\)], "Input"], Cell[BoxData[ RowBox[{ SubsuperscriptBox["\[Integral]", "\[Xi]", InterpretationBox["\[Infinity]", DirectedInfinity[ 1]]], \(\(1\/\@\(\((\(-1\) + \[Lambda]\^2)\)\ \((\(-e\^2\) + \ \[Lambda]\^2)\)\)\) \[DifferentialD]\[Lambda]\)}]], "Output"] }, Open ]], Cell[BoxData[ StyleBox[\(No, it\ does\ not . \ \ The\ integral\ above\ is\ listed\ in\ all\ tables, \ \ even\ small\ \(\(ones\)\(.\)\)\), "Subtitle"]], "Input"], Cell["\<\ Even this integral requires a hypergeometric function. \ \>", "MathCaption", CellTags->{"S1.5.3", "3.9"}], Cell[CellGroupData[{ Cell["Integrate[(1 - x^2)^n, x]", "Input", CellTags->"S1.5.3"], Cell[BoxData[ \(x\ Hypergeometric2F1[1\/2, \(-n\), 3\/2, x\^2]\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Well, does ", StyleBox["Mathematica", FontSlant->"Italic"], " know that in case of an integer n, the hypergeometric function reduces to \ a finite sum?" }], "Subtitle"], Cell[CellGroupData[{ Cell[BoxData[ \(FullSimplify[%, {n \[Element] Integers}]\)], "Input"], Cell[BoxData[ \(x\ Hypergeometric2F1[1\/2, \(-n\), 3\/2, x\^2]\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ No, it does not. It is though able to write correct result for a specific \ integer n.\ \>", "Subtitle"], Cell[CellGroupData[{ Cell[BoxData[ \(FullSimplify[%] /. n \[Rule] 5\)], "Input"], Cell[BoxData[ \(x\ \((1 - \(5\ x\^2\)\/3 + 2\ x\^4 - \(10\ x\^6\)\/7 + \(5\ x\^8\)\/9 - x\^10\/11)\)\)], "Output"] }, Open ]] }, Open ]] }, FrontEndVersion->"4.2 for Microsoft Windows", ScreenRectangle->{{0, 800}, {0, 504}}, WindowSize->{691, 422}, WindowMargins->{{0, Automatic}, {Automatic, 0}} ] (******************************************************************* 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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