Services & Resources / Wolfram Forums
-----
 /
MathGroup Archive
2004
*January
*February
*March
*April
*May
*June
*July
*August
*September
*October
*November
*December
*Archive Index
*Ask about this page
*Print this page
*Give us feedback
*Sign up for the Wolfram Insider

MathGroup Archive 2004

[Date Index] [Thread Index] [Author Index]

Search the Archive

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.  The cache data will then be recreated when
you save this file from within Mathematica.
*******************************************************************)

(*CellTagsOutline
CellTagsIndex->{
  "S1.5.3"->{
    Cell[9784, 286, 119, 5, 59, "MathCaption",
      CellTags->{"S1.5.3", "3.9"}],
    Cell[9928, 295, 64, 1, 30, "Input",
      CellTags->"S1.5.3"]},
  "3.9"->{
    Cell[9784, 286, 119, 5, 59, "MathCaption",
      CellTags->{"S1.5.3", "3.9"}]}
  }
*)

(*CellTagsIndex
CellTagsIndex->{
  {"S1.5.3", 11501, 356},
  {"3.9", 11663, 361}
  }
*)

(*NotebookFileOutline
Notebook[{

Cell[CellGroupData[{
Cell[1776, 53, 459, 12, 183, "Subtitle"],
Cell[2238, 67, 652, 14, 90, "Input"]
}, Open  ]],

Cell[CellGroupData[{
Cell[2927, 86, 177, 5, 94, "Subtitle"],
Cell[3107, 93, 614, 9, 157, "Input"]
}, Open  ]],

Cell[CellGroupData[{
Cell[3758, 107, 344, 7, 181, "Subtitle"],
Cell[4105, 116, 26, 0, 30, "Input"],
Cell[4134, 118, 836, 19, 90, "Input"],
Cell[4973, 139, 184, 4, 77, "Input"],
Cell[5160, 145, 230, 3, 66, "Input"],

Cell[CellGroupData[{
Cell[5415, 152, 48, 1, 30, "Input"],
Cell[5466, 155, 672, 12, 205, "Output"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{
Cell[6187, 173, 90, 2, 64, "Subtitle"],
Cell[6280, 177, 927, 20, 90, "Input"]
}, Open  ]],

Cell[CellGroupData[{
Cell[7244, 202, 53, 0, 64, "Subtitle"],
Cell[7300, 204, 599, 13, 205, "Input"]
}, Open  ]],

Cell[CellGroupData[{
Cell[7936, 222, 220, 8, 97, "Subtitle"],
Cell[8159, 232, 329, 9, 30, "Input"],
Cell[8491, 243, 385, 8, 50, "Input"]
}, Open  ]],

Cell[CellGroupData[{
Cell[8913, 256, 139, 5, 65, "Subtitle"],

Cell[CellGroupData[{
Cell[9077, 265, 229, 4, 113, "Input"],
Cell[9309, 271, 280, 6, 47, "Output"]
}, Open  ]],
Cell[9604, 280, 177, 4, 115, "Input"],
Cell[9784, 286, 119, 5, 59, "MathCaption",
  CellTags->{"S1.5.3", "3.9"}],

Cell[CellGroupData[{
Cell[9928, 295, 64, 1, 30, "Input",
  CellTags->"S1.5.3"],
Cell[9995, 298, 80, 1, 42, "Output"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{
Cell[10124, 305, 195, 6, 94, "Subtitle"],

Cell[CellGroupData[{
Cell[10344, 315, 73, 1, 30, "Input"],
Cell[10420, 318, 80, 1, 42, "Output"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{
Cell[10549, 325, 115, 3, 93, "Subtitle"],

Cell[CellGroupData[{
Cell[10689, 332, 63, 1, 30, "Input"],
Cell[10755, 335, 129, 2, 45, "Output"]
}, Open  ]]
}, Open  ]]
}
]
*)



Alex


  • Prev by Date: Re: Compile
  • Next by Date: convert notebook into batch file
  • Previous by thread: RE: multiple list plot and frameticks
  • Next by thread: convert notebook into batch file