(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.2' 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. To get the notebook into a Mathematica-compatible application, do one of 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 clipboard, then use the Paste menu command inside the application. 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 or CRLF (Unix, Macintosh or MS-DOS style). 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As is well known, the perimeter, ", Cell[BoxData[ FormBox[ StyleBox["\[ScriptCapitalP]", FontSlant->"Plain"], TraditionalForm]]], ", of an ellipse with semimajor axis ", Cell[BoxData[ \(TraditionalForm\`a\)]], " and semiminor axis ", Cell[BoxData[ \(TraditionalForm\`b\)]], " ", Cell[BoxData[ FormBox[ StyleBox[Cell[""], "SubsectionNoSpace"], TraditionalForm]]], "can be expressed exactly as a complete elliptic integral of the second \ kind, which can also be written as a Gaussian hypergeometric function," }], "Abstract"], Cell[BoxData[ FormBox[ RowBox[{ "\[ScriptCapitalP]", "\[LongEqual]", \(4\ a\ \(E(1 - b\^2\/a\^2)\)\), "\[LongEqual]", RowBox[{"2", "\[Pi]", " ", "a", " ", RowBox[{ TagBox[ TagBox[ RowBox[{\(\(\[InvisiblePrefixScriptBase]\_2\)\(F\_1\)\), "\[InvisibleApplication]", RowBox[{"(", RowBox[{ TagBox[ TagBox[ RowBox[{ TagBox[\(1\/2\), Hypergeometric2F1, Editable->True], ",", TagBox[\(-\(1\/2\)\), Hypergeometric2F1, Editable->True]}], InterpretTemplate[ { SlotSequence[ 1]}&]], Hypergeometric2F1, Editable->False], ";", TagBox[ TagBox[ TagBox["1", Hypergeometric2F1, Editable->True], InterpretTemplate[ { SlotSequence[ 1]}&]], Hypergeometric2F1, Editable->False], ";", TagBox[\(1 - b\^2\/a\^2\), Hypergeometric2F1, Editable->True]}], ")"}]}], InterpretTemplate[ HypergeometricPFQ[ #, #2, #3]&], Editable->False], Hypergeometric2F1], "."}]}]}], TraditionalForm]], "NumberedEquation"], Cell[TextData[{ "What is less well known is that the various exact forms attributed to \ Maclaurin, Gauss-Kummer, and Euler, are related via quadratic transformation \ formulae for hypergeometric functions. In this way we obtain additional \ identities, including a particularly elegant formula, symmetric in ", Cell[BoxData[ \(TraditionalForm\`a\)]], " and ", Cell[BoxData[ \(TraditionalForm\`b\)]], "," }], "Abstract"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"\[ScriptCapitalP]", "\[LongEqual]", RowBox[{"2", " ", "\[Pi]", " ", \(\@\(a\ b\)\), " ", RowBox[{ SubscriptBox[ TagBox["P", LegendreP], \(1\/2\)], "(", \(\(a\^2 + b\^2\)\/\(2\ a\ b\)\), ")"}]}]}], ","}], TraditionalForm]], "NumberedEquation"], Cell[TextData[{ "where ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox[ TagBox["P", LegendreP], StyleBox["\[Nu]", FontSlant->"Plain"]], "(", "z", ")"}], TraditionalForm]]], " is a Legendre function." }], "Abstract"], Cell[TextData[{ "Approximate formulae can be obtained by truncating the series \ representations of exact formulas. For example, Kepler used the geometric \ mean, ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["\[ScriptCapitalP]", FontSlant->"Plain"], StyleBox["\[TildeTilde]", FontSlant->"Plain"], RowBox[{ StyleBox["2", FontSlant->"Plain"], " ", StyleBox["\[Pi]", FontSlant->"Plain"], " ", \(\@\(a\ b\)\)}]}], TraditionalForm]]], ". In this paper, we examine the properties of a number of approximate \ formulas, using series methods, polynomial interpolation, rational polynomial \ approximants, and minimax methods. " }], "Abstract"], Cell["", "Abstract"], Cell[CellGroupData[{ Cell["Cartesian Equation", "Section"], Cell[TextData[{ "The Cartesian equation for an ellipse with centre at ", Cell[BoxData[ \(TraditionalForm\`\((0, 0)\)\)]], ", semimajor axis ", Cell[BoxData[ \(TraditionalForm\`a\)]], ", and semiminor axis ", Cell[BoxData[ \(TraditionalForm\`b\)]], " reads" }], "Text"], Cell[BoxData[ \(TraditionalForm\`\(\[ScriptCapitalE](x_, y_) = \((x\/a)\)\^2 + \((y\/b)\)\^2 \[LongEqual] 1;\)\)], "Input"], Cell[TextData[{ "Introducing the parameter ", Cell[BoxData[ \(TraditionalForm\`\[Phi]\)]], " into the Cartesian coordinates, as ", Cell[BoxData[ \(TraditionalForm\`\((x = a\ \(sin(\[Phi])\), y = b\ \(cos(\[Phi])\))\)\)]], ", one verifies that the ellipse equation is satisfied." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(TraditionalForm\`Simplify[\[ScriptCapitalE](a\ \(sin(\[Phi])\), b\ \(cos(\[Phi])\))]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`True\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Arclength", "Section"], Cell["In general, the parametric arclength is defined by", "Text"], Cell[BoxData[ \(TraditionalForm\`\[ScriptCapitalL] = \[Integral]\_\(\[Phi]\_1\)\%\(\ \[Phi]\_2\)\(\@\(\((\[PartialD]x\/\[PartialD]\[Phi])\)\^2 + \((\[PartialD]y\/\ \[PartialD]\[Phi])\)\^2\)\) \[DifferentialD]\[Phi]\)], "NumberedEquation"], Cell[TextData[{ "The arclength of an ellipse as a function of the parameter ", Cell[BoxData[ \(TraditionalForm\`\[Phi]\)]], " is an (incomplete) elliptic integral of the second kind." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(TraditionalForm\`\[ScriptCapitalL](\[Phi]_) = With[{x = a\ \(sin(\[Phi])\), y = b\ \(cos(\[Phi])\)}, Simplify[\[Integral]\(\@\(\((\[PartialD]x\/\[PartialD]\[Phi])\)\^2 + \ \((\[PartialD]y\/\[PartialD]\[Phi])\)\^2\)\) \[DifferentialD]\[Phi], a > b > 0 \[And] 0 < \[Phi] < \[Pi]\/2]]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`a\ \(E(\[Phi] \[VerticalSeparator] 1 - b\^2\/a\^2)\)\)], "Output"] }, Open ]], Cell["Since,", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(TraditionalForm\`\[ScriptCapitalL](0) \[LongEqual] 0\)], "Input"], Cell[BoxData[ \(TraditionalForm\`True\)], "Output"] }, Open ]], Cell["the arclenth of the ellipse is", "Text"], Cell[BoxData[ \(TraditionalForm\`\[ScriptCapitalL](\[Phi]) = a\ \(E(\[Phi] \[VerticalSeparator] e\^2)\)\)], "NumberedEquation"], Cell[TextData[{ "where the eccentricity, ", Cell[BoxData[ \(TraditionalForm\`e\)]], ", is defined by" }], "Text"], Cell[BoxData[ \(TraditionalForm\`\(e(a_, b_) = \@\(1 - b\^2\/a\^2\);\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Perimeter", "Section"], Cell[TextData[{ "Since the parameter ranges over ", Cell[BoxData[ \(TraditionalForm\`0 \[LessEqual] \[Phi] \[LessEqual] \[Pi]/2\)]], " for one quarter of the ellipse, the perimeter of the ellipse is" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(TraditionalForm\`\(\[ScriptCapitalP]\_1\)(a_, b_) = 4 \( \[ScriptCapitalL](\[Pi]\/2)\)\)], "Input"], Cell[BoxData[ \(TraditionalForm\`4\ a\ \(E(1 - b\^2\/a\^2)\)\)], "Output"] }, Open ]], Cell[TextData[{ "That is ", Cell[BoxData[ \(TraditionalForm\`\[ScriptCapitalP] = 4\ a\ \(E(e\^2)\)\)]], " where ", Cell[BoxData[ \(TraditionalForm\`E(m)\)]], " is the complete elliptic integral of the second kind." }], "Text"], Cell[CellGroupData[{ Cell["Alternative Expressions for the Perimeter ", "Subsection"], Cell[TextData[{ "The above expression for the perimeter of the ellipse is ", StyleBox["unsymmetrical", FontSlant->"Italic"], " with respect to the parameters ", Cell[BoxData[ \(TraditionalForm\`a\)]], " and ", Cell[BoxData[ \(TraditionalForm\`b\)]], ". This is \[OpenCurlyDoubleQuote]unphysical\[CloseCurlyDoubleQuote] in \ that both parameters, being lengths of the (major and minor) axes, should be \ on the same footing. We can expect that a ", StyleBox["symmetric", FontSlant->"Italic"], " formula, when truncated, will more accurately approximate the perimeter \ for both ", Cell[BoxData[ \(TraditionalForm\`a \[GreaterEqual] b\)]], " and ", Cell[BoxData[ \(TraditionalForm\`a \[LessEqual] b\)]], "." }], "Text"], Cell["\<\ Noting that the complete elliptic integral is a gaussian \ hypergeometric function,\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ FormBox[ TagBox[ TagBox[ RowBox[{\(\(\[InvisiblePrefixScriptBase]\_2\)\(F\_1\)\), "\[InvisibleApplication]", RowBox[{"(", RowBox[{ TagBox[ TagBox[ RowBox[{ TagBox[\(1\/2\), Hypergeometric2F1, Editable->True], ",", TagBox[\(-\(1\/2\)\), Hypergeometric2F1, Editable->True]}], InterpretTemplate[ { SlotSequence[ 1]}&]], Hypergeometric2F1, Editable->False], ";", TagBox[ TagBox[ TagBox["1", Hypergeometric2F1, Editable->True], InterpretTemplate[ { SlotSequence[ 1]}&]], Hypergeometric2F1, Editable->False], ";", TagBox["z", Hypergeometric2F1, Editable->True]}], ")"}]}], InterpretTemplate[ HypergeometricPFQ[ #, #2, #3]&], Editable->False], Hypergeometric2F1], TraditionalForm]], "Input"], Cell[BoxData[ \(TraditionalForm\`\(2\ \(E(z)\)\)\/\[Pi]\)], "Output"] }, Open ]], Cell[TextData[{ "one obtains Maclaurin's 1742 formula (see [", CounterBox["Reference", "Michon"], "])" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ FormBox[ RowBox[{\(\(\[ScriptCapitalP]\_1\)(a, b)\), "\[LongEqual]", RowBox[{"2", "\[Pi]", " ", "a", " ", TagBox[ TagBox[ RowBox[{\(\(\[InvisiblePrefixScriptBase]\_2\)\(F\_1\)\), "\[InvisibleApplication]", RowBox[{"(", RowBox[{ TagBox[ TagBox[ RowBox[{ TagBox[\(1\/2\), Hypergeometric2F1, Editable->True], ",", TagBox[\(-\(1\/2\)\), Hypergeometric2F1, Editable->True]}], InterpretTemplate[ { SlotSequence[ 1]}&]], Hypergeometric2F1, Editable->False], ";", TagBox[ TagBox[ TagBox["1", Hypergeometric2F1, Editable->True], InterpretTemplate[ { SlotSequence[ 1]}&]], Hypergeometric2F1, Editable->False], ";", TagBox[\(\(e(a, b)\)\^2\), Hypergeometric2F1, Editable->True]}], ")"}]}], InterpretTemplate[ HypergeometricPFQ[ #, #2, #3]&], Editable->False], Hypergeometric2F1]}]}], TraditionalForm]], "Input"], Cell[BoxData[ \(TraditionalForm\`True\)], "Output"] }, Open ]], Cell[TextData[{ "Equivalent alternative expressions for the perimeter of the ellipse can be \ obtained from quadratic transformation formul\[AE] for gaussian \ hypergeometric functions. For example, using ", ButtonBox["functions.wolfram.com/07.23.17.0106.01", ButtonData:>{ URL[ "http://functions.wolfram.com/07.23.17.0106.01"], None}, ButtonStyle->"Hyperlink"], ", " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ FormBox[ RowBox[{"Simplify", "[", RowBox[{ RowBox[{ RowBox[{ TagBox[ TagBox[ RowBox[{\(\(\[InvisiblePrefixScriptBase]\_2\)\(F\_1\)\), "\[InvisibleApplication]", RowBox[{"(", RowBox[{ TagBox[ TagBox[ RowBox[{ TagBox["\[Alpha]", Hypergeometric2F1, Editable->True], ",", TagBox["\[Beta]", Hypergeometric2F1, Editable->True]}], InterpretTemplate[ { SlotSequence[ 1]}&]], Hypergeometric2F1, Editable->False], ";", TagBox[ TagBox[ TagBox[\(2\ \[Beta]\), Hypergeometric2F1, Editable->True], InterpretTemplate[ { SlotSequence[ 1]}&]], Hypergeometric2F1, Editable->False], ";", TagBox["z", Hypergeometric2F1, Editable->True]}], ")"}]}], InterpretTemplate[ HypergeometricPFQ[ #, #2, #3]&], Editable->False], Hypergeometric2F1], "\[LongEqual]", FractionBox[ TagBox[ TagBox[ RowBox[{\(\(\[InvisiblePrefixScriptBase]\_2\)\(F\_1\)\), "\[InvisibleApplication]", RowBox[{"(", RowBox[{ TagBox[ TagBox[ RowBox[{ TagBox["\[Alpha]", Hypergeometric2F1, Editable->True], ",", TagBox[\(\[Alpha] - 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1)\)\ \(K(h)\))\)\)], "Output"] }, Open ]], Cell["\<\ Explicitly, the Gauss-Kummer series reads \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(TraditionalForm\`\(\[ScriptCapitalP]\_3\)(a_, b_) = FullSimplify[\(\[ScriptCapitalP]\_2\)(a, b) // FunctionExpand, a > b > 0]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`4\ \((a + b)\)\ \(E( 1 - \(4\ a\ b\)\/\((a + b)\)\^2)\) - \(8\ a\ b\ \(K(1 - \(4\ a\ b\)\ \/\((a + b)\)\^2)\)\)\/\(a + b\)\)], "Output"] }, Open ]], Cell[TextData[{ "Instead, using ", ButtonBox["functions.wolfram.com/07.23.17.0103.01", ButtonData:>{ URL[ "http://functions.wolfram.com/07.23.17.0103.01"], None}, ButtonStyle->"Hyperlink"], ", one obtains Euler's 1773 formula (see also [", CounterBox["Reference", "Michon"], "]):" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{ TagBox[ TagBox[ RowBox[{\(\(\[InvisiblePrefixScriptBase]\_2\)\(F\_1\)\), "\[InvisibleApplication]", RowBox[{"(", RowBox[{ TagBox[ TagBox[ RowBox[{ TagBox["\[Alpha]", Hypergeometric2F1, Editable->True], ",", TagBox["\[Beta]", Hypergeometric2F1, Editable->True]}], InterpretTemplate[ { SlotSequence[ 1]}&]], Hypergeometric2F1, Editable->False], ";", TagBox[ TagBox[ TagBox[\(2 \[Beta]\), Hypergeometric2F1, Editable->True], InterpretTemplate[ { SlotSequence[ 1]}&]], Hypergeometric2F1, Editable->False], ";", TagBox["z", Hypergeometric2F1, Editable->True]}], ")"}]}], InterpretTemplate[ HypergeometricPFQ[ #, #2, #3]&], Editable->False], Hypergeometric2F1], "\[LongEqual]", FractionBox[ TagBox[ TagBox[ RowBox[{\(\(\[InvisiblePrefixScriptBase]\_2\)\(F\_1\)\), "\[InvisibleApplication]", RowBox[{"(", RowBox[{ TagBox[ TagBox[ RowBox[{ TagBox[\(\[Alpha]\/2\), Hypergeometric2F1, Editable->True], ",", TagBox[\(\(\[Alpha] + 1\)\/2\), Hypergeometric2F1, Editable->True]}], InterpretTemplate[ { SlotSequence[ 1]}&]], Hypergeometric2F1, Editable->False], ";", TagBox[ TagBox[ TagBox[\(\[Beta] + 1\/2\), Hypergeometric2F1, Editable->True], InterpretTemplate[ { SlotSequence[ 1]}&]], Hypergeometric2F1, Editable->False], ";", TagBox[\(z\^2\/\((2 - 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Cell[CellGroupData[{ Cell["Chebyshev polynomial Approximant", "Subsubsection"], Cell[TextData[{ "Sampling the Gauss-Kummer function at the zeros of ", Cell[BoxData[ \(TraditionalForm\`\(T\_n\)(2\ x - 1)\)]], ", which are at ", Cell[BoxData[ \(TraditionalForm\`x\_m = \(cos\^2\)(\((m + 1/4)\) \[Pi]\/n)\)]], ", yields a Chebyshev polynomial approximant." }], "Text"], Cell[BoxData[ \(TraditionalForm\`Chebyshevpoly[n_] := \(Chebyshevpoly[n] = Function[h, Evaluate@ InterpolatingPolynomial[ N@Join[{{0, GaussKummer[0]}, {1, GaussKummer[1]}}, Table[{\(cos\^2\)(\((m + 1/4)\) \[Pi]\/n), GaussKummer[\(cos\^2\)(\((m + 1/4)\) \[Pi]\/n)]}, {m, n}]], \ h]]\)\)], "Input"], Cell[TextData[{ "The ", Cell[BoxData[ \(TraditionalForm\`8\^th\)]], "-order approximant has a maximum absolute relative error of ", Cell[BoxData[ \(TraditionalForm\`\(\(\[LessTilde]\)\(7\[Times]10\^\(-6\)\)\)\)]], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(TraditionalForm\`Plot[ 10\^6\ \((1 - 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"properties of special functions\[LongDash]such as identities and \ transformations\[LongDash]are available at MathWorld [", CounterBox["Reference", "Arithmetic-Geometric Mean"], "] and the Wolfram functions Site [", CounterBox["Reference", "functions"], "] and, because these properties are expressed in ", StyleBox["Mathematica", FontSlant->"Italic"], " syntax, can be used directly;" }], "BulletedList"], Cell["\<\ relevant built-in numerical methods include rational polynomial \ approximants, minimax methods, and numerical optimization for arbitrary \ norms;\ \>", "BulletedList"], Cell["\<\ visualization of approximants can be used to estimate the quality \ of approximants; and\ \>", "BulletedList"], Cell["\<\ combining these approaches is straightforward and leads, in a \ natural way, to optimal approximants.\ \>", "BulletedList"], Cell["\<\ This paper uses the exercise of computing the perimeter of an \ ellipse using a simple set of approximants to illustrate these points.\ \>", \ 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