(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 4.0, MathReader 4.0, or any compatible application. 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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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@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[ 201417, 5664]*) (*NotebookOutlinePosition[ 203380, 5731]*) (* CellTagsIndexPosition[ 203162, 5720]*) (*WindowFrame->Normal*) Notebook[{ Cell["Non-trivial asymptotic formulas by symbolic computation", "Title"], Cell["\<\ Giuliano Gargiulo Saverio Salerno\ \>", "Author"], Cell[TextData[{ "Abstract: we want to study the asymptotic behavior of the complete \ elliptic integral of the first kind ", Cell[BoxData[ \(TraditionalForm\`K(m)\)], "InlineFormula"], " when m\[Rule]1. This is motivated, for example, by the occurrence of \ K(m) as capacitance of a circular capacitor with slit (or similar geometries \ - see e.g. [9]), m being essentially the ratio between the slit's length and \ the radius of the circle.\nWe show that the analysis of the asymptotic \ behavior can be done in several ways (including series expansion and \ summation, symbolic integration and computation of limits) using both the \ numerical and symbolical capabilities of state-of-the-art Computer Algebra \ Systems. " }], "Abstract"], Cell["Introduction", "SectionFirst"], Cell[TextData[StyleBox["We want to prove that the complete elliptic integral \ of the first kind K(m) has a logarithmic behavior when m approaches 1. More \ precisely", FontSize->16, FontWeight->"Bold"]], "Text"], Cell[BoxData[ \(TraditionalForm \`lim\+\(m \[Rule] 1\)\[ThinSpace]\(K(m) + 1\/2\ \(log(1 - m)\)\) == 2\ \(log(2)\)\)], "NumberedEquation", CellFrame->False, FormatType->StandardForm], Cell[TextData[StyleBox["where K(m) is defined by", FontSize->16, FontWeight->"Bold"]], "Text"], Cell[BoxData[ \(TraditionalForm \`K(m) = \[Integral]\_0\%1 \(\([\((1 - t\^2)\) \((1 - m t\^2)\)]\)\^\(\(-1\)/2\)\) \[DifferentialD]t\)], "NumberedEquation", FormatType->StandardForm], Cell[TextData[StyleBox["Moreover, we show that (1) is equivalent to ", FontSize->16, FontWeight->"Bold"]], "Text"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"(", FormBox[ RowBox[{ RowBox[{\(\[Sum]\+\(n = 0\)\%\[Infinity]\), RowBox[{"(", RowBox[{ FractionBox[ RowBox[{"\[Pi]", " ", SuperscriptBox[ RowBox[{"(", TagBox[ RowBox[{"(", GridBox[{ {\(2\ n\)}, {"n"} }], ")"}], Binomial, Editable->False], ")"}], "2"]}], \(2\ 16\^n\)], "-", \(1\/\(2\ n + 1\)\)}], ")"}]}], "==", \(log(2)\)}], "TraditionalForm"], ")"}], " "}], TraditionalForm]], "NumberedEquation", CellFrame->False, FormatType->StandardForm], Cell[TextData[{ StyleBox[ "In doing this, both symbolic and numeric power of a Computer Algebra \ System will be exploited, namely ", FontSize->16, FontWeight->"Bold"], StyleBox["Mathematica", FontSize->16, FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" by Wolfram Research. 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