(************** 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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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[ 20257, 785]*) (*NotebookOutlinePosition[ 21324, 820]*) (* CellTagsIndexPosition[ 21280, 816]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Transform Methods", "Title", Evaluatable->False, TextAlignment->Center, AspectRatioFixed->True], Cell[TextData[{ "Sean Mauch\nsean@caltech.edu\n", ButtonBox["http://www.its.caltech.edu/~sean", ButtonData:>{ URL[ "http://www.its.caltech.edu/~sean"], None}, ButtonStyle->"Hyperlink"], "\n", "This work is distributed under the GNU FDL. See ", ButtonBox["license.nb ", ButtonData:>{"license.nb", None}, ButtonStyle->"Hyperlink"], "for details." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["The Laplace Transform", "Section"], Cell[CellGroupData[{ Cell["Laplace Transform", "Subsection"], Cell[TextData[{ "To use the Laplace transform load the ", StyleBox["LaplaceTransform", "Input"], " package." }], "Text", TextAlignment->Left, TextJustification->1], Cell[BoxData[ \(<< Calculus`LaplaceTransform`\)], "Input"], Cell[TextData[{ "The ", Cell[BoxData[ FormBox[ StyleBox[\(LaplaceTransform[]\), "Input"], TraditionalForm]]], " function gives the Laplace transform of a function." }], "Text", TextAlignment->Left, TextJustification->1], Cell[BoxData[ \(TraditionalForm \`F \((s)\) = \[Integral]\_0\%\[Infinity]\( \[ExponentialE]\^\(\(-s\)\ t\)\) \(f(t)\) \[DifferentialD]t\)], "DisplayFormula", TextAlignment->Center, TextJustification->0], Cell[BoxData[ \(\(?LaplaceTransform\)\)], "Input"], Cell[TextData[{ "The Laplace transform of ", Cell[BoxData[ \(TraditionalForm\`t\ \[ExponentialE]\^t\)]], " is" }], "Text", TextAlignment->Left, TextJustification->1], Cell[BoxData[ \(LaplaceTransform[t\ Exp[t], t, s]\)], "Input"], Cell[TextData[{ "The Laplace transform of ", Cell[BoxData[ \(TraditionalForm\`\(f'\)' \((t)\)\)]], " is" }], "Text", TextAlignment->Left, TextJustification->1], Cell[BoxData[ \(LaplaceTransform[\(\(f'\)'\)[t], t, s]\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Inverse Laplace Transform", "Subsection"], Cell[TextData[{ "The ", Cell[BoxData[ FormBox[ StyleBox[\(InverseLaplaceTransform[]\), "Input"], TraditionalForm]]], " function gives the inverse Laplace transform." }], "Text", TextAlignment->Left, TextJustification->1], Cell[BoxData[ \(TraditionalForm \`f \((t)\) = \(1\/\(2 \[Pi]\ \[ImaginaryI]\)\) \(\[Integral]\_\(c - \[ImaginaryI]\[Infinity]\)\%\(c + \[ImaginaryI]\[Infinity]\)\(\[ExponentialE]\^\(s\ t\)\) \(F(s)\) \(\[DifferentialD]s . \)\)\)], "DisplayFormula", TextAlignment->Center, TextJustification->0], Cell[TextData[{ "Here ", Cell[BoxData[ \(TraditionalForm\`c\)]], " is to the right of the singularities of ", Cell[BoxData[ \(TraditionalForm\`F(s)\)]], ". 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