(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 7.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 165192, 6244] NotebookOptionsPosition[ 141359, 5449] NotebookOutlinePosition[ 142966, 5502] CellTagsIndexPosition[ 142746, 5493] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Prepare Notebook", "Subsubsection"], Cell[CellGroupData[{ Cell["email", "Text"], Cell[TextData[StyleBox["Robert Zimmerman\nUniversity of Oregon\nInstitute of \ Theoretical Science\nEugene, OR 97405\nbob@zim.uoregon.edu\n\ http://darkwing.uoregon.edu/~phys600", FontSize->9, FontWeight->"Bold"]], "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Reference ", FontFamily->"Times New Roman", FontVariations->{"CompatibilityType"->0}], StyleBox[" ", FontFamily->"Times New Roman"] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Arial"], Cell[TextData[{ " Robert L Zimmerman, 2006: Permission to copy and use the mathematica \ notebook for internal use is granted, provided that the work is properly \ referenced. These notebooks are used for the authors course on ", StyleBox["Mathematica", FontSlant->"Italic"], " and will be submitted for publication after they have been polished so \ please send remarks and observations to the author at bob@zim.uoregon.edu." }], "Text", Evaluatable->False, AspectRatioFixed->True, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Clear the memory and turn off the spell check message for the rest of \ this notebook. \ \>", "Text"], Cell[BoxData[ RowBox[{"Clear", "[", "\"\\"", "]"}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"Off", "[", RowBox[{"General", "::", "spell"}], "]"}], ";", " ", RowBox[{"Off", "[", RowBox[{"General", "::", "spell1"}], "]"}], ";"}]], "Input"] }, Open ]] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{"date", "=", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"NumberForm", "[", RowBox[{"#1", ",", "4"}], "]"}], "&"}], ")"}], "[", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"TableForm", "[", RowBox[{"#1", ",", RowBox[{"TableSpacing", "\[Rule]", RowBox[{"{", RowBox[{"1", ",", FractionBox["1", "4"]}], "}"}]}]}], "]"}], "&"}], ")"}], "[", RowBox[{"Transpose", "[", RowBox[{"{", RowBox[{ RowBox[{"Drop", "[", RowBox[{ RowBox[{"DateList", "[", "]"}], ",", RowBox[{"-", "2"}]}], "]"}], ",", RowBox[{"{", RowBox[{ "\"\\"", ",", "\"\\"", ",", "\"\\"", ",", "\"\\""}], "}"}]}], "}"}], "]"}], "]"}], "]"}]}], ";"}]], "Input"], Cell[CellGroupData[{ Cell[BoxData["date"], "Input"], Cell[BoxData[ TagBox[ TagBox[GridBox[{ { InterpretationBox["\<\"2006\"\>", 2006, Editable->False], "\<\"year\"\>"}, { InterpretationBox["\<\"1\"\>", 1, Editable->False], "\<\"month\"\>"}, { InterpretationBox["\<\"25\"\>", 25, Editable->False], "\<\"day\"\>"}, { InterpretationBox["\<\"13\"\>", 13, Editable->False], "\<\"hour\"\>"} }, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.175]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], Function[BoxForm`e$, TableForm[BoxForm`e$, TableSpacing -> {1, Rational[1, 4]}]]], NumberForm[#, 4]& ]], "Output"] }, Closed]], Cell[TextData[{ StyleBox[" Notebook #3: Fundamental Calculations", FontFamily->"Brush Script MT", FontSize->16, FontColor->GrayLevel[0]], StyleBox[" ", FontFamily->"Monotype Corsiva", FontColor->GrayLevel[0]] }], "Title", Evaluatable->False, TextAlignment->Center, TextJustification->0, AspectRatioFixed->True, FontColor->RGBColor[1, 0, 1]], Cell[TextData[{ StyleBox["Using ", FontFamily->"Brush Script MT", FontSize->10, FontColor->GrayLevel[0]], StyleBox["Mathematica ", FontFamily->"Brush Script MT", FontSize->10, FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["\n", FontFamily->"Brush Script MT", FontSize->10, FontColor->GrayLevel[0]], StyleBox["Robert L. 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The position of b x follows from Position[eq1,b x]\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{" ", RowBox[{"Position", "[", RowBox[{"eq1", ",", RowBox[{"b", " ", "x"}]}], " ", "]"}]}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{"{", "3", "}"}], "}"}]], "Output"] }, Open ]], Cell["Extracting the part at position 3 gives", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{" ", RowBox[{"{", " ", RowBox[{ RowBox[{"eq1", "[", RowBox[{"[", "3", "]"}], "]"}], ",", RowBox[{"(*", " ", "or", " ", "*)"}], RowBox[{"Part", "[", RowBox[{"eq1", ",", "3"}], "]"}]}], " ", "}"}]}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"b", " ", "x"}], ",", RowBox[{"b", " ", "x"}]}], "}"}]], "Output"] }, Open ]], Cell["Another example is the x/y term", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{" ", RowBox[{"Position", "[", RowBox[{"eq1", ",", RowBox[{"x", "/", "y"}]}], " ", "]"}], " "}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{"4", ",", "1"}], "}"}], "}"}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"eq1", "[", RowBox[{"[", RowBox[{"4", ",", "1"}], " ", "]"}], "]"}], ",", RowBox[{"Part", "[", RowBox[{"eq1", ",", " ", "4", ",", "1"}], " ", "]"}]}], " ", "}"}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ FractionBox["x", "y"], ",", FractionBox["x", "y"]}], "}"}]], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Replace parts in Expressions ", "Subsection"], Cell["\<\ The command ReplacePart[expr, new, n] replaces the nth part of expr with \ new. For example, consider\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["eq2= a+c/(b+c Sqrt[(x+b)^2+a]-e)", "Input", AspectRatioFixed->True], Cell[BoxData[ RowBox[{"a", "+", FractionBox["c", RowBox[{"b", "-", "e", "+", RowBox[{"c", " ", SqrtBox[ RowBox[{"a", "+", SuperscriptBox[ RowBox[{"(", RowBox[{"b", "+", "x"}], ")"}], "2"]}]]}]}]]}]], "Output"] }, Open ]], Cell["\<\ and replace (x + b)^2 +a with its expanded form. The position of the term \ is \ \>", "Text"], Cell[CellGroupData[{ Cell["\<\ part= Position[eq2,(x+b)^2+a ] \ \>", "Input"], Cell[BoxData[ RowBox[{ RowBox[{"General", "::", "\<\"spell\"\>"}], RowBox[{ ":", " "}], "\<\"Possible spelling error: new symbol name \\\"\\!\\(part\\)\ \\\" is similar to existing symbols \\!\\({Apart, Part}\\). \ \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", ButtonStyle->\\\"RefGuideLinkText\ \\\", ButtonFrame->None, ButtonData:>\\\"General::spell\\\"]\\)\"\>"}]], \ "Message"], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{"2", ",", "2", ",", "1", ",", "3", ",", "2", ",", "1"}], "}"}], "}"}]], "Output"] }, Open ]], Cell["so it follows that ", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"ReplacePart", "[", RowBox[{"eq2", ",", RowBox[{"Expand", "[", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"x", " ", "+", " ", "b"}], ")"}], "^", "2"}], "+", "a"}], "]"}], ",", "part"}], "]"}]], "Input"], Cell[BoxData[ RowBox[{"a", "+", FractionBox["c", RowBox[{"b", "-", "e", "+", RowBox[{"c", " ", SqrtBox[ RowBox[{"a", "+", SuperscriptBox["b", "2"], "+", RowBox[{"2", " ", "b", " ", "x"}], "+", SuperscriptBox["x", "2"]}]]}]}]]}]], "Output"] }, Open ]], Cell["An alternate solution follows from ExpandALL", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"eq2", "//", "ExpandAll"}]], "Input"], Cell[BoxData[ RowBox[{"a", "+", FractionBox["c", RowBox[{"b", "-", "e", "+", RowBox[{"c", " ", SqrtBox[ RowBox[{"a", "+", SuperscriptBox["b", "2"], "+", RowBox[{"2", " ", "b", " ", "x"}], "+", SuperscriptBox["x", "2"]}]]}]}]]}]], "Output"] }, Open ]], Cell["\<\ ReplacePart[expr, new, {{i1, j1, ...}, {i2, j2, ...}, ...}] replaces \ several terms with new. \ \>", "Text", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell["Selecting Parts of an expression using Patterns", "Subsection"], Cell["\<\ The command Cases[{e1, e2, ... }, pattern] gives a list of the ei that \ match the pattern. Consider the expression\ \>", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"exp", "=", " ", RowBox[{"a", " ", "+", " ", RowBox[{"Exp", "[", RowBox[{"-", RowBox[{"x", "^", "2"}]}], "]"}], " ", "+", RowBox[{"w", "^", RowBox[{"(", RowBox[{"5", "^", "x"}], ")"}]}], " ", "+", RowBox[{"u", "^", "2"}], " ", "+", RowBox[{"1", "/", RowBox[{"z", "^", "2"}]}], "+", "1.4"}]}], ";"}]], "Input"], Cell["and match those terms that have the pattern x_^2 or _^2, ", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"Cases", "[", RowBox[{"exp", ",", RowBox[{"x_", "^", "2"}]}], "]"}], " ", ",", RowBox[{"Cases", "[", RowBox[{"exp", ",", " ", RowBox[{"_", "^", "2"}]}], " ", "]"}]}], "}"}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", SuperscriptBox["u", "2"], "}"}], ",", RowBox[{"{", SuperscriptBox["u", "2"], "}"}]}], "}"}]], "Output"] }, Open ]], Cell["\<\ This is not all the terms because some patterns are located deeper in the \ expression like Exp[-x^2]. You get these terms by specifying the level. \ Cases[expr, pattern, levspec] gives a list of all parts of expr on levels \ specified by levspec which match the pattern. Increase the level for the \ previous example\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"{", " ", RowBox[{ RowBox[{"Cases", "[", RowBox[{"exp", ",", RowBox[{"x_", "^", "2"}], " ", ",", "1"}], "]"}], ",", " ", RowBox[{"Cases", "[", RowBox[{"exp", ",", " ", RowBox[{"x_", "^", "2"}], " ", ",", "3"}], "]"}], ",", " ", RowBox[{"Cases", "[", RowBox[{"exp", ",", " ", RowBox[{"x_", "^", "2"}], ",", "Infinity"}], "]"}]}], " ", "}"}], " "}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", SuperscriptBox["u", "2"], "}"}], ",", RowBox[{"{", RowBox[{ SuperscriptBox["x", "2"], ",", SuperscriptBox["u", "2"]}], "}"}], ",", RowBox[{"{", RowBox[{ SuperscriptBox["x", "2"], ",", SuperscriptBox["u", "2"]}], "}"}]}], "}"}]], "Output"] }, Open ]], Cell["To extract all terms of the form x_^y_ or _^ _ enter", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"Cases", "[", RowBox[{"exp", ",", RowBox[{"x_", "^", "y_"}], " ", ",", "Infinity"}], "]"}], ",", RowBox[{"(*", "or", "*)"}], " ", RowBox[{"Cases", "[", RowBox[{"exp", ",", " ", RowBox[{"_", "^", " ", "_"}], " ", ",", "Infinity"}], "]"}]}], "}"}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ SuperscriptBox["x", "2"], ",", SuperscriptBox["\[ExponentialE]", RowBox[{"-", SuperscriptBox["x", "2"]}]], ",", SuperscriptBox["u", "2"], ",", SuperscriptBox["5", "x"], ",", SuperscriptBox["w", SuperscriptBox["5", "x"]], ",", FractionBox["1", SuperscriptBox["z", "2"]]}], "}"}], ",", RowBox[{"{", RowBox[{ SuperscriptBox["x", "2"], ",", SuperscriptBox["\[ExponentialE]", RowBox[{"-", SuperscriptBox["x", "2"]}]], ",", SuperscriptBox["u", "2"], ",", SuperscriptBox["5", "x"], ",", SuperscriptBox["w", SuperscriptBox["5", "x"]], ",", FractionBox["1", SuperscriptBox["z", "2"]]}], "}"}]}], "}"}]], "Output"] }, Open ]], Cell[" Consider another example ", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ RowBox[{ RowBox[{"eq1", " ", "=", RowBox[{ RowBox[{"a", " ", RowBox[{"Cos", "[", "x", "]"}]}], "+", RowBox[{"b", " ", RowBox[{"Cos", "[", "x", "]"}]}], "-", RowBox[{"b", " ", SuperscriptBox["c", "2"], " ", RowBox[{"Cos", "[", RowBox[{"2", " ", "x"}], "]"}]}], "-", RowBox[{"2", " ", "b", " ", "c", " ", "d", " ", RowBox[{"Cos", "[", RowBox[{"2", " ", "x"}], "]"}]}], "-", RowBox[{"b", " ", SuperscriptBox["d", "2"], " ", RowBox[{"Cos", "[", RowBox[{"2", " ", "x"}], "]"}]}], "+", RowBox[{"c", " ", RowBox[{"Cos", "[", RowBox[{"x", "-", "y"}], "]"}]}]}]}], ";"}]], "Input"], Cell["Assume you want the Cos[x] terms then enter ", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"Cases", "[", RowBox[{"eq1", ",", RowBox[{"Cos", "[", "x", " ", "]"}], ",", "2"}], "]"}], ",", RowBox[{"(*", "or", "*)"}], " ", RowBox[{"Cases", "[", RowBox[{"eq1", ",", RowBox[{"Cos", "[", "x", " ", "]"}], ",", "Infinity"}], "]"}]}], "}"}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Cos", "[", "x", "]"}], ",", RowBox[{"Cos", "[", "x", "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"Cos", "[", "x", "]"}], ",", RowBox[{"Cos", "[", "x", "]"}]}], "}"}]}], "}"}]], "Output"] }, Open ]], Cell["\<\ You must go to level 2 or above to get all the Cos[x] terms, If you want all Cos [] terms, regardless of the argument, then enter \ the pattern Cos[x_] \ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ RowBox[{"Column", "[", RowBox[{"{", RowBox[{ RowBox[{"Cases", "[", RowBox[{"eq1", ",", RowBox[{"Cos", "[", "x_", "]"}], ",", "2"}], "]"}], ",", RowBox[{"Cases", "[", RowBox[{"eq1", ",", RowBox[{"Cos", "[", "x_", "]"}], ",", "\[Infinity]"}], "]"}]}], "}"}], "]"}]], "Input"], Cell[BoxData[ InterpretationBox[GridBox[{ { RowBox[{"{", RowBox[{ RowBox[{"Cos", "[", "x", "]"}], ",", RowBox[{"Cos", "[", "x", "]"}], ",", RowBox[{"Cos", "[", RowBox[{"2", " ", "x"}], "]"}], ",", RowBox[{"Cos", "[", RowBox[{"2", " ", "x"}], "]"}], ",", RowBox[{"Cos", "[", RowBox[{"2", " ", "x"}], "]"}], ",", RowBox[{"Cos", "[", RowBox[{"x", "-", "y"}], "]"}]}], "}"}]}, { RowBox[{"{", RowBox[{ RowBox[{"Cos", "[", "x", "]"}], ",", RowBox[{"Cos", "[", "x", "]"}], ",", RowBox[{"Cos", "[", RowBox[{"2", " ", "x"}], "]"}], ",", RowBox[{"Cos", "[", RowBox[{"2", " ", "x"}], "]"}], ",", RowBox[{"Cos", "[", RowBox[{"2", " ", "x"}], "]"}], ",", RowBox[{"Cos", "[", RowBox[{"x", "-", "y"}], "]"}]}], "}"}]} }, BaselinePosition->{Baseline, {1, 1}}, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}], ColumnForm[{{ Cos[x], Cos[x], Cos[2 x], Cos[2 x], Cos[2 x], Cos[x - y]}, { Cos[x], Cos[x], Cos[2 x], Cos[2 x], Cos[2 x], Cos[x - y]}}], Editable->False]], "Output"], Cell["\<\ Likewise, you can identify terms of the form 2 x using the pattern 2_,\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"temp2", "=", " ", RowBox[{"Cases", "[", RowBox[{"eq1", ",", RowBox[{"Cos", "[", RowBox[{"2", "_"}], "]"}], ",", "Infinity"}], " ", "]"}], " "}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"Cos", "[", RowBox[{"2", " ", "x"}], "]"}], ",", RowBox[{"Cos", "[", RowBox[{"2", " ", "x"}], "]"}], ",", RowBox[{"Cos", "[", RowBox[{"2", " ", "x"}], "]"}]}], "}"}]], "Output"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Simplifying Expressions", FontWeight->"Bold"]], "Section"], Cell[TextData[{ " One of the powers of ", StyleBox["Mathematica", FontSlant->"Italic"], " is its ability to simplify expressions. There are many commands that can \ be used to simplify algebraic expressions. Some of these commands are:\ni. \ Simplify, FullSimplify,\nii. Factor, FactorTerms, Together, Collect, Cancel \n\ iii. ComplexExpand, Expand, ExpandAll, ExpandDenominator, ExpandNumerator, \ PowerExpand, FunctionExpand, LogicalExpand \niv. Apart, ApartSquareFree, \ Coefficient, CoefficientList, Denominator, Numerator, Select,\nv. TrigToExp, \ ExpToTrig ,TrigExpand, TrigFactor, TrigReduce\nIn this section we illustrate \ the most common commands. " }], "Text"], Cell[CellGroupData[{ Cell[" Simplify and FullSimplify ", "Subsection"], Cell[CellGroupData[{ Cell[" Simplify ", "Subsubsection"], Cell[TextData[{ " The most useful simplifying commands are Simplify and FullSimplify. \ Simplify tries to reduce the expression to the smallest number of terms. For \ example, consider", StyleBox[" ", FontFamily->"Times New Roman", FontVariations->{"CompatibilityType"->0}] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"eq1", "=", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"4", "+", RowBox[{"3", " ", "x"}], " ", "+", RowBox[{"2", " ", RowBox[{"x", "^", "2"}]}], "+", RowBox[{"x", "^", "3"}]}], ")"}], "/", RowBox[{"(", RowBox[{"1", "+", "x", "+", RowBox[{"x", "^", "2"}], "+", RowBox[{"x", "^", "3"}], "+", RowBox[{"x", "^", "4"}]}], ")"}]}], " ", "+", RowBox[{ RowBox[{"1", "/", RowBox[{"(", RowBox[{"1", "-", "x"}], ")"}]}], " "}]}]}]], "Input"], Cell[BoxData[ RowBox[{ FractionBox["1", RowBox[{"1", "-", "x"}]], "+", FractionBox[ RowBox[{"4", "+", RowBox[{"3", " ", "x"}], "+", RowBox[{"2", " ", SuperscriptBox["x", "2"]}], "+", SuperscriptBox["x", "3"]}], RowBox[{"1", "+", "x", "+", SuperscriptBox["x", "2"], "+", SuperscriptBox["x", "3"], "+", SuperscriptBox["x", "4"]}]]}]], "Output"] }, Closed]], Cell["If you apply Simplify the expression reduces to", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{" ", RowBox[{"{", RowBox[{ RowBox[{"Simplify", "[", "eq1", "]"}], ",", RowBox[{"eq1", "//", " ", "Simplify"}]}], "}"}]}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"-", FractionBox["5", RowBox[{ RowBox[{"-", "1"}], "+", SuperscriptBox["x", "5"]}]]}], ",", RowBox[{"-", FractionBox["5", RowBox[{ RowBox[{"-", "1"}], "+", SuperscriptBox["x", "5"]}]]}]}], "}"}]], "Output"] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["Options for Simplify", "Subsubsection"], Cell[" Simplify has several options", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Options", "[", "Simplify", "]"}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"Assumptions", "\[RuleDelayed]", "$Assumptions"}], ",", RowBox[{"ComplexityFunction", "\[Rule]", "Automatic"}], ",", RowBox[{"TimeConstraint", "\[Rule]", "300"}], ",", RowBox[{"TransformationFunctions", "\[Rule]", "Automatic"}], ",", RowBox[{"Trig", "\[Rule]", "True"}]}], "}"}]], "Output"] }, Closed]], Cell["To illustrate the Assumptions consider the expression", "Text"], Cell[BoxData[ RowBox[{"Clear", "[", RowBox[{"k", ",", "r", ",", "theta"}], "]"}]], "Input"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"eq1", "=", " ", RowBox[{"Sqrt", "[", " ", RowBox[{ RowBox[{"k", "^", "2"}], " ", RowBox[{"r", "^", "2"}], " ", RowBox[{ RowBox[{"Cos", "[", "\[Theta]", "]"}], "^", "2"}]}], "]"}], " "}]], "Input"], Cell[BoxData[ SqrtBox[ RowBox[{ SuperscriptBox["k", "2"], " ", SuperscriptBox["r", "2"], " ", SuperscriptBox[ RowBox[{"Cos", "[", "\[Theta]", "]"}], "2"]}]]], "Output"] }, Closed]], Cell["Nothing happens if you apply simplify", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"eq1", "//", "Simplify"}]], "Input"], Cell[BoxData[ SqrtBox[ RowBox[{ SuperscriptBox["k", "2"], " ", SuperscriptBox["r", "2"], " ", SuperscriptBox[ RowBox[{"Cos", "[", "\[Theta]", "]"}], "2"]}]]], "Output"] }, Closed]], Cell["\<\ If the range of the variable are included as an option then \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"{", " ", RowBox[{ RowBox[{"eq1", "//", RowBox[{ RowBox[{"Simplify", "[", RowBox[{"#", ",", " ", RowBox[{"{", " ", RowBox[{ RowBox[{"r", ">", "0"}], ",", RowBox[{ RowBox[{"Pi", "/", "2"}], "<", "\[Theta]", "<", "Pi"}]}], " ", "}"}]}], "]"}], "&"}]}], " ", "\[IndentingNewLine]", RowBox[{"(*", "or", "*)"}], ",", "\[IndentingNewLine]", " ", RowBox[{"eq1", "//", RowBox[{ RowBox[{"Simplify", "[", RowBox[{"#", ",", RowBox[{"Assumptions", "->", RowBox[{"{", " ", RowBox[{ RowBox[{"r", ">", "0"}], ",", RowBox[{ RowBox[{"Pi", "/", "2"}], "<", "\[Theta]", "<", "Pi"}]}], " ", "}"}]}]}], "]"}], "&"}]}]}], "}"}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"-", SqrtBox[ SuperscriptBox["k", "2"]]}], " ", "r", " ", RowBox[{"Cos", "[", "\[Theta]", "]"}]}], ",", RowBox[{ RowBox[{"-", SqrtBox[ SuperscriptBox["k", "2"]]}], " ", "r", " ", RowBox[{"Cos", "[", "\[Theta]", "]"}]}]}], "}"}]], "Output"] }, Closed]], Cell[TextData[{ "Another example is Cos[ n Pi] or Sin[n Pi]. If you know n is an integer \ then the expressions can be simplified. The command Element[x,dom] or n\ \[Element]dom asserts that x is an element of dom, where \[Element] is found \ ", StyleBox["in the menu under File->Palettes.", FontVariations->{"CompatibilityType"->0}], " In this case dom=Integer. If we include this assumption in Simplify then \ it reduces to a simple expression. Entering the command in serval equivalent \ ways it follows " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"Simplify", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Cos", "[", " ", RowBox[{"n", " ", "Pi"}], "]"}], ",", RowBox[{"Sin", "[", RowBox[{"n", " ", "Pi"}], "]"}]}], "}"}], ",", "\[IndentingNewLine]", " ", RowBox[{"Assumptions", "->", RowBox[{"{", RowBox[{"Element", "[", RowBox[{"n", ",", "Integers"}], "]"}], "}"}]}]}], "]"}], ",", "\[IndentingNewLine]", RowBox[{"Simplify", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Cos", "[", " ", RowBox[{"n", " ", "Pi"}], "]"}], ",", RowBox[{"Sin", "[", RowBox[{"n", " ", "Pi"}], "]"}]}], "}"}], ",", "\[IndentingNewLine]", " ", RowBox[{"Assumptions", "->", RowBox[{"{", " ", RowBox[{"n", "\[Element]", "Integers"}], " ", "}"}]}]}], "]"}], ",", " ", "\[IndentingNewLine]", RowBox[{"Simplify", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Cos", "[", " ", RowBox[{"n", " ", "Pi"}], "]"}], ",", RowBox[{"Sin", "[", RowBox[{"n", " ", "Pi"}], "]"}]}], "}"}], ",", "\[IndentingNewLine]", " ", RowBox[{"{", RowBox[{"Element", "[", RowBox[{"n", ",", "Integers"}], "]"}], "}"}]}], "]"}], ",", "\[IndentingNewLine]", RowBox[{"Simplify", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Cos", "[", " ", RowBox[{"n", " ", "Pi"}], "]"}], ",", RowBox[{"Sin", "[", RowBox[{"n", " ", "Pi"}], "]"}]}], "}"}], ",", "\[IndentingNewLine]", " ", RowBox[{"{", " ", RowBox[{"n", "\[Element]", "Integers"}], " ", "}"}]}], "]"}]}], " ", "}"}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], "n"], ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], "n"], ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], "n"], ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], "n"], ",", "0"}], "}"}]}], "}"}]], "Output"] }, Closed]] }, Open ]], Cell["Consider the additional example", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"eq1", "=", RowBox[{"Sqrt", "[", RowBox[{ RowBox[{"x", "^", "2"}], "+", RowBox[{"2", "*", "x", "*", RowBox[{ RowBox[{"Cos", "[", "x", "]"}], "^", "2"}]}], "+", RowBox[{ RowBox[{"Cos", "[", "x", "]"}], "^", "4"}], "+", RowBox[{ RowBox[{"Sin", "[", "x", "]"}], "^", "4"}], "+", RowBox[{"2", "*", "x", "*", RowBox[{ RowBox[{"Sin", "[", RowBox[{ RowBox[{"2", "*", "n", "*", "Pi"}], "+", "x"}], "]"}], "^", "2"}]}], "+", RowBox[{"2", "*", RowBox[{ RowBox[{"Cos", "[", "x", "]"}], "^", "2"}], "*", RowBox[{ RowBox[{"Sin", "[", RowBox[{ RowBox[{"2", "*", "n", "*", "Pi"}], "+", "x"}], "]"}], "^", "2"}]}]}], "]"}]}], " ", ";"}]], "Input"], Cell["and simplify for integer n and x>-1: ", "Text"], Cell[BoxData[ RowBox[{"Column", "[", RowBox[{"{", RowBox[{ RowBox[{"Simplify", "[", "eq1", "]"}], ",", RowBox[{"Simplify", "[", RowBox[{"eq1", ",", RowBox[{"n", "\[Element]", "Integers"}]}], "]"}], ",", RowBox[{"Simplify", "[", RowBox[{"eq1", ",", RowBox[{ RowBox[{"n", "\[Element]", "Integers"}], "&&", RowBox[{"x", ">", RowBox[{"-", "1"}]}]}]}], "]"}]}], "}"}], "]"}]], "Input"], Cell[BoxData[ InterpretationBox[GridBox[{ { SqrtBox[ RowBox[{ SuperscriptBox[ RowBox[{"Cos", "[", "x", "]"}], "4"], "+", SuperscriptBox[ RowBox[{"Sin", "[", "x", "]"}], "4"], "+", RowBox[{"2", " ", SuperscriptBox[ RowBox[{"Cos", "[", "x", "]"}], "2"], " ", RowBox[{"(", RowBox[{"x", "+", SuperscriptBox[ RowBox[{"Sin", "[", RowBox[{ RowBox[{"2", " ", "n", " ", "\[Pi]"}], "+", "x"}], "]"}], "2"]}], ")"}]}], "+", RowBox[{"x", " ", RowBox[{"(", RowBox[{"x", "+", RowBox[{"2", " ", SuperscriptBox[ RowBox[{"Sin", "[", RowBox[{ RowBox[{"2", " ", "n", " ", "\[Pi]"}], "+", "x"}], "]"}], "2"]}]}], ")"}]}]}]]}, { SqrtBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", "x"}], ")"}], "2"]]}, { RowBox[{"1", "+", "x"}]} }, BaselinePosition->{Baseline, {1, 1}}, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}], ColumnForm[{(Cos[x]^4 + Sin[x]^4 + 2 Cos[x]^2 (x + Sin[2 n Pi + x]^2) + x (x + 2 Sin[2 n Pi + x]^2))^Rational[1, 2], ((1 + x)^2)^Rational[1, 2], 1 + x}], Editable->False]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["FullSimplify", "Subsubsection"], Cell["\<\ If Simplify does not work then try FullSimplify. Two examples where \ Simplify does not do anything but FullSimplify does are \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"{", " ", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{"Gamma", "[", "\[Omega]", "]"}], RowBox[{"Gamma", "[", RowBox[{"1", "-", "\[Omega]"}], "]"}]}], "//", "Simplify"}], ",", "\[IndentingNewLine]", " ", RowBox[{ RowBox[{ RowBox[{"Gamma", "[", "\[Omega]", "]"}], RowBox[{"Gamma", "[", RowBox[{"1", "-", "\[Omega]"}], "]"}]}], "//", "FullSimplify"}]}], "}"}], ",", "\[IndentingNewLine]", " ", RowBox[{"{", " ", RowBox[{ RowBox[{ RowBox[{ RowBox[{"Abs", "[", "z", "]"}], " ", RowBox[{"Exp", "[", RowBox[{"I", " ", RowBox[{"Arg", "[", "z", "]"}]}], "]"}]}], "//", "Simplify"}], ",", "\[IndentingNewLine]", " ", RowBox[{ RowBox[{ RowBox[{"Abs", "[", "z", "]"}], " ", RowBox[{"Exp", "[", RowBox[{"I", " ", RowBox[{"Arg", "[", "z", "]"}]}], "]"}]}], "//", "FullSimplify"}]}], "}"}]}], "}"}], " "}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"Gamma", "[", RowBox[{"1", "-", "\[Omega]"}], "]"}], " ", RowBox[{"Gamma", "[", "\[Omega]", "]"}]}], ",", RowBox[{"\[Pi]", " ", RowBox[{"Csc", "[", RowBox[{"\[Pi]", " ", "\[Omega]"}], "]"}]}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{"\[ImaginaryI]", " ", RowBox[{"Arg", "[", "z", "]"}]}]], " ", RowBox[{"Abs", "[", "z", "]"}]}], ",", "z"}], "}"}]}], "}"}]], "Output"] }, Closed]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[" Factor and Expand ", "Subsection"], Cell[CellGroupData[{ Cell["Factor ", "Subsubsection"], Cell["\<\ Factor tries to simplify an expression by factoring it. Consider the \ expression,\ \>", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"eq1", "=", RowBox[{"1", "+", " ", RowBox[{"3", " ", RowBox[{"x", "^", "2"}]}], "+", " ", RowBox[{"3", " ", RowBox[{"x", "^", "4"}]}], "+", RowBox[{"x", "^", "6"}]}]}], " ", ";"}]], "Input"], Cell["and apply Factor,", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Factor", "[", "eq1", "]"}]], "Input"], Cell[BoxData[ SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", SuperscriptBox["x", "2"]}], ")"}], "3"]], "Output"] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Factor Options Commands: GaussianIntegers, Extension\ \>", "Subsubsection"], Cell[" Further reductions occurs using the options for Factor,", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["\<\ Options[Factor] \ \>", "Input", AspectRatioFixed->True], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"Extension", "\[Rule]", "None"}], ",", RowBox[{"GaussianIntegers", "\[Rule]", "False"}], ",", RowBox[{"Modulus", "\[Rule]", "0"}], ",", RowBox[{"Trig", "\[Rule]", "False"}]}], "}"}]], "Output"] }, Closed]], Cell["\<\ To factor eq1 over the complex numbers turn on the GaussianIntegers option, \ \ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Factor[eq1,GaussianIntegers->True] ", "Input", AspectRatioFixed->True], Cell[BoxData[ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"-", "\[ImaginaryI]"}], "+", "x"}], ")"}], "3"], " ", SuperscriptBox[ RowBox[{"(", RowBox[{"\[ImaginaryI]", "+", "x"}], ")"}], "3"]}]], "Output"] }, Closed]], Cell["\<\ You can also factor over complex integers by including the complex symbol \ \[ImaginaryI] with the Extension option, \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Factor", "[", " ", RowBox[{"eq1", ",", RowBox[{"Extension", "\[Rule]", "\[ImaginaryI]"}]}], "]"}]], "Input"], Cell[BoxData[ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"-", "\[ImaginaryI]"}], "+", "x"}], ")"}], "3"], " ", SuperscriptBox[ RowBox[{"(", RowBox[{"\[ImaginaryI]", "+", "x"}], ")"}], "3"]}]], "Output"] }, Closed]], Cell[" Consider an example using the Extension option. Enter ", "Text"], Cell[BoxData[{ RowBox[{ RowBox[{"eq2", "=", RowBox[{ RowBox[{"x", "^", "2"}], "-", "2"}]}], ";"}], "\[IndentingNewLine]", RowBox[{" "}]}], "Input"], Cell[TextData[{ "Factor does not extract any factors from x^2-2 but if ", StyleBox["you use the option ", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ RowBox[{"Extension", "->", SqrtBox["2"]}]], FontFamily->"Times", FontVariations->{"CompatibilityType"->0}], StyleBox[" then ", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["x", "2"], "-", "2"}], TraditionalForm]], FontVariations->{"CompatibilityType"->0}], StyleBox[" is factored", FontVariations->{"CompatibilityType"->0}] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Factor", "[", RowBox[{ RowBox[{ RowBox[{"x", "^", "2"}], " ", "-", " ", "2"}], ",", " ", RowBox[{"Extension", "->", SqrtBox["2"]}]}], "]"}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"-", RowBox[{"(", RowBox[{ SqrtBox["2"], "-", "x"}], ")"}]}], " ", RowBox[{"(", RowBox[{ SqrtBox["2"], "+", "x"}], ")"}]}]], "Output"] }, Closed]], Cell[" Consider another example", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"eq3", "=", RowBox[{ RowBox[{"x", "^", "4"}], "-", RowBox[{"2", " ", RowBox[{"Sqrt", "[", "3", "]"}], " ", RowBox[{"x", "^", "2"}]}], "+", "3"}]}], ";"}]], "Input"], Cell["Applying Factor with various extensions, it follows", "Text"], Cell[BoxData[ RowBox[{"Column", "[", RowBox[{"{", RowBox[{ RowBox[{"Factor", "[", RowBox[{"eq3", ",", RowBox[{"Extension", "\[Rule]", "Automatic"}]}], "]"}], ",", RowBox[{"Factor", "[", RowBox[{"eq3", ",", RowBox[{"Extension", "\[Rule]", RowBox[{"{", SuperscriptBox["3", RowBox[{"1", "/", "4"}]], "}"}]}]}], "]"}], ",", RowBox[{"Factor", "[", "eq3", "]"}]}], "}"}], "]"}]], "Input"], Cell[BoxData[ InterpretationBox[GridBox[{ { SuperscriptBox[ RowBox[{"(", RowBox[{ SqrtBox["3"], "-", SuperscriptBox["x", "2"]}], ")"}], "2"]}, { RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{ SuperscriptBox["3", RowBox[{"1", "/", "4"}]], "-", "x"}], ")"}], "2"], " ", SuperscriptBox[ RowBox[{"(", RowBox[{ SuperscriptBox["3", RowBox[{"1", "/", "4"}]], "+", "x"}], ")"}], "2"]}]}, { RowBox[{"3", "-", RowBox[{"2", " ", SqrtBox["3"], " ", SuperscriptBox["x", "2"]}], "+", SuperscriptBox["x", "4"]}]} }, BaselinePosition->{Baseline, {1, 1}}, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}], ColumnForm[{(3^Rational[1, 2] - x^2)^2, (3^Rational[1, 4] - x)^2 (3^Rational[1, 4] + x)^2, 3 - 2 3^Rational[1, 2] x^2 + x^4}], Editable->False]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Expand", "Subsubsection"], Cell["\<\ Instead of factoring you can expand expressions. Consider \ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ RowBox[{ RowBox[{"eq1", "=", RowBox[{ RowBox[{"(", RowBox[{"1", "+", RowBox[{"x", "^", "2"}]}], ")"}], "^", "5"}]}], ";"}]], "Input"], Cell["and apply Expand, ", "Text"], Cell[CellGroupData[{ Cell["eq1= Expand[(1+ x^2)^5] ", "Input", AspectRatioFixed->True], Cell[BoxData[ RowBox[{"1", "+", RowBox[{"5", " ", SuperscriptBox["x", "2"]}], "+", RowBox[{"10", " ", SuperscriptBox["x", "4"]}], "+", RowBox[{"10", " ", SuperscriptBox["x", "6"]}], "+", RowBox[{"5", " ", SuperscriptBox["x", "8"]}], "+", SuperscriptBox["x", "10"]}]], "Output"] }, Closed]], Cell["\<\ You have additional control of Expand by adding a pattern. The command \ Expand[expr, patt ] expands expr avoiding those parts that do not contain \ terms matching patt. For example, \ \>", "Text"], Cell[CellGroupData[{ Cell["Expand[(x+1)^5 (y +1)^2,y]", "Input"], Cell[BoxData[ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", "x"}], ")"}], "5"], "+", RowBox[{"2", " ", SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", "x"}], ")"}], "5"], " ", "y"}], "+", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", "x"}], ")"}], "5"], " ", SuperscriptBox["y", "2"]}]}]], "Output"] }, Closed]], Cell["\<\ To expand only the denominator or numerator then use ExpandDenominator \ or ExpandNumerator,\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"ExpandDenominator", "[", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"1", "+", "x"}], ")"}], "^", "2"}], "/", " ", RowBox[{ RowBox[{"(", RowBox[{"1", "-", RowBox[{"a", " ", "x"}]}], ")"}], "^", "3"}]}], " ", "]"}], ",", "\[IndentingNewLine]", RowBox[{"ExpandNumerator", "[", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"1", "+", "x"}], ")"}], "^", "2"}], "/", RowBox[{ RowBox[{"(", RowBox[{"1", "-", RowBox[{"a", " ", "x"}]}], ")"}], "^", "3"}]}], "]"}]}], "}"}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", "x"}], ")"}], "2"], RowBox[{"1", "-", RowBox[{"3", " ", "a", " ", "x"}], "+", RowBox[{"3", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["x", "2"]}], "-", RowBox[{ SuperscriptBox["a", "3"], " ", SuperscriptBox["x", "3"]}]}]], ",", FractionBox[ RowBox[{"1", "+", RowBox[{"2", " ", "x"}], "+", SuperscriptBox["x", "2"]}], SuperscriptBox[ RowBox[{"(", RowBox[{"1", "-", RowBox[{"a", " ", "x"}]}], ")"}], "3"]]}], "}"}]], "Output"] }, Closed]], Cell["Expand Used in a Rule", "Subsubsection"], Cell["\<\ Consider an example were you want to apply Expand in a rule. Enter\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Sqrt", "[", RowBox[{ RowBox[{"(", RowBox[{"a", "+", "b"}], ")"}], RowBox[{"(", RowBox[{"c", "+", "d"}], ")"}]}], "]"}], " "}]], "Input"], Cell[BoxData[ SqrtBox[ RowBox[{ RowBox[{"(", RowBox[{"a", "+", "b"}], ")"}], " ", RowBox[{"(", RowBox[{"c", "+", "d"}], ")"}]}]]], "Output"] }, Closed]], Cell["\<\ and expand the terms under the square root. Expand won't work by the \ following delayed rule will\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Sqrt", "[", RowBox[{ RowBox[{"(", RowBox[{"a", "+", "b"}], ")"}], RowBox[{"(", RowBox[{"c", "+", "d"}], ")"}]}], "]"}], "/.", RowBox[{ RowBox[{"Sqrt", "[", "x_", " ", "]"}], "\[RuleDelayed]", RowBox[{"Sqrt", "[", RowBox[{"Expand", "[", "x", " ", "]"}], "]"}]}]}]], "Input"], Cell[BoxData[ SqrtBox[ RowBox[{ RowBox[{"a", " ", "c"}], "+", RowBox[{"b", " ", "c"}], "+", RowBox[{"a", " ", "d"}], "+", RowBox[{"b", " ", "d"}]}]]], "Output"] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["PowerExpand", "Subsubsection"], Cell[TextData[{ StyleBox[" Mathematica does not automatically simplify ", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ SqrtBox[ SuperscriptBox["z", "2"]]], FontFamily->"Times", FontVariations->{"CompatibilityType"->0}], StyleBox[" to z because the simplification is valid only for nonnegative \ real values of z,", FontVariations->{"CompatibilityType"->0}] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{" ", RowBox[{ RowBox[{"Sqrt", "[", RowBox[{"z", "^", "2"}], "]"}], "//", "Expand", " "}]}]], "Input"], Cell[BoxData[ SqrtBox[ SuperscriptBox["z", "2"]]], "Output"] }, Closed]], Cell["PowerExpand works,", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{" ", RowBox[{ RowBox[{"Sqrt", "[", RowBox[{"z", "^", "2"}], "]"}], "//", "PowerExpand"}]}]], "Input"], Cell[BoxData["z"], "Output"] }, Closed]], Cell["\<\ You must be careful when you use PowerExpand because the command does not pay \ attention to branch cuts so it may return the wrong answer. Other examples of \ PowerExpand[] are\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{" ", RowBox[{ RowBox[{ RowBox[{"{", " ", RowBox[{ RowBox[{"Sqrt", "[", RowBox[{"5", " ", RowBox[{"x", "^", "2"}]}], "]"}], ",", RowBox[{"Log", "[", RowBox[{ RowBox[{"x", "^", "3"}], " ", RowBox[{"y", "/", "z"}]}], "]"}], ",", RowBox[{ RowBox[{"(", RowBox[{"A", " ", "B"}], ")"}], "^", "c"}]}], "}"}], "//", "PowerExpand"}], " ", "\[IndentingNewLine]", " "}]}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ SqrtBox["5"], " ", "x"}], ",", RowBox[{ RowBox[{"3", " ", RowBox[{"Log", "[", "x", "]"}]}], "+", RowBox[{"Log", "[", "y", "]"}], "-", RowBox[{"Log", "[", "z", "]"}]}], ",", RowBox[{ SuperscriptBox["A", "c"], " ", SuperscriptBox["B", "c"]}]}], "}"}]], "Output"] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["FunctionExpand", "Subsubsection"], Cell["\<\ FunctionExpand[expr] tries to expand special functions. Examples are\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"n", "!"}], ",", RowBox[{"1", "/", RowBox[{"n", "!!"}]}]}], "}"}], "//", "FunctionExpand"}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"Gamma", "[", RowBox[{"1", "+", "n"}], "]"}], ",", FractionBox[ RowBox[{ SuperscriptBox["2", RowBox[{ RowBox[{"-", FractionBox["1", "4"]}], "-", FractionBox["n", "2"], "+", RowBox[{ FractionBox["1", "4"], " ", RowBox[{"Cos", "[", RowBox[{"n", " ", "\[Pi]"}], "]"}]}]}]], " ", SuperscriptBox["\[Pi]", RowBox[{ FractionBox["1", "4"], "-", RowBox[{ FractionBox["1", "4"], " ", RowBox[{"Cos", "[", RowBox[{"n", " ", "\[Pi]"}], "]"}]}]}]]}], RowBox[{"Gamma", "[", RowBox[{"1", "+", FractionBox["n", "2"]}], "]"}]]}], "}"}]], "Output"] }, Closed]], Cell[TextData[{ "Notice you can not sum ", Cell[BoxData[ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"n", "=", "0"}], "\[Infinity]"], FractionBox["1", RowBox[{"n", "!!"}]]}]]], " but you can sum ", Cell[BoxData[ RowBox[{ RowBox[{"FunctionExpand", "[", FractionBox["1", RowBox[{"n", "!!"}]], "]"}], ":"}]]] }], "Text"], Cell[BoxData[ RowBox[{ RowBox[{"Column", "[", RowBox[{"{", RowBox[{ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"n", "=", "0"}], "\[Infinity]"], FractionBox["1", RowBox[{"n", "!!"}]]}], ",", RowBox[{"FunctionExpand", "[", FractionBox["1", RowBox[{"n", "!!"}]], "]"}], ",", RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"n", "=", "0"}], "\[Infinity]"], RowBox[{"FunctionExpand", "[", FractionBox["1", RowBox[{"n", "!!"}]], "]"}]}]}], "}"}], "]"}], "\[IndentingNewLine]"}]], "Input"], Cell[BoxData[ InterpretationBox[GridBox[{ { RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"n", "=", "0"}], "\[Infinity]"], FractionBox["1", RowBox[{"n", "!!"}]]}]}, { FractionBox[ RowBox[{ SuperscriptBox["2", RowBox[{ RowBox[{"-", FractionBox["1", "4"]}], "-", FractionBox["n", "2"], "+", RowBox[{ FractionBox["1", "4"], " ", RowBox[{"Cos", "[", RowBox[{"n", " ", "\[Pi]"}], "]"}]}]}]], " ", SuperscriptBox["\[Pi]", RowBox[{ FractionBox["1", "4"], "-", RowBox[{ FractionBox["1", "4"], " ", RowBox[{"Cos", "[", RowBox[{"n", " ", "\[Pi]"}], "]"}]}]}]]}], RowBox[{"Gamma", "[", RowBox[{"1", "+", FractionBox["n", "2"]}], "]"}]]}, { RowBox[{ FractionBox["1", "2"], " ", SuperscriptBox[ RowBox[{"(", FractionBox["\[Pi]", "2"], ")"}], RowBox[{"1", "/", "4"}]], " ", RowBox[{"(", RowBox[{ RowBox[{"2", " ", SqrtBox["\[ExponentialE]"], " ", SuperscriptBox[ RowBox[{"(", FractionBox["2", "\[Pi]"], ")"}], RowBox[{"1", "/", "4"}]]}], "+", RowBox[{ SuperscriptBox["2", RowBox[{"3", "/", "4"}]], " ", SqrtBox["\[ExponentialE]"], " ", SuperscriptBox["\[Pi]", RowBox[{"1", "/", "4"}]], " ", RowBox[{"Erf", "[", FractionBox["1", SqrtBox["2"]], "]"}]}]}], ")"}]}]} }, BaselinePosition->{Baseline, {1, 1}}, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}], ColumnForm[{ Sum[Factorial2[n]^(-1), {n, 0, DirectedInfinity[1]}], 2^(Rational[-1, 4] + Rational[-1, 2] n + Rational[1, 4] Cos[n Pi]) Pi^(Rational[1, 4] + Rational[-1, 4] Cos[n Pi])/Gamma[ 1 + Rational[1, 2] n], Rational[1, 2] (Rational[1, 2] Pi)^Rational[1, 4] ( 2 E^Rational[1, 2] (2/Pi)^Rational[1, 4] + 2^Rational[3, 4] E^Rational[1, 2] Pi^Rational[1, 4] Erf[2^Rational[-1, 2]])}], Editable->False]], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[" Other Commands: Cancel, Apart and Together", "Subsection"], Cell[" Cancel divides out common factors. Consider ", "Text"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"eq1", "=", RowBox[{ RowBox[{"(", RowBox[{"1", "+", RowBox[{"x", "^", "9"}]}], ")"}], "/", RowBox[{"(", RowBox[{"1", "+", "x"}], ")"}]}]}], ";"}], " "}]], "Input"], Cell["Notice what Cancel and Factor do , ", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{" ", RowBox[{ RowBox[{"eq1", " ", "//", "Cancel"}], "//", "Factor", " "}]}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"(", RowBox[{"1", "-", "x", "+", SuperscriptBox["x", "2"]}], ")"}], " ", RowBox[{"(", RowBox[{"1", "-", SuperscriptBox["x", "3"], "+", SuperscriptBox["x", "6"]}], ")"}]}]], "Output"] }, Closed]], Cell["\<\ Apart performs partial-fraction decomposition of a rational expression.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"eq3", "=", RowBox[{"Apart", "[", RowBox[{"1", "/", " ", RowBox[{"(", RowBox[{"x", "-", RowBox[{"x", "^", "5"}]}], ")"}]}], " ", "]"}]}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"-", FractionBox["1", RowBox[{"4", " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "x"}], ")"}]}]]}], "+", FractionBox["1", "x"], "-", FractionBox["1", RowBox[{"4", " ", RowBox[{"(", RowBox[{"1", "+", "x"}], ")"}]}]], "-", FractionBox["x", RowBox[{"2", " ", RowBox[{"(", RowBox[{"1", "+", SuperscriptBox["x", "2"]}], ")"}]}]]}]], "Output"] }, Closed]], Cell["\<\ Together does the opposite: it puts two or more rational expressions over a \ common denominator,\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Together", "[", "eq3", "]"}]], "Input"], Cell[BoxData[ RowBox[{"-", FractionBox["1", RowBox[{"x", " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", SuperscriptBox["x", "4"]}], ")"}]}]]}]], "Output"] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ " ", StyleBox["Simplifying ", FontWeight->"Bold"], "Trig ", StyleBox["Expressions", FontWeight->"Bold"], " " }], "Subsection"], Cell[CellGroupData[{ Cell[" TrigExpand", "Subsection"], Cell["\<\ There are several ways to reduce trigonometric functions. A summary of \ the major commands are TrigExpand[expr]---Expand sums and multiple angles TrigFactor[expr]---Factor expr in a product form TrigReduce[expr]---Write expr to a sum with no products or powers TrigToExp[expr]---Trigonometric functions to complex exponentials ExpToTrig[expr]---Complex exponentials to trigonometric functions Consider the example ofTrigExpand \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{" ", RowBox[{ RowBox[{"Sin", "[", " ", RowBox[{"\[Alpha]", "+", "\[Beta]", "+", "\[Gamma]"}], "]"}], " ", "//", "TrigExpand"}]}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"Cos", "[", "\[Beta]", "]"}], " ", RowBox[{"Cos", "[", "\[Gamma]", "]"}], " ", RowBox[{"Sin", "[", "\[Alpha]", "]"}]}], "+", RowBox[{ RowBox[{"Cos", "[", "\[Alpha]", "]"}], " ", RowBox[{"Cos", "[", "\[Gamma]", "]"}], " ", RowBox[{"Sin", "[", "\[Beta]", "]"}]}], "+", RowBox[{ RowBox[{"Cos", "[", "\[Alpha]", "]"}], " ", RowBox[{"Cos", "[", "\[Beta]", "]"}], " ", RowBox[{"Sin", "[", "\[Gamma]", "]"}]}], "-", RowBox[{ RowBox[{"Sin", "[", "\[Alpha]", "]"}], " ", RowBox[{"Sin", "[", "\[Beta]", "]"}], " ", RowBox[{"Sin", "[", "\[Gamma]", "]"}]}]}]], "Output"] }, Open ]], Cell["\<\ TrigExpands sums and the angles. For other examples consider the list of \ trigonometric expression,\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"list", "=", RowBox[{"Table", "[", RowBox[{ RowBox[{"Cos", "[", RowBox[{"n", " ", RowBox[{"ArcSin", "[", "x", "]"}]}], "]"}], ",", RowBox[{"{", RowBox[{"n", ",", "0", ",", "4"}], "}"}]}], "]"}]}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{"1", ",", SqrtBox[ RowBox[{"1", "-", SuperscriptBox["x", "2"]}]], ",", RowBox[{"Cos", "[", RowBox[{"2", " ", RowBox[{"ArcSin", "[", "x", "]"}]}], "]"}], ",", RowBox[{"Cos", "[", RowBox[{"3", " ", RowBox[{"ArcSin", "[", "x", "]"}]}], "]"}], ",", RowBox[{"Cos", "[", RowBox[{"4", " ", RowBox[{"ArcSin", "[", "x", "]"}]}], "]"}]}], "}"}]], "Output"] }, Closed]], Cell["Applying TrigExpand, you get", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"list", "//", "TrigExpand"}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{"1", ",", SqrtBox[ RowBox[{"1", "-", SuperscriptBox["x", "2"]}]], ",", RowBox[{"1", "-", RowBox[{"2", " ", SuperscriptBox["x", "2"]}]}], ",", RowBox[{ RowBox[{ RowBox[{"-", "3"}], " ", SuperscriptBox["x", "2"], " ", SqrtBox[ RowBox[{"1", "-", SuperscriptBox["x", "2"]}]]}], "+", SuperscriptBox[ RowBox[{"(", RowBox[{"1", "-", SuperscriptBox["x", "2"]}], ")"}], RowBox[{"3", "/", "2"}]]}], ",", RowBox[{"1", "-", RowBox[{"8", " ", SuperscriptBox["x", "2"]}], "+", RowBox[{"8", " ", SuperscriptBox["x", "4"]}]}]}], "}"}]], "Output"] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell[" TrigToExp and ExpToTrig", "Subsection"], Cell["\<\ TrigToExp converts trigonometric expressions into exponential form,\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"TrigToExp", "[", " ", RowBox[{ RowBox[{"Cos", "[", "z", "]"}], "+", RowBox[{"I", " ", RowBox[{"Sin", "[", "z", "]"}]}]}], "]"}]], "Input"], Cell[BoxData[ SuperscriptBox["\[ExponentialE]", RowBox[{"\[ImaginaryI]", " ", "z"}]]], "Output"] }, Closed]], Cell["Example applied to inverse trigonometric functions are", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"list", "=", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"ArcCos", "[", "x", "]"}], ",", RowBox[{"ArcCosh", "[", "x", "]"}], ",", RowBox[{"ArcSec", "[", "x", "]"}], ",", RowBox[{"ArcSech", "[", "x", "]"}], ",", RowBox[{"ArcSin", "[", "x", "]"}], ",", " ", RowBox[{"ArcSinh", "[", "x", "]"}]}], "}"}], "//", RowBox[{ RowBox[{"TableForm", "[", RowBox[{ RowBox[{"#", "//", "TrigToExp"}], ",", RowBox[{"TableHeadings", "\[Rule]", RowBox[{"{", " ", "#", " ", "}"}]}]}], "]"}], "&"}]}]}]], "Input"], Cell[BoxData[ InterpretationBox[GridBox[{ { RowBox[{"ArcCos", "[", "x", "]"}], RowBox[{ FractionBox["\[Pi]", "2"], "+", RowBox[{"\[ImaginaryI]", " ", RowBox[{"Log", "[", RowBox[{ RowBox[{"\[ImaginaryI]", " ", "x"}], "+", SqrtBox[ RowBox[{"1", "-", SuperscriptBox["x", "2"]}]]}], "]"}]}]}]}, { RowBox[{"ArcCosh", "[", "x", "]"}], RowBox[{"Log", "[", RowBox[{"x", "+", RowBox[{ SqrtBox[ RowBox[{ RowBox[{"-", "1"}], "+", "x"}]], " ", SqrtBox[ RowBox[{"1", "+", "x"}]]}]}], "]"}]}, { RowBox[{"ArcSec", "[", "x", "]"}], RowBox[{ FractionBox["\[Pi]", "2"], "+", RowBox[{"\[ImaginaryI]", " ", RowBox[{"Log", "[", RowBox[{ SqrtBox[ RowBox[{"1", "-", FractionBox["1", SuperscriptBox["x", "2"]]}]], "+", FractionBox["\[ImaginaryI]", "x"]}], "]"}]}]}]}, { RowBox[{"ArcSech", "[", "x", "]"}], RowBox[{"Log", "[", RowBox[{ RowBox[{ SqrtBox[ RowBox[{ RowBox[{"-", "1"}], "+", FractionBox["1", "x"]}]], " ", SqrtBox[ RowBox[{"1", "+", FractionBox["1", "x"]}]]}], "+", FractionBox["1", "x"]}], "]"}]}, { RowBox[{"ArcSin", "[", "x", "]"}], RowBox[{ RowBox[{"-", "\[ImaginaryI]"}], " ", RowBox[{"Log", "[", RowBox[{ RowBox[{"\[ImaginaryI]", " ", "x"}], "+", SqrtBox[ RowBox[{"1", "-", SuperscriptBox["x", "2"]}]]}], "]"}]}]}, { RowBox[{"ArcSinh", "[", "x", "]"}], RowBox[{"Log", "[", RowBox[{"x", "+", SqrtBox[ RowBox[{"1", "+", SuperscriptBox["x", "2"]}]]}], "]"}]} }, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[2.0999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], TableForm[{ Rational[1, 2] Pi + Complex[0, 1] Log[Complex[0, 1] x + (1 - x^2)^Rational[1, 2]], Log[x + (-1 + x)^Rational[1, 2] (1 + x)^Rational[1, 2]], Rational[1, 2] Pi + Complex[0, 1] Log[(1 - x^(-2))^Rational[1, 2] + Complex[0, 1]/x], Log[(-1 + x^(-1))^Rational[1, 2] (1 + x^(-1))^Rational[1, 2] + x^(-1)], Complex[0, -1] Log[Complex[0, 1] x + (1 - x^2)^Rational[1, 2]], Log[x + (1 + x^2)^Rational[1, 2]]}, TableHeadings -> {{ ArcCos[x], ArcCosh[x], ArcSec[x], ArcSech[x], ArcSin[x], ArcSinh[x]}}]]], "Output"] }, Closed]], Cell[TextData[{ StyleBox[" ", "Input"], "ExpToTrig takes exponential functions and converts them to trig \ functions" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"{", " ", RowBox[{ RowBox[{"E", "^", RowBox[{"(", RowBox[{"I", " ", "z"}], ")"}]}], ",", " ", RowBox[{"Exp", "[", "z", "]"}]}], " ", "}"}], "//", "ExpToTrig"}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"Cos", "[", "z", "]"}], "+", RowBox[{"\[ImaginaryI]", " ", RowBox[{"Sin", "[", "z", "]"}]}]}], ",", RowBox[{ RowBox[{"Cosh", "[", "z", "]"}], "+", RowBox[{"Sinh", "[", "z", "]"}]}]}], "}"}]], "Output"] }, Closed]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ " ", StyleBox["Trig Options", FontWeight->"Bold"] }], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ Trig options in built-in functions will also simplify Trig expressions. \ Some of the built-in functions that have the Trig option are: Apart, Cancel, Expand, ExpandAll, ExpandDenominator, ExpandNumerator, Factor, FullSimplify, Simplify, Together. Simplify and FullSimplify have the default setting (Trig->True) and all \ other functions above have the default setting (Trig->False). Some examples \ are\[NonBreakingSpace] \ \>", "Text"], Cell[BoxData[ RowBox[{"Column", "[", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"Factor", "[", RowBox[{ RowBox[{ SuperscriptBox[ RowBox[{"Cos", "[", RowBox[{"4", " ", "x"}], "]"}], "2"], "-", RowBox[{"2", " ", RowBox[{"Cos", "[", RowBox[{"4", " ", "x"}], "]"}], " ", RowBox[{"Sin", "[", RowBox[{"3", " ", "x"}], "]"}]}], "+", SuperscriptBox[ RowBox[{"Sin", "[", RowBox[{"3", " ", "x"}], "]"}], "2"]}], ",", RowBox[{"Trig", "\[Rule]", "True"}]}], "]"}], " ", RowBox[{"Expand", "[", RowBox[{ SuperscriptBox[ RowBox[{"Sin", "[", RowBox[{"3", " ", "x"}], "]"}], "2"], ",", RowBox[{"Trig", "\[Rule]", "True"}]}], "]"}]}], ",", RowBox[{"Apart", "[", RowBox[{ FractionBox["1", RowBox[{"Sin", "[", RowBox[{"3", " ", "x"}], "]"}]], ",", RowBox[{"Trig", "\[Rule]", "True"}]}], "]"}]}], "}"}], "]"}]], "Input"], Cell[BoxData[ InterpretationBox[GridBox[{ { RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"Cos", "[", FractionBox["x", "2"], "]"}], "+", RowBox[{"Sin", "[", FractionBox["x", "2"], "]"}]}], ")"}], "4"], " ", RowBox[{"(", RowBox[{ FractionBox["1", "2"], "-", FractionBox[ SuperscriptBox[ RowBox[{"Cos", "[", "x", "]"}], "6"], "2"], "+", RowBox[{ FractionBox["15", "2"], " ", SuperscriptBox[ RowBox[{"Cos", "[", "x", "]"}], "4"], " ", SuperscriptBox[ RowBox[{"Sin", "[", "x", "]"}], "2"]}], "-", RowBox[{ FractionBox["15", "2"], " ", SuperscriptBox[ RowBox[{"Cos", "[", "x", "]"}], "2"], " ", SuperscriptBox[ RowBox[{"Sin", "[", "x", "]"}], "4"]}], "+", FractionBox[ SuperscriptBox[ RowBox[{"Sin", "[", "x", "]"}], "6"], "2"]}], ")"}], " ", SuperscriptBox[ RowBox[{"(", RowBox[{"1", "-", RowBox[{"2", " ", RowBox[{"Cos", "[", RowBox[{"2", " ", "x"}], "]"}]}], "-", RowBox[{"2", " ", RowBox[{"Sin", "[", "x", "]"}]}], "+", RowBox[{"2", " ", RowBox[{"Sin", "[", RowBox[{"3", " ", "x"}], "]"}]}]}], ")"}], "2"]}]}, { RowBox[{ FractionBox[ RowBox[{"Csc", "[", "x", "]"}], "3"], "+", FractionBox[ RowBox[{"4", " ", RowBox[{"Sin", "[", "x", "]"}]}], RowBox[{"3", " ", RowBox[{"(", RowBox[{"1", "+", RowBox[{"2", " ", RowBox[{"Cos", "[", RowBox[{"2", " ", "x"}], "]"}]}]}], ")"}]}]]}]} }, BaselinePosition->{Baseline, {1, 1}}, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}], ColumnForm[{(Cos[Rational[1, 2] x] + Sin[Rational[1, 2] x])^4 ( Rational[1, 2] + Rational[-1, 2] Cos[x]^6 + Rational[15, 2] Cos[x]^4 Sin[x]^2 + Rational[-15, 2] Cos[x]^2 Sin[x]^4 + Rational[1, 2] Sin[x]^6) (1 - 2 Cos[2 x] - 2 Sin[x] + 2 Sin[3 x])^2, Rational[1, 3] Csc[x] + Rational[4, 3] (1 + 2 Cos[2 x])^(-1) Sin[x]}], Editable->False]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ " ", StyleBox["Simplifying ", FontWeight->"Bold"], " Complex ", StyleBox["Expressions", FontWeight->"Bold"], " " }], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["ComplexExpand", "Subsubsection"], Cell["\<\ The command ComplexExpand simplifies complex expressions. Consider the \ complex expression\ \>", "Text", Evaluatable->False, AspectRatioFixed->True, CellTags->{"Complex Numbers", "Arg", "Phase Angle"}], Cell["eq2= (a + b I)(x - y I);", "Input", AspectRatioFixed->True], Cell["and apply ComplexExpand ", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["ComplexExpand[eq2] ", "Input", AspectRatioFixed->True], Cell[BoxData[ RowBox[{ RowBox[{"a", " ", "x"}], "+", RowBox[{"b", " ", "y"}], "+", RowBox[{"\[ImaginaryI]", " ", RowBox[{"(", RowBox[{ RowBox[{"b", " ", "x"}], "-", RowBox[{"a", " ", "y"}]}], ")"}]}]}]], "Output"] }, Closed]], Cell["\<\ Complex Expand assumes all variables are real and expands. The alternate \ form ComplexExpand[expr, {x1, x2, ...}] assumes that variables matching the \ xi are complex. 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Take[list, n] or Take[list, -n], gives the first or last n elements 2. Take[list, {m, n}] or Take[list, {m, n, s}], gives elements m through n in \ steps of s 3. Take[list, seq1, seq2, ... ], takes elements seq. 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Make a table \ of the year vs the Log of the debt. Restrict the number of digits to four. 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Consider\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ RowBox[{ RowBox[{"eq1", "=", RowBox[{ RowBox[{"a", "/", RowBox[{"x", "^", "2"}]}], " ", RowBox[{"Sin", "[", "x", " ", "]"}], RowBox[{"(", RowBox[{"b", "-", RowBox[{"c", " ", RowBox[{"x", "^", "2"}]}]}], ")"}]}]}], ";"}]], "Input"], Cell["and integrate", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[" eq2=Integrate[eq1,x] ", "Input", AspectRatioFixed->True], Cell[BoxData[ RowBox[{ RowBox[{"a", " ", "c", " ", RowBox[{"Cos", "[", "x", "]"}]}], "+", RowBox[{"a", " ", "b", " ", RowBox[{"CosIntegral", "[", "x", "]"}]}], "-", FractionBox[ RowBox[{"a", " ", "b", " ", RowBox[{"Sin", "[", "x", "]"}]}], "x"]}]], "Output"] }, Closed]], Cell["\<\ The command Integrate[f,{x,xmin,xmax}] gives the definite integral where \ the integration over x is from xmin to xmax. For example, integrating eq1 \ between {x,1,2}\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{" ", RowBox[{"Integrate", "[", RowBox[{"eq1", ",", RowBox[{"{", RowBox[{"x", ",", "1", ",", "2"}], "}"}]}], "]"}]}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"-", "a"}], " ", "c", " ", RowBox[{"(", RowBox[{ RowBox[{"Cos", "[", "1", "]"}], "-", RowBox[{"Cos", "[", "2", "]"}]}], ")"}]}], "+", RowBox[{"a", " ", "b", " ", RowBox[{"(", RowBox[{ RowBox[{"-", RowBox[{"CosIntegral", "[", "1", "]"}]}], "+", RowBox[{"CosIntegral", "[", "2", "]"}], "+", RowBox[{"Sin", "[", "1", "]"}], "-", FractionBox[ RowBox[{"Sin", "[", "2", "]"}], "2"]}], ")"}]}]}]], "Output"] }, Closed]], Cell["\<\ The Integrate command also calculates multiple dimensional integrals. \ Consider the expression \ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell["eq1=Sqrt[x^2+y^2];", "Input", AspectRatioFixed->True], Cell["and integrate y from 0 to x and x from 0 to r: ", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[" Integrate[eq1, {x,0,r},{y,0,x}]", "Input", AspectRatioFixed->True], Cell[BoxData[ RowBox[{ FractionBox["1", "6"], " ", SuperscriptBox["r", "3"], " ", RowBox[{"(", RowBox[{ SqrtBox["2"], "+", RowBox[{"ArcSinh", "[", "1", "]"}]}], ")"}]}]], "Output"] }, Closed]], Cell["\<\ The outer variable y is integrated first followed by the integration over x.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Options Commands: Assumptions, Element[n,Integers]], n\[Element]Integers\ \>", "Subsubsection"], Cell["Integration has the following options", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Options", "[", "Integrate", "]"}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"Assumptions", "\[RuleDelayed]", "$Assumptions"}], ",", RowBox[{"GenerateConditions", "\[Rule]", "Automatic"}], ",", RowBox[{"PrincipalValue", "\[Rule]", "False"}]}], "}"}]], "Output"] }, Closed]], Cell["Consider", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"eq2", "=", RowBox[{ RowBox[{"Cos", "[", RowBox[{"n", " ", "\[Theta]"}], "]"}], " ", RowBox[{"Sin", "[", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"2", " ", "p"}], "+", "1"}], ")"}], " ", "\[Theta]"}], "]"}], " ", RowBox[{ RowBox[{"Sin", "[", "\[Theta]", "]"}], "/", RowBox[{"Sin", "[", "\[Theta]", "]"}]}]}]}], ";"}]], "Input"], Cell["\<\ and integrate over \[Theta] from 0 to Pi with the assumption that n is an \ integer\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{" ", RowBox[{"{", RowBox[{ RowBox[{"Integrate", "[", RowBox[{"eq2", ",", RowBox[{"{", RowBox[{"\[Theta]", ",", "0", ",", "\[Pi]"}], "}"}], ",", RowBox[{"Assumptions", "\[Rule]", RowBox[{"Element", "[", RowBox[{"n", ",", "Integers"}], "]"}]}]}], "]"}], ",", RowBox[{"Integrate", "[", RowBox[{"eq2", ",", RowBox[{"{", RowBox[{"\[Theta]", ",", "0", ",", "\[Pi]"}], "}"}], ",", RowBox[{"Assumptions", "\[Rule]", RowBox[{"n", "\[Element]", "Integers"}]}]}], "]"}]}], " ", "}"}], " "}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"a", " ", "c", " ", "\[Pi]", " ", RowBox[{"Cos", "[", "x", "]"}]}], "+", RowBox[{"a", " ", "b", " ", "\[Pi]", " ", RowBox[{"CosIntegral", "[", "x", "]"}]}], "-", FractionBox[ RowBox[{"a", " ", "b", " ", "\[Pi]", " ", RowBox[{"Sin", "[", "x", "]"}]}], "x"]}], ",", RowBox[{ RowBox[{"a", " ", "c", " ", "\[Pi]", " ", RowBox[{"Cos", "[", "x", "]"}]}], "+", RowBox[{"a", " ", "b", " ", "\[Pi]", " ", RowBox[{"CosIntegral", "[", "x", "]"}]}], "-", FractionBox[ RowBox[{"a", " ", "b", " ", "\[Pi]", " ", RowBox[{"Sin", "[", "x", "]"}]}], "x"]}]}], "}"}]], "Output"] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["Numerical integrations", "Subsubsection"], Cell["\<\ If an analytic solution is not found and you know the values of all the \ parameters then you can numerically integrate the function. Consider the \ expression Sin[Cos[x I]] and integrate x from 0 to Pi. There is no analytic \ solution \ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{" ", RowBox[{"Integrate", "[", RowBox[{ RowBox[{"Sin", "[", RowBox[{"Cos", "[", RowBox[{"x", " ", "I"}], "]"}], "]"}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "Pi"}], "}"}]}], "]"}], " "}]], "Input"], Cell[BoxData[ RowBox[{ SubsuperscriptBox["\[Integral]", "0", "\[Pi]"], RowBox[{ RowBox[{"Sin", "[", RowBox[{"Cosh", "[", "x", "]"}], "]"}], RowBox[{"\[DifferentialD]", "x"}]}]}]], "Output"] }, Closed]], Cell["but applying NIntegrate gives,", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{" ", RowBox[{"NIntegrate", "[", RowBox[{ RowBox[{"Sin", "[", RowBox[{"Cos", "[", RowBox[{"x", " ", "I"}], "]"}], "]"}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "Pi"}], "}"}]}], "]"}], " "}]], "Input"], Cell[BoxData["1.1599933373590443`"], "Output"] }, Closed]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["2. Derivative ", "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Partial Derivative", "Subsubsection"], Cell["\<\ The partial derivative of an expression follows from D. D[f,x] gives the \ partial derivative of f with respect to x. D[f,{x,n}] gives the nth partial \ derivative. D[f,x1,x2,...] gives a mixed derivative. An alternate notation is \ f'[x],f''[x],f'''[x],f''''[x],... Examples of first derivatives expressed \ with different syntax, are\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"D", "[", RowBox[{ RowBox[{"Sin", "[", "x", "]"}], ",", "x"}], "]"}], ",", RowBox[{ RowBox[{"Tan", "'"}], "[", "x", "]"}], ",", RowBox[{"D", "[", RowBox[{ RowBox[{"Csc", "[", "x", "]"}], ",", RowBox[{"{", RowBox[{"x", ",", "1"}], "}"}]}], "]"}]}], "}"}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"Cos", "[", "x", "]"}], ",", SuperscriptBox[ RowBox[{"Sec", "[", "x", "]"}], "2"], ",", RowBox[{ RowBox[{"-", RowBox[{"Cot", "[", "x", "]"}]}], " ", RowBox[{"Csc", "[", "x", "]"}]}]}], "}"}]], "Output"] }, Closed]], Cell["Another example is", "Text"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"f", "=", RowBox[{ RowBox[{"x", "^", "3"}], " ", RowBox[{"y", "^", "2"}], " ", RowBox[{ RowBox[{"Cosh", "[", RowBox[{"2", RowBox[{"x", "/", "y"}]}], "]"}], "/", RowBox[{"(", RowBox[{"1", "+", RowBox[{ RowBox[{"x", "^", "4"}], " ", RowBox[{"y", "^", 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