(* Content-type: application/mathematica *)

(*** Wolfram Notebook File ***)
(* http://www.wolfram.com/nb *)

(* CreatedBy='Mathematica 6.0' *)

(*CacheID: 234*)
(* Internal cache information:
NotebookFileLineBreakTest
NotebookFileLineBreakTest
NotebookDataPosition[       145,          7]
NotebookDataLength[    111499,       2348]
NotebookOptionsPosition[    106454,       2203]
NotebookOutlinePosition[    107271,       2228]
CellTagsIndexPosition[    107228,       2225]
WindowFrame->Normal
ContainsDynamic->False*)

(* Beginning of Notebook Content *)
Notebook[{

Cell[CellGroupData[{
Cell[TextData[StyleBox["Tutorial 1: Fundamentals of pattern matching",
 FontColor->GrayLevel[0.5]]], "Title",
 CellChangeTimes->{{3.4033384525984616`*^9, 3.4033384766691866`*^9}, {
  3.4033386741995125`*^9, 3.403338677967826*^9}, {3.4033388210729246`*^9, 
  3.403338835799521*^9}, {3.403340373121048*^9, 3.4033404065573845`*^9}}],

Cell[TextData[StyleBox["Mathematica is very powerful, flexible and efficient \
in applying the knowledge you give it. At present, however, it knows nothing \
-- not even that H is a Hamiltonian. Our job now is to provide it with \
knowledge (which it will trust completely, be it right, wrong or \
approximate). It will then be able to apply that knowledge with astronomical \
speed to analytical problems that may be too large to be treated by a human.",
 
 FontSize->16,
 FontSlant->"Italic"]], "Text",
 CellChangeTimes->{{3.403339183467698*^9, 3.4033392359096546`*^9}, {
  3.403339932198097*^9, 3.4033400761273613`*^9}, {3.4033401098673897`*^9, 
  3.403340209649089*^9}, {3.4033402586449766`*^9, 3.403340308463077*^9}, {
  3.403340434347598*^9, 3.403340534071684*^9}, {3.403358505284734*^9, 
  3.4033585592197585`*^9}, {3.403459660577959*^9, 3.403459715684784*^9}, {
  3.4034598067941866`*^9, 3.403459839937573*^9}}],

Cell[CellGroupData[{

Cell["\<\
Getting started 1: declaring patterns and associating them with symbols or \
operations\
\>", "Section",
 CellChangeTimes->{{3.4033384525984616`*^9, 3.4033384766691866`*^9}, {
  3.4033386741995125`*^9, 3.403338677967826*^9}, {3.4033388210729246`*^9, 
  3.403338835799521*^9}, {3.403460216154704*^9, 3.403460217132181*^9}, {
  3.4035123588456793`*^9, 3.4035123642291236`*^9}, {3.403607839787242*^9, 
  3.403607861744821*^9}, {3.4036268563679543`*^9, 3.4036268565339594`*^9}}],

Cell[TextData[{
 "At the outset, the ",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 " kernel knows very little. For example, is not aware of the fact that our \
operators are linear and would not open the brackets (press Shift-Enter here \
and below to evaluate the cells):"
}], "Text",
 CellChangeTimes->{{3.403339183467698*^9, 3.4033392359096546`*^9}, {
  3.4034588882689915`*^9, 3.4034589007310843`*^9}, {3.403510475861706*^9, 
  3.403510493251218*^9}, {3.4035300608337727`*^9, 3.4035300994279823`*^9}, {
  3.403592874696241*^9, 3.403592875413968*^9}},
 FontSize->16,
 FontSlant->"Italic"],

Cell[BoxData[
 RowBox[{"H", ".", 
  RowBox[{"(", 
   RowBox[{
    RowBox[{"2", "*", "Lx"}], "+", 
    RowBox[{"3", "*", "Ly"}], "-", 
    RowBox[{"5", "*", "Lz"}]}], ")"}]}]], "Input",
 CellChangeTimes->{{3.4033394772301464`*^9, 3.403339497844062*^9}, {
  3.403339713462097*^9, 3.4033397140147963`*^9}, {3.4034588202460012`*^9, 
  3.4034588555288997`*^9}, {3.4036099649646983`*^9, 3.403609978123036*^9}}],

Cell[TextData[StyleBox["Here the dot '.' is a symbol for the matrix (and any \
other non-commutative) multiplication, whereas the star '*' denotes \
commutative multiplication. In order to inform the kernel that our operators \
are linear, we need to declare a linearity pattern:",
 FontSize->16,
 FontSlant->"Italic"]], "Text",
 CellChangeTimes->{{3.403339183467698*^9, 3.4033392844085035`*^9}, {
  3.4034589236007147`*^9, 3.403459034973469*^9}, {3.4035123921628823`*^9, 
  3.403512395737849*^9}, {3.4035124260786805`*^9, 3.403512428550202*^9}, {
  3.4035300851652236`*^9, 3.403530105002821*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"H", "/:", 
   RowBox[{"H", ".", 
    RowBox[{"(", 
     RowBox[{"A_", "+", "B_"}], ")"}]}], ":=", 
   RowBox[{
    RowBox[{"H", ".", "A"}], "+", 
    RowBox[{"H", ".", "B"}]}]}], ";"}]], "Input",
 CellChangeTimes->{{3.40333849785826*^9, 3.4033386620020514`*^9}, {
   3.4033387042942657`*^9, 3.4033387308121*^9}, {3.403338767848792*^9, 
   3.40333880508469*^9}, {3.4033389357433195`*^9, 3.403338980331286*^9}, {
   3.4033390263742375`*^9, 3.4033390555071383`*^9}, {3.4033396498616753`*^9, 
   3.403339650917272*^9}, 3.4035105148670216`*^9, 3.4035105456160297`*^9}],

Cell[TextData[{
 StyleBox["This is a simple replacement pattern. To read it literally: \
\"Pattern to be associated with the H symbol. Upon encountering a match to ",
  FontSize->16,
  FontSlant->"Italic"],
 "H.(A+B)",
 StyleBox[", just replace it with ",
  FontSize->16,
  FontSlant->"Italic"],
 "H.A + H.B, asking no questions about what those symbols actually are",
 StyleBox["\". Now the brackets will open:",
  FontSize->16,
  FontSlant->"Italic"]
}], "Text",
 CellChangeTimes->{{3.403339183467698*^9, 3.4033392844085035`*^9}, {
  3.4033393996452684`*^9, 3.403339411131838*^9}, {3.4033394689425907`*^9, 
  3.4033394737733364`*^9}, {3.4034590645955963`*^9, 3.403459065147319*^9}, {
  3.4034593099002085`*^9, 3.4034594088128233`*^9}, {3.403512464362363*^9, 
  3.4035125452087507`*^9}}],

Cell[BoxData[
 RowBox[{"H", ".", 
  RowBox[{"(", 
   RowBox[{
    RowBox[{"2", "*", "Lx"}], "+", 
    RowBox[{"3", "*", "Ly"}], "-", 
    RowBox[{"5", "*", "Lz"}]}], ")"}]}]], "Input",
 CellChangeTimes->{{3.4033388838511333`*^9, 3.4033389251746597`*^9}, {
   3.4033389858963594`*^9, 3.403338986256688*^9}, 3.4033395056218843`*^9, {
   3.403339721221366*^9, 3.4033397217242637`*^9}, 3.4036099843462706`*^9}],

Cell["\<\
However, the scalars are not taken out, because the kernel does not know that \
multiplication by a complex scalar commutes with matrix multiplication. We \
can tell it to check whether one of the multipliers is a complex number and, \
if such a number is encountered, to take it out. For that we would need a \
conditional pattern.\
\>", "Text",
 CellChangeTimes->{{3.4034591705136223`*^9, 3.403459273991374*^9}, {
  3.4035124501650295`*^9, 3.4035124523328595`*^9}, {3.403626867316472*^9, 
  3.4036268675811043`*^9}},
 FontSize->16,
 FontSlant->"Italic"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Getting started 2: putting constraints on patterns", "Section",
 CellChangeTimes->{{3.4033384525984616`*^9, 3.4033384766691866`*^9}, {
  3.4033386741995125`*^9, 3.403338677967826*^9}, {3.4033388210729246`*^9, 
  3.403338835799521*^9}, {3.403460216154704*^9, 3.403460217132181*^9}, {
  3.4035123588456793`*^9, 3.4035123642291236`*^9}, {3.403607839787242*^9, 
  3.403607861744821*^9}, {3.4036268563679543`*^9, 3.4036268873952656`*^9}}],

Cell[TextData[StyleBox["The '/;' symbol is used, which can be interpreted as \
'whenever':",
 FontSize->16,
 FontSlant->"Italic"]], "Text",
 CellChangeTimes->{{3.403339183467698*^9, 3.4033392844085035`*^9}, {
  3.4033393996452684`*^9, 3.403339411131838*^9}, {3.4033394689425907`*^9, 
  3.4033394737733364`*^9}, {3.4034590645955963`*^9, 3.403459065147319*^9}, {
  3.4034593099002085`*^9, 3.4034594088128233`*^9}, {3.4034594530062838`*^9, 
  3.4034595138715286`*^9}, {3.403512700896052*^9, 3.4035127022709637`*^9}, {
  3.4036271744960303`*^9, 3.403627206348484*^9}}],

Cell[BoxData[
 RowBox[{"H", "/:", 
  RowBox[{"H", ".", 
   RowBox[{"(", 
    RowBox[{"A_", "*", "B_"}], ")"}]}], ":=", 
  RowBox[{
   RowBox[{"A", "*", 
    RowBox[{"H", ".", "B"}]}], "/;", 
   RowBox[{"A", "\[Element]", "Complexes"}]}]}]], "Input",
 CellChangeTimes->{{3.40333849785826*^9, 3.4033386620020514`*^9}, {
  3.4033387042942657`*^9, 3.4033387308121*^9}, {3.403338767848792*^9, 
  3.40333880508469*^9}, {3.4033389357433195`*^9, 3.403338980331286*^9}, {
  3.4033390263742375`*^9, 3.4033390555071383`*^9}, {3.4033395936035576`*^9, 
  3.403339647165559*^9}, {3.403625801883688*^9, 3.4036258724768257`*^9}}],

Cell[TextData[{
 StyleBox["In human language, this reads: \"Pattern to be associated with the \
H symbol. Upon encountering a match to ",
  FontSize->16,
  FontSlant->"Italic"],
 "H.(A*B)",
 StyleBox[", take A out, but only if it's a complex scalar\". ",
  FontSize->16,
  FontSlant->"Italic"],
 "Now the scalars will be taken out:"
}], "Text",
 CellChangeTimes->{{3.403339183467698*^9, 3.4033392844085035`*^9}, {
  3.4033393996452684`*^9, 3.403339411131838*^9}, {3.4033394689425907`*^9, 
  3.4033394737733364`*^9}, {3.4034590645955963`*^9, 3.403459065147319*^9}, {
  3.4034593099002085`*^9, 3.4034594088128233`*^9}, {3.4034594530062838`*^9, 
  3.4034595138715286`*^9}, {3.403512700896052*^9, 3.4035127022709637`*^9}, {
  3.4036272129818487`*^9, 3.403627217420041*^9}}],

Cell[BoxData[
 RowBox[{"H", ".", 
  RowBox[{"(", 
   RowBox[{
    RowBox[{"2", "*", "Lx"}], "+", 
    RowBox[{"3", "*", "Ly"}], "-", 
    RowBox[{"5", "*", "Lz"}]}], ")"}]}]], "Input",
 CellChangeTimes->{{3.403339729748164*^9, 3.40333973030184*^9}, 
   3.4036099883382025`*^9}],

Cell[TextData[StyleBox["We can actually declare the linearity for the \
operator dot product (denoted A.B or Dot[A,B]) in full generality. By \
default, dot product is protected, but we know what we are doing, so we \
remove the protection:",
 FontSize->16,
 FontSlant->"Italic"]], "Text",
 CellChangeTimes->{{3.403339183467698*^9, 3.4033392844085035`*^9}, {
  3.4033393996452684`*^9, 3.403339411131838*^9}, {3.4033394689425907`*^9, 
  3.4033394737733364`*^9}, {3.4034590645955963`*^9, 3.403459065147319*^9}, {
  3.4034593099002085`*^9, 3.4034594088128233`*^9}, {3.4034594530062838`*^9, 
  3.4034595138715286`*^9}, {3.4034598973626084`*^9, 3.4034599978327403`*^9}, {
  3.4034603019636655`*^9, 3.4034603667739706`*^9}, {3.4034605051928453`*^9, 
  3.403460533461544*^9}, {3.4034606662645674`*^9, 3.4034606827908535`*^9}, {
  3.403510788011648*^9, 3.4035108104887257`*^9}, {3.403511083995634*^9, 
  3.403511289284252*^9}, {3.40351184111905*^9, 3.403511855726514*^9}, {
  3.403629502008193*^9, 3.4036295141539*^9}}],

Cell[BoxData[{
 RowBox[{
  RowBox[{"Unprotect", "[", "Dot", "]"}], ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"Dot", "/:", 
   RowBox[{"Dot", "[", 
    RowBox[{"A_", ",", 
     RowBox[{"B_", "+", "C_"}]}], "]"}], ":=", 
   RowBox[{
    RowBox[{"A", ".", "B"}], "+", 
    RowBox[{"A", ".", "C"}]}]}], ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"Dot", "/:", 
   RowBox[{"Dot", "[", 
    RowBox[{
     RowBox[{"A_", "+", "B_"}], ",", "C_"}], "]"}], ":=", 
   RowBox[{
    RowBox[{"A", ".", "C"}], "+", 
    RowBox[{"B", ".", "C"}]}]}], ";"}], "\[IndentingNewLine]", 
 RowBox[{"Dot", "/:", 
  RowBox[{"Dot", "[", 
   RowBox[{"A_", ",", 
    RowBox[{"B_", "*", "C_"}]}], "]"}], ":=", 
  RowBox[{
   RowBox[{"B", "*", 
    RowBox[{"A", ".", "C"}]}], "/;", 
   RowBox[{"B", "\[Element]", "Complexes"}]}]}], "\[IndentingNewLine]", 
 RowBox[{"Dot", "/:", 
  RowBox[{"Dot", "[", 
   RowBox[{
    RowBox[{"A_", "*", "B_"}], ",", "C_"}], "]"}], ":=", 
  RowBox[{
   RowBox[{"A", "*", 
    RowBox[{"B", ".", "C"}]}], "/;", 
   RowBox[{"A", "\[Element]", "Complexes"}]}]}]}], "Input",
 CellChangeTimes->{{3.4035108580530643`*^9, 3.403510914447892*^9}, {
  3.403510984080154*^9, 3.403511049938244*^9}, {3.403511536653186*^9, 
  3.4035115582494593`*^9}, {3.4035116471304893`*^9, 3.4035117588352537`*^9}, {
  3.40351260845275*^9, 3.4035126195955915`*^9}, {3.403625810508136*^9, 
  3.403625815020542*^9}, {3.4036258764648523`*^9, 3.4036258835493593`*^9}}],

Cell[TextData[StyleBox["Linearity is now specified in full. Mathematica still \
has no idea what the symbols might actually mean, but it knows what it can do \
with them. We generally want to keep it that way -- the processing then \
becomes very fast because no time is spent in unnecessary transformation and \
simplification attempts. For example:",
 FontSize->16,
 FontSlant->"Italic"]], "Text",
 CellChangeTimes->{{3.403339183467698*^9, 3.4033392844085035`*^9}, {
  3.4033393996452684`*^9, 3.403339411131838*^9}, {3.4033394689425907`*^9, 
  3.4033394737733364`*^9}, {3.4034590645955963`*^9, 3.403459065147319*^9}, {
  3.4034593099002085`*^9, 3.4034594088128233`*^9}, {3.4034594530062838`*^9, 
  3.4034595138715286`*^9}, {3.4034598973626084`*^9, 3.4034599978327403`*^9}, {
  3.4034603019636655`*^9, 3.4034603667739706`*^9}, {3.4034605051928453`*^9, 
  3.403460533461544*^9}, {3.4034606662645674`*^9, 3.4034606827908535`*^9}, {
  3.403510788011648*^9, 3.4035108104887257`*^9}, {3.403511083995634*^9, 
  3.403511289284252*^9}, {3.403511331032557*^9, 3.403511367971599*^9}, {
  3.4035119249173975`*^9, 3.4035120132437754`*^9}, {3.4035122566510477`*^9, 
  3.4035122749750705`*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"(", 
   RowBox[{
    RowBox[{"3", " ", "Lx"}], "+", " ", "Ly", "-", "Lz"}], ")"}], ".", 
  RowBox[{"(", 
   RowBox[{
    RowBox[{"3", " ", "Sx"}], "+", "Sy"}], ")"}]}]], "Input",
 CellChangeTimes->{{3.4035113794816046`*^9, 3.4035113898100452`*^9}, {
  3.4035116294167795`*^9, 3.403511631534808*^9}, {3.4035116909294443`*^9, 
  3.403511694416526*^9}, {3.4035117741125965`*^9, 3.403511777927782*^9}, {
  3.4035118908668423`*^9, 3.4035118948988113`*^9}}]
}, Open  ]],

Cell[CellGroupData[{

Cell["Getting started 3: crunching huge commutators", "Section",
 CellChangeTimes->{{3.4033384525984616`*^9, 3.4033384766691866`*^9}, {
  3.4033386741995125`*^9, 3.403338677967826*^9}, {3.4033388210729246`*^9, 
  3.403338835799521*^9}, {3.403460216154704*^9, 3.4034602378095684`*^9}, {
  3.4035123336002245`*^9, 3.403512346502719*^9}, {3.403629842114354*^9, 
  3.4036298464090014`*^9}}],

Cell[TextData[StyleBox["Ever encountered enormous commutators that are \
painful to even look at? We will now specify a few additional patterns that \
would enable Mathematica to reduce analytically commutators of any size. \
First of all, we will define a symbol for the commutator. The brackets are \
reserved by the system, but we can use something else, a diamond for \
example:",
 FontSize->16,
 FontSlant->"Italic"]], "Text",
 CellChangeTimes->{{3.403339183467698*^9, 3.4033392844085035`*^9}, {
  3.4033393996452684`*^9, 3.403339411131838*^9}, {3.4033394689425907`*^9, 
  3.4033394737733364`*^9}, {3.4034590645955963`*^9, 3.403459065147319*^9}, {
  3.4034593099002085`*^9, 3.4034594088128233`*^9}, {3.4034594530062838`*^9, 
  3.4034595138715286`*^9}, {3.4034598973626084`*^9, 3.4034599978327403`*^9}, {
  3.4034603019636655`*^9, 3.4034603667739706`*^9}, {3.4034605051928453`*^9, 
  3.403460533461544*^9}, {3.4034606662645674`*^9, 3.4034606827908535`*^9}, {
  3.403510788011648*^9, 3.4035108104887257`*^9}, {3.403511083995634*^9, 
  3.403511289284252*^9}, {3.403511331032557*^9, 3.403511367971599*^9}, {
  3.4035119249173975`*^9, 3.4035120132437754`*^9}, {3.4035122566510477`*^9, 
  3.4035122749750705`*^9}, {3.4035127511223297`*^9, 3.4035128513991146`*^9}, {
  3.403513482390955*^9, 3.4035135283850813`*^9}, {3.403513665178084*^9, 
  3.4035136711767235`*^9}, {3.403513738601119*^9, 3.4035137423450203`*^9}}],

Cell[BoxData[{
 RowBox[{
  RowBox[{"Diamond", "/:", 
   TagBox[
    StyleBox[
     RowBox[{"Diamond", "[", 
      RowBox[{"A_", ",", "B_"}], "]"}],
     ShowSpecialCharacters->False,
     ShowStringCharacters->True,
     NumberMarks->True],
    FullForm], ":=", 
   RowBox[{
    RowBox[{"A", ".", "B"}], "-", 
    RowBox[{"B", ".", "A"}]}]}], ";"}], "\[IndentingNewLine]", 
 RowBox[{"Lx", "\[Diamond]", "Ly"}]}], "Input",
 CellChangeTimes->{{3.403513601191945*^9, 3.4035137231323824`*^9}, {
  3.4035206154861655`*^9, 3.403520730464158*^9}, {3.4035207702360263`*^9, 
  3.403520804943766*^9}, {3.4035208397725916`*^9, 3.403520910052273*^9}}],

Cell[TextData[StyleBox["There are many ways to specify patterns required for \
commutator simplification. The general rule is that they shoud be as simple \
as possible. We will use a simple multiplication table here, associated with \
the dot product:",
 FontSize->16,
 FontSlant->"Italic"]], "Text",
 CellChangeTimes->{{3.403339183467698*^9, 3.4033392844085035`*^9}, {
  3.4033393996452684`*^9, 3.403339411131838*^9}, {3.4033394689425907`*^9, 
  3.4033394737733364`*^9}, {3.4034590645955963`*^9, 3.403459065147319*^9}, {
  3.4034593099002085`*^9, 3.4034594088128233`*^9}, {3.4034594530062838`*^9, 
  3.4034595138715286`*^9}, {3.4034598973626084`*^9, 3.4034599978327403`*^9}, {
  3.4034603019636655`*^9, 3.4034603667739706`*^9}, {3.4034605051928453`*^9, 
  3.403460533461544*^9}, {3.4034606662645674`*^9, 3.4034606827908535`*^9}, {
  3.403510788011648*^9, 3.4035108104887257`*^9}, {3.403511083995634*^9, 
  3.403511289284252*^9}, {3.403511331032557*^9, 3.403511367971599*^9}, {
  3.4035119249173975`*^9, 3.4035120132437754`*^9}, {3.4035122566510477`*^9, 
  3.4035122749750705`*^9}, {3.4035127511223297`*^9, 3.4035128513991146`*^9}, {
  3.403513482390955*^9, 3.4035135283850813`*^9}, {3.403513665178084*^9, 
  3.4035136711767235`*^9}, {3.403513738601119*^9, 3.4035137423450203`*^9}, {
  3.403520963517601*^9, 3.403521050756158*^9}}],

Cell[BoxData[{
 RowBox[{
  RowBox[{"Dot", "/:", 
   RowBox[{"Dot", "[", 
    RowBox[{"Lx", ",", "Ly"}], "]"}], ":=", 
   RowBox[{
    FractionBox["\[ImaginaryI]", "2"], " ", "Lz"}]}], 
  ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"Dot", "/:", 
   RowBox[{"Dot", "[", 
    RowBox[{"Lx", ",", "Lz"}], "]"}], ":=", 
   RowBox[{
    RowBox[{"-", 
     FractionBox["\[ImaginaryI]", "2"]}], " ", "Ly"}]}], 
  ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"Dot", "/:", 
   RowBox[{"Dot", "[", 
    RowBox[{"Ly", ",", "Lx"}], "]"}], ":=", 
   RowBox[{
    RowBox[{"-", 
     FractionBox["\[ImaginaryI]", "2"]}], " ", "Lz"}]}], 
  ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"Dot", "/:", 
   RowBox[{"Dot", "[", 
    RowBox[{"Ly", ",", "Lz"}], "]"}], ":=", 
   RowBox[{
    FractionBox["\[ImaginaryI]", "2"], " ", "Lx"}]}], 
  ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"Dot", "/:", 
   RowBox[{"Dot", "[", 
    RowBox[{"Lz", ",", "Lx"}], "]"}], ":=", 
   RowBox[{
    FractionBox["\[ImaginaryI]", "2"], " ", "Ly"}]}], 
  ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"Dot", "/:", 
   RowBox[{"Dot", "[", 
    RowBox[{"Lz", ",", "Ly"}], "]"}], ":=", 
   RowBox[{
    RowBox[{"-", 
     FractionBox["\[ImaginaryI]", "2"]}], " ", "Lx"}]}], 
  ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"Lp", ":=", 
   RowBox[{"Lx", "+", 
    RowBox[{"\[ImaginaryI]", " ", "Ly"}]}]}], ";", 
  RowBox[{"Lm", ":=", 
   RowBox[{"Lx", "-", 
    RowBox[{"\[ImaginaryI]", " ", "Ly"}]}]}], ";", 
  "\[IndentingNewLine]"}]}], "Input",
 CellChangeTimes->{{3.4035210597155457`*^9, 3.4035210689630003`*^9}, {
  3.403521292572712*^9, 3.4035213490202713`*^9}, {3.4035214013362584`*^9, 
  3.403521405534232*^9}, {3.403521467600572*^9, 3.4035215210903125`*^9}, {
  3.403521596227105*^9, 3.4035216251666594`*^9}, {3.4035929394518623`*^9, 
  3.40359294413125*^9}}],

Cell[TextData[StyleBox["Now for the nested eighteen-fold commutator (any \
volunteers to do it by hand?):",
 FontSize->16,
 FontSlant->"Italic"]], "Text",
 CellChangeTimes->{{3.403339183467698*^9, 3.4033392844085035`*^9}, {
   3.4033393996452684`*^9, 3.403339411131838*^9}, {3.4033394689425907`*^9, 
   3.4033394737733364`*^9}, {3.4034590645955963`*^9, 3.403459065147319*^9}, {
   3.4034593099002085`*^9, 3.4034594088128233`*^9}, {3.4034594530062838`*^9, 
   3.4034595138715286`*^9}, {3.4034598973626084`*^9, 
   3.4034599978327403`*^9}, {3.4034603019636655`*^9, 
   3.4034603966744003`*^9}, {3.403460587840876*^9, 3.4034606532986*^9}, 
   3.403510570739422*^9, {3.403521571362486*^9, 3.4035215772253914`*^9}, {
   3.4035216546091104`*^9, 3.40352169316426*^9}, {3.403521760462687*^9, 
   3.4035217732196827`*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"(", 
   RowBox[{"Lp", "\[Diamond]", 
    RowBox[{"(", 
     RowBox[{"Lz", "\[Diamond]", 
      RowBox[{"(", 
       RowBox[{"Ly", "\[Diamond]", 
        RowBox[{"(", 
         RowBox[{"Lp", "\[Diamond]", 
          RowBox[{"(", 
           RowBox[{"Lz", "\[Diamond]", 
            RowBox[{"(", 
             RowBox[{"Ly", "\[Diamond]", 
              RowBox[{"(", 
               RowBox[{"Lp", "\[Diamond]", 
                RowBox[{"(", 
                 RowBox[{"Lz", "\[Diamond]", 
                  RowBox[{"(", 
                   RowBox[{"Ly", "\[Diamond]", 
                    RowBox[{"(", 
                    RowBox[{"Lp", "\[Diamond]", 
                    RowBox[{"(", 
                    RowBox[{"Lz", "\[Diamond]", 
                    RowBox[{"(", 
                    RowBox[{"Ly", "\[Diamond]", 
                    RowBox[{"(", 
                    RowBox[{"Lp", "\[Diamond]", 
                    RowBox[{"(", 
                    RowBox[{"Lz", "\[Diamond]", 
                    RowBox[{"(", 
                    RowBox[{"Ly", "\[Diamond]", 
                    RowBox[{"(", 
                    RowBox[{"Lp", "\[Diamond]", 
                    RowBox[{"(", 
                    RowBox[{"Lz", "\[Diamond]", "Ly"}], ")"}]}], ")"}]}], 
                    ")"}]}], ")"}]}], ")"}]}], ")"}]}], ")"}]}], ")"}]}], 
                   ")"}]}], ")"}]}], ")"}]}], ")"}]}], ")"}]}], ")"}]}], 
       ")"}]}], ")"}]}], ")"}], "//", "Timing"}]], "Input",
 CellChangeTimes->{{3.403521632449396*^9, 3.4035217057269325`*^9}, {
  3.4035218845934143`*^9, 3.403521960496759*^9}, {3.4035223608451586`*^9, 
  3.4035223624524775`*^9}}],

Cell[TextData[StyleBox["No matrix multiplication and no arithmetic has been \
done above: only patterns were detected and replaced by other patterns -- \
just as a human would do. The timing figure above is likely to be zero -- \
that means it is less than a millisecond and the system could not measure it \
accurately.",
 FontSize->16,
 FontSlant->"Italic"]], "Text",
 CellChangeTimes->{{3.403339183467698*^9, 3.4033392844085035`*^9}, {
   3.4033393996452684`*^9, 3.403339411131838*^9}, {3.4033394689425907`*^9, 
   3.4033394737733364`*^9}, {3.4034590645955963`*^9, 3.403459065147319*^9}, {
   3.4034593099002085`*^9, 3.4034594088128233`*^9}, {3.4034594530062838`*^9, 
   3.4034595138715286`*^9}, {3.4034598973626084`*^9, 
   3.4034599978327403`*^9}, {3.4034603019636655`*^9, 
   3.4034603966744003`*^9}, {3.403460587840876*^9, 3.4034606532986*^9}, 
   3.403510570739422*^9, {3.403521571362486*^9, 3.4035215772253914`*^9}, {
   3.4035216546091104`*^9, 3.40352169316426*^9}, {3.403521760462687*^9, 
   3.4035217732196827`*^9}, {3.403522729482697*^9, 3.403522762078267*^9}, {
   3.4035229702719727`*^9, 3.403523051118361*^9}, {3.4035929741810846`*^9, 
   3.4035929747318306`*^9}}]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData[StyleBox["Exercise 1: a better pattern set for commutators",
 FontColor->GrayLevel[0]]], "Section",
 CellChangeTimes->{{3.4033384525984616`*^9, 3.4033384766691866`*^9}, {
  3.4033386741995125`*^9, 3.403338677967826*^9}, {3.4033388210729246`*^9, 
  3.403338835799521*^9}, {3.403460216154704*^9, 3.4034602378095684`*^9}, {
  3.4035123336002245`*^9, 3.403512346502719*^9}, {3.403522074171124*^9, 
  3.403522075610485*^9}, {3.4035221059269037`*^9, 3.403522146059101*^9}}],

Cell[TextData[StyleBox["In Lie algebras (such as spin state space) only the \
commutators are defined well -- the products we used above might actually \
change between the spin systems. Overloading the definition table for the dot \
product is also undesirable, as it may slow down the system (the full list is \
checked every time dot is used). In a separate worksheet, program an \
alternative pattern set, associated with Diamond operator and based on \
directly matching commutators without opening them. Make sure you quit the \
kernel (Menu > Evaluation > Quit Kernel > Local) to make it forget everything \
we told it above. Double-click the arrow on the right of the section below \
for one of the possible answers.",
 FontSize->16,
 FontSlant->"Italic"]], "Text",
 CellChangeTimes->{{3.403339183467698*^9, 3.4033392844085035`*^9}, {
  3.4033393996452684`*^9, 3.403339411131838*^9}, {3.4033394689425907`*^9, 
  3.4033394737733364`*^9}, {3.4034590645955963`*^9, 3.403459065147319*^9}, {
  3.4034593099002085`*^9, 3.4034594088128233`*^9}, {3.4034594530062838`*^9, 
  3.4034595138715286`*^9}, {3.4034598973626084`*^9, 3.4034599978327403`*^9}, {
  3.4034603019636655`*^9, 3.4034603667739706`*^9}, {3.4034605051928453`*^9, 
  3.403460533461544*^9}, {3.4034606662645674`*^9, 3.4034606827908535`*^9}, {
  3.403510788011648*^9, 3.4035108104887257`*^9}, {3.403511083995634*^9, 
  3.403511289284252*^9}, {3.403511331032557*^9, 3.403511367971599*^9}, {
  3.4035119249173975`*^9, 3.4035120132437754`*^9}, {3.4035122566510477`*^9, 
  3.4035122749750705`*^9}, {3.4035127511223297`*^9, 3.4035128513991146`*^9}, {
  3.403513482390955*^9, 3.4035135283850813`*^9}, {3.403513665178084*^9, 
  3.4035136711767235`*^9}, {3.403513738601119*^9, 3.4035137423450203`*^9}, {
  3.4035221495940313`*^9, 3.403522241344018*^9}, {3.403522462896244*^9, 
  3.4035224719513283`*^9}, {3.4035226048959446`*^9, 3.4035226995363717`*^9}, {
  3.403529323207156*^9, 3.4035293302701807`*^9}, {3.4035299214003143`*^9, 
  3.403530013157137*^9}}],

Cell[CellGroupData[{

Cell["Answer", "Text",
 CellGroupingRules->{GroupTogetherGrouping, 10000.},
 CellChangeTimes->{{3.403528988628961*^9, 3.403529017208186*^9}, {
  3.403529279737282*^9, 3.403529287722122*^9}}],

Cell[BoxData[{
 RowBox[{
  RowBox[{"Diamond", "/:", 
   RowBox[{"Diamond", "[", 
    RowBox[{"A_", ",", 
     RowBox[{"B_", "+", "C_"}]}], "]"}], ":=", 
   RowBox[{
    RowBox[{"A", "\[Diamond]", "B"}], "+", 
    RowBox[{"A", "\[Diamond]", "C"}]}]}], ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"Diamond", "/:", 
   RowBox[{"Diamond", "[", 
    RowBox[{
     RowBox[{"A_", "+", "B_"}], ",", "C_"}], "]"}], ":=", 
   RowBox[{
    RowBox[{"A", "\[Diamond]", "C"}], "+", 
    RowBox[{"B", "\[Diamond]", "C"}]}]}], ";"}], "\[IndentingNewLine]", 
 RowBox[{"Diamond", "/:", 
  RowBox[{"Diamond", "[", 
   RowBox[{"A_", ",", 
    RowBox[{"B_", "*", "C_"}]}], "]"}], ":=", 
  RowBox[{
   RowBox[{"B", "*", 
    RowBox[{"A", "\[Diamond]", "C"}]}], "/;", 
   RowBox[{"B", "\[Element]", "Complexes"}]}]}], "\[IndentingNewLine]", 
 RowBox[{"Diamond", "/:", 
  RowBox[{"Diamond", "[", 
   RowBox[{
    RowBox[{"A_", "*", "B_"}], ",", "C_"}], "]"}], ":=", 
  RowBox[{
   RowBox[{"A", "*", 
    RowBox[{"B", "\[Diamond]", "C"}]}], "/;", 
   RowBox[{"A", "\[Element]", "Complexes"}]}]}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{
   RowBox[{"Diamond", "/:", 
    RowBox[{"Diamond", "[", 
     RowBox[{"A_", ",", "A_"}], "]"}], ":=", "0"}], ";"}], 
  "\[IndentingNewLine]"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"Diamond", "/:", 
   TagBox[
    StyleBox[
     RowBox[{"Diamond", "[", 
      RowBox[{"Lx", ",", "Ly"}], "]"}],
     ShowSpecialCharacters->False,
     ShowStringCharacters->True,
     NumberMarks->True],
    FullForm], ":=", 
   RowBox[{"\[ImaginaryI]", " ", "Lz"}]}], ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"Diamond", "/:", 
   TagBox[
    StyleBox[
     RowBox[{"Diamond", "[", 
      RowBox[{"Ly", ",", "Lx"}], "]"}],
     ShowSpecialCharacters->False,
     ShowStringCharacters->True,
     NumberMarks->True],
    FullForm], ":=", 
   RowBox[{
    RowBox[{"-", "\[ImaginaryI]"}], " ", "Lz"}]}], 
  ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"Diamond", "/:", 
   TagBox[
    StyleBox[
     RowBox[{"Diamond", "[", 
      RowBox[{"Lx", ",", "Lz"}], "]"}],
     ShowSpecialCharacters->False,
     ShowStringCharacters->True,
     NumberMarks->True],
    FullForm], ":=", 
   RowBox[{
    RowBox[{"-", "\[ImaginaryI]"}], " ", "Ly"}]}], 
  ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"Diamond", "/:", 
   TagBox[
    StyleBox[
     RowBox[{"Diamond", "[", 
      RowBox[{"Lz", ",", "Lx"}], "]"}],
     ShowSpecialCharacters->False,
     ShowStringCharacters->True,
     NumberMarks->True],
    FullForm], ":=", 
   RowBox[{"\[ImaginaryI]", " ", "Ly"}]}], ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"Diamond", "/:", 
   TagBox[
    StyleBox[
     RowBox[{"Diamond", "[", 
      RowBox[{"Ly", ",", "Lz"}], "]"}],
     ShowSpecialCharacters->False,
     ShowStringCharacters->True,
     NumberMarks->True],
    FullForm], ":=", 
   RowBox[{"\[ImaginaryI]", " ", "Lx"}]}], ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"Diamond", "/:", 
   TagBox[
    StyleBox[
     RowBox[{"Diamond", "[", 
      RowBox[{"Lz", ",", "Ly"}], "]"}],
     ShowSpecialCharacters->False,
     ShowStringCharacters->True,
     NumberMarks->True],
    FullForm], ":=", 
   RowBox[{
    RowBox[{"-", "\[ImaginaryI]"}], " ", "Lx"}]}], 
  ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"Lp", ":=", 
   RowBox[{"Lx", "+", 
    RowBox[{"\[ImaginaryI]", " ", "Ly"}]}]}], ";", 
  RowBox[{"Lm", ":=", 
   RowBox[{"Lx", "-", 
    RowBox[{"\[ImaginaryI]", " ", "Ly"}]}]}], ";", 
  "\[IndentingNewLine]"}]}], "Input",
 CellGroupingRules->{GroupTogetherGrouping, 10000.},
 CellChangeTimes->{{3.4035108580530643`*^9, 3.403510914447892*^9}, {
   3.403510984080154*^9, 3.403511049938244*^9}, {3.403511536653186*^9, 
   3.4035115582494593`*^9}, {3.4035116471304893`*^9, 
   3.4035117588352537`*^9}, {3.40351260845275*^9, 3.4035126195955915`*^9}, {
   3.4035243552839127`*^9, 3.4035244021578655`*^9}, {3.4035266998047523`*^9, 
   3.403526750531974*^9}, {3.4035285080122614`*^9, 3.4035285152442207`*^9}, {
   3.4035287899688244`*^9, 3.403528805392642*^9}, 3.403528881270598*^9, {
   3.4035290015705156`*^9, 3.4035290127836647`*^9}, {3.4035290532674017`*^9, 
   3.403529067368062*^9}, {3.403529161903027*^9, 3.4035291878915977`*^9}, {
   3.403529222239009*^9, 3.4035292416205807`*^9}, {3.403529279737282*^9, 
   3.403529287722122*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"(", 
   RowBox[{"Lp", "\[Diamond]", 
    RowBox[{"(", 
     RowBox[{"Lz", "\[Diamond]", 
      RowBox[{"(", 
       RowBox[{"Ly", "\[Diamond]", 
        RowBox[{"(", 
         RowBox[{"Lp", "\[Diamond]", 
          RowBox[{"(", 
           RowBox[{"Lz", "\[Diamond]", 
            RowBox[{"(", 
             RowBox[{"Ly", "\[Diamond]", 
              RowBox[{"(", 
               RowBox[{"Lp", "\[Diamond]", 
                RowBox[{"(", 
                 RowBox[{"Lz", "\[Diamond]", 
                  RowBox[{"(", 
                   RowBox[{"Ly", "\[Diamond]", 
                    RowBox[{"(", 
                    RowBox[{"Lp", "\[Diamond]", 
                    RowBox[{"(", 
                    RowBox[{"Lz", "\[Diamond]", 
                    RowBox[{"(", 
                    RowBox[{"Ly", "\[Diamond]", 
                    RowBox[{"(", 
                    RowBox[{"Lp", "\[Diamond]", 
                    RowBox[{"(", 
                    RowBox[{"Lz", "\[Diamond]", 
                    RowBox[{"(", 
                    RowBox[{"Ly", "\[Diamond]", 
                    RowBox[{"(", 
                    RowBox[{"Lp", "\[Diamond]", 
                    RowBox[{"(", 
                    RowBox[{"Lz", "\[Diamond]", "Ly"}], ")"}]}], ")"}]}], 
                    ")"}]}], ")"}]}], ")"}]}], ")"}]}], ")"}]}], ")"}]}], 
                   ")"}]}], ")"}]}], ")"}]}], ")"}]}], ")"}]}], ")"}]}], 
       ")"}]}], ")"}]}], ")"}], "//", "Timing"}]], "Input",
 CellGroupingRules->{GroupTogetherGrouping, 10000.},
 CellChangeTimes->{3.4035292877230988`*^9}],

Cell[TextData[StyleBox["The patterns above are declared by simple enumeration \
of cases. In later tutorials we will learn how to declare them much more \
compactly, using membership checks.",
 FontSize->16,
 FontSlant->"Italic"]], "Text",
 CellGroupingRules->{GroupTogetherGrouping, 10000.},
 CellChangeTimes->{{3.403339183467698*^9, 3.4033392844085035`*^9}, {
   3.4033393996452684`*^9, 3.403339411131838*^9}, {3.4033394689425907`*^9, 
   3.4033394737733364`*^9}, {3.4034590645955963`*^9, 3.403459065147319*^9}, {
   3.4034593099002085`*^9, 3.4034594088128233`*^9}, {3.4034594530062838`*^9, 
   3.4034595138715286`*^9}, {3.4034598973626084`*^9, 
   3.4034599978327403`*^9}, {3.4034603019636655`*^9, 
   3.4034603667739706`*^9}, {3.4034605051928453`*^9, 3.403460533461544*^9}, {
   3.4034606662645674`*^9, 3.4034606827908535`*^9}, {3.403510788011648*^9, 
   3.4035108104887257`*^9}, {3.403511083995634*^9, 3.403511289284252*^9}, {
   3.403511331032557*^9, 3.403511367971599*^9}, {3.4035119249173975`*^9, 
   3.4035120132437754`*^9}, {3.4035122566510477`*^9, 
   3.4035122749750705`*^9}, {3.4035127511223297`*^9, 
   3.4035128513991146`*^9}, {3.403513482390955*^9, 3.4035135283850813`*^9}, {
   3.403513665178084*^9, 3.4035136711767235`*^9}, {3.403513738601119*^9, 
   3.4035137423450203`*^9}, {3.4035221495940313`*^9, 3.403522241344018*^9}, {
   3.403522462896244*^9, 3.4035224719513283`*^9}, {3.4035226048959446`*^9, 
   3.4035226995363717`*^9}, {3.403528834107601*^9, 3.403528881269622*^9}, {
   3.4035289687093377`*^9, 3.4035290127836647`*^9}, {3.4035290447337685`*^9, 
   3.4035290452610784`*^9}, 3.4035293514035935`*^9, {3.403529885442655*^9, 
   3.4035298906581416`*^9}}]
}, Closed]]
}, Open  ]],

Cell[CellGroupData[{

Cell["Getting started 4: building an integrator", "Section",
 CellChangeTimes->{{3.4033384525984616`*^9, 3.4033384766691866`*^9}, {
  3.4033386741995125`*^9, 3.403338677967826*^9}, {3.4033388210729246`*^9, 
  3.403338835799521*^9}, {3.403460216154704*^9, 3.4034602378095684`*^9}, {
  3.403512348662737*^9, 3.4035123487584343`*^9}, {3.4035992010149636`*^9, 
  3.403599201127261*^9}, {3.4036298572247157`*^9, 3.403629860679572*^9}}],

Cell[TextData[StyleBox["Many analyical spin dynamics problems lead to \
integrals of highly specialized functions. Furthermore, the exact answer \
might be unknown in principle, and the integral (for example of a correlation \
function) must simply replaced by another symbol. In relaxation theory we \
frequently encounter integrals of this form:",
 FontSize->16,
 FontSlant->"Italic"]], "Text",
 CellChangeTimes->{{3.403339183467698*^9, 3.4033392844085035`*^9}, {
  3.4033393996452684`*^9, 3.403339411131838*^9}, {3.4033394689425907`*^9, 
  3.4033394737733364`*^9}, {3.4034590645955963`*^9, 3.403459065147319*^9}, {
  3.4034593099002085`*^9, 3.4034594088128233`*^9}, {3.4034594530062838`*^9, 
  3.4034595138715286`*^9}, {3.4034598973626084`*^9, 3.4034599978327403`*^9}, {
  3.4034603019636655`*^9, 3.4034603667739706`*^9}, {3.4034605051928453`*^9, 
  3.403460533461544*^9}, {3.4034606662645674`*^9, 3.4034606827908535`*^9}, {
  3.403510788011648*^9, 3.4035108104887257`*^9}, {3.403511083995634*^9, 
  3.403511289284252*^9}, {3.403511331032557*^9, 3.403511367971599*^9}, {
  3.4035119249173975`*^9, 3.4035120132437754`*^9}, {3.4035122566510477`*^9, 
  3.4035122749750705`*^9}, {3.4035127511223297`*^9, 3.4035128513991146`*^9}, {
  3.403513482390955*^9, 3.4035135283850813`*^9}, {3.403513665178084*^9, 
  3.4035136711767235`*^9}, {3.403513738601119*^9, 3.4035137423450203`*^9}, {
  3.4035295479994087`*^9, 3.403529729804179*^9}, {3.4035298109874597`*^9, 
  3.403529827081156*^9}, {3.403588761194874*^9, 3.4035887620346637`*^9}, {
  3.40359196144024*^9, 3.4035920228083825`*^9}}],

Cell[BoxData[
 RowBox[{
  SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], 
  RowBox[{
   RowBox[{"g", "[", "\[Tau]", "]"}], " ", 
   SuperscriptBox["\[ExponentialE]", 
    RowBox[{
     RowBox[{"-", "\[ImaginaryI]"}], " ", "\[Omega]", " ", "\[Tau]"}]], 
   RowBox[{"\[DifferentialD]", "\[Tau]"}]}]}]], "Input",
 CellChangeTimes->{{3.4035307857785845`*^9, 3.4035308128256817`*^9}}],

Cell[TextData[{
 StyleBox["Because the system doesn't know (and neither do we) what the \
correlation function g[",
  FontSize->16,
  FontSlant->"Italic"],
 "\[Tau]",
 StyleBox["] is, no amount of tinkering will help -- the integral will stay \
unevaluated. We do, however, know from relaxation theory that we can simply \
assume that this integral evaluates to a certain function (so-called spectral \
density function) and leave it to the experimentalists to mesure. We \
therefore need (1) to program an integrator that knows this, and (2) to carry \
the result along without telling the kernel anything about it.",
  FontSize->16,
  FontSlant->"Italic"]
}], "Text",
 CellChangeTimes->{{3.403339183467698*^9, 3.4033392844085035`*^9}, {
  3.4033393996452684`*^9, 3.403339411131838*^9}, {3.4033394689425907`*^9, 
  3.4033394737733364`*^9}, {3.4034590645955963`*^9, 3.403459065147319*^9}, {
  3.4034593099002085`*^9, 3.4034594088128233`*^9}, {3.4034594530062838`*^9, 
  3.4034595138715286`*^9}, {3.4034598973626084`*^9, 3.4034599978327403`*^9}, {
  3.4034603019636655`*^9, 3.4034603667739706`*^9}, {3.4034605051928453`*^9, 
  3.403460533461544*^9}, {3.4034606662645674`*^9, 3.4034606827908535`*^9}, {
  3.403510788011648*^9, 3.4035108104887257`*^9}, {3.403511083995634*^9, 
  3.403511289284252*^9}, {3.403511331032557*^9, 3.403511367971599*^9}, {
  3.4035119249173975`*^9, 3.4035120132437754`*^9}, {3.4035122566510477`*^9, 
  3.4035122749750705`*^9}, {3.4035127511223297`*^9, 3.4035128513991146`*^9}, {
  3.403513482390955*^9, 3.4035135283850813`*^9}, {3.403513665178084*^9, 
  3.4035136711767235`*^9}, {3.403513738601119*^9, 3.4035137423450203`*^9}, {
  3.4035295479994087`*^9, 3.403529729804179*^9}, {3.4035298109874597`*^9, 
  3.403529827081156*^9}, {3.4035308264068437`*^9, 3.403531026826633*^9}, {
  3.4035886411673994`*^9, 3.403588684834526*^9}, {3.4035888188611045`*^9, 
  3.4035888221567917`*^9}, {3.4035920441898265`*^9, 3.4035920500214844`*^9}}],

Cell[BoxData[{
 RowBox[{
  RowBox[{
   RowBox[{"Int", "[", 
    RowBox[{"A_", "+", "B_"}], "]"}], ":=", 
   RowBox[{
    RowBox[{"Int", "[", "A", "]"}], "+", 
    RowBox[{"Int", "[", "B", "]"}]}]}], ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"Int", "[", 
   RowBox[{"A_", " ", "B_"}], "]"}], ":=", 
  RowBox[{
   RowBox[{"A", " ", 
    RowBox[{"Int", "[", "B", "]"}]}], "/;", 
   RowBox[{"FreeQ", "[", 
    RowBox[{"A", ",", "\[Tau]"}], "]"}]}]}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{
   RowBox[{"Int", "[", 
    RowBox[{
     SuperscriptBox["\[ExponentialE]", 
      RowBox[{"Times", "[", 
       RowBox[{
        RowBox[{"Complex", "[", 
         RowBox[{"0", ",", "a_"}], "]"}], ",", "\[Tau]", ",", "\[Omega]"}], 
       "]"}]], " ", 
     RowBox[{"g", "[", "\[Tau]", "]"}]}], "]"}], ":=", 
   RowBox[{"J", "[", 
    RowBox[{"a", " ", "\[Omega]"}], "]"}]}], ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{
   RowBox[{"Int", "[", 
    RowBox[{"g", "[", "\[Tau]", "]"}], "]"}], ":=", 
   RowBox[{"J", "[", "0", "]"}]}], ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{
   RowBox[{"Int", "[", "0", "]"}], ":=", "0"}], ";"}]}], "Input",
 CellChangeTimes->{
  3.403624586699746*^9, 3.4036246335385447`*^9, {3.40362467047954*^9, 
   3.4036246708164325`*^9}, {3.403629185956156*^9, 3.403629196715233*^9}}],

Cell[TextData[StyleBox["The first two patterns prescribe the integrator Int \
to be linear. The FreeQ condition checks whether the integration variable \
appears in the expression A before it's taken out. The third pattern is the \
integration pattern itself (multiplication operation is written explicitly as \
Times[a,b,c] and the complex scalar as Complex[a,b]), and the last two deal \
with the special cases of missing exponential and zero integrand. We can now \
easily integrate any number of correlation functions with any multipliers:",
 FontSize->16,
 FontSlant->"Italic"]], "Text",
 CellChangeTimes->{{3.403339183467698*^9, 3.4033392844085035`*^9}, {
  3.4033393996452684`*^9, 3.403339411131838*^9}, {3.4033394689425907`*^9, 
  3.4033394737733364`*^9}, {3.4034590645955963`*^9, 3.403459065147319*^9}, {
  3.4034593099002085`*^9, 3.4034594088128233`*^9}, {3.4034594530062838`*^9, 
  3.4034595138715286`*^9}, {3.4034598973626084`*^9, 3.4034599978327403`*^9}, {
  3.4034603019636655`*^9, 3.4034603667739706`*^9}, {3.4034605051928453`*^9, 
  3.403460533461544*^9}, {3.4034606662645674`*^9, 3.4034606827908535`*^9}, {
  3.403510788011648*^9, 3.4035108104887257`*^9}, {3.403511083995634*^9, 
  3.403511289284252*^9}, {3.403511331032557*^9, 3.403511367971599*^9}, {
  3.4035119249173975`*^9, 3.4035120132437754`*^9}, {3.4035122566510477`*^9, 
  3.4035122749750705`*^9}, {3.4035127511223297`*^9, 3.4035128513991146`*^9}, {
  3.403513482390955*^9, 3.4035135283850813`*^9}, {3.403513665178084*^9, 
  3.4035136711767235`*^9}, {3.403513738601119*^9, 3.4035137423450203`*^9}, {
  3.4035295479994087`*^9, 3.403529729804179*^9}, {3.4035298109874597`*^9, 
  3.403529827081156*^9}, {3.4035308264068437`*^9, 3.403531026826633*^9}, {
  3.4035886411673994`*^9, 3.403588684834526*^9}, {3.4035888188611045`*^9, 
  3.4035888221567917`*^9}, {3.403590674188645*^9, 3.403590780904471*^9}, {
  3.403624739246623*^9, 3.4036247814988017`*^9}, {3.403629350873335*^9, 
  3.40362942252207*^9}}],

Cell[BoxData[
 RowBox[{"Int", "[", 
  RowBox[{
   RowBox[{
    RowBox[{"g", "[", "\[Tau]", "]"}], " ", 
    SuperscriptBox["\[ExponentialE]", 
     RowBox[{
      RowBox[{"-", "\[ImaginaryI]"}], " ", "\[Omega]", " ", "\[Tau]"}]]}], 
   "+", 
   RowBox[{"15", " ", 
    RowBox[{"g", "[", "\[Tau]", "]"}], " ", 
    SuperscriptBox["\[ExponentialE]", 
     RowBox[{
      RowBox[{"-", "2"}], " ", "\[ImaginaryI]", " ", "\[Omega]", " ", 
      "\[Tau]"}]]}], "+", 
   RowBox[{
    RowBox[{"(", 
     RowBox[{"1", "-", 
      RowBox[{"3", " ", 
       RowBox[{
        RowBox[{"Cos", "[", "\[Theta]", "]"}], "^", "2"}]}]}], ")"}], " ", 
    RowBox[{"g", "[", "\[Tau]", "]"}], " ", 
    SuperscriptBox["\[ExponentialE]", 
     RowBox[{"3", " ", "\[ImaginaryI]", " ", "\[Omega]", " ", "\[Tau]"}]]}], 
   "-", 
   RowBox[{"4", " ", 
    RowBox[{"g", "[", "\[Tau]", "]"}]}]}], "]"}]], "Input",
 CellChangeTimes->{{3.4035907848876143`*^9, 3.403590832813258*^9}, 
   3.4035908984106216`*^9, 3.4035909757035265`*^9, {3.403629296906086*^9, 
   3.403629326105389*^9}, 3.403629433860211*^9}],

Cell[TextData[StyleBox["See page 281 of the Mathematica manual for a slightly \
more complicated hand-crafted analytical integrator.",
 FontSize->16,
 FontSlant->"Italic"]], "Text",
 CellChangeTimes->{{3.403339183467698*^9, 3.4033392844085035`*^9}, {
  3.4033393996452684`*^9, 3.403339411131838*^9}, {3.4033394689425907`*^9, 
  3.4033394737733364`*^9}, {3.4034590645955963`*^9, 3.403459065147319*^9}, {
  3.4034593099002085`*^9, 3.4034594088128233`*^9}, {3.4034594530062838`*^9, 
  3.4034595138715286`*^9}, {3.4034598973626084`*^9, 3.4034599978327403`*^9}, {
  3.4034603019636655`*^9, 3.4034603667739706`*^9}, {3.4034605051928453`*^9, 
  3.403460533461544*^9}, {3.4034606662645674`*^9, 3.4034606827908535`*^9}, {
  3.403510788011648*^9, 3.4035108104887257`*^9}, {3.403511083995634*^9, 
  3.403511289284252*^9}, {3.403511331032557*^9, 3.403511367971599*^9}, {
  3.4035119249173975`*^9, 3.4035120132437754`*^9}, {3.4035122566510477`*^9, 
  3.4035122749750705`*^9}, {3.4035127511223297`*^9, 3.4035128513991146`*^9}, {
  3.403513482390955*^9, 3.4035135283850813`*^9}, {3.403513665178084*^9, 
  3.4035136711767235`*^9}, {3.403513738601119*^9, 3.4035137423450203`*^9}, {
  3.4035295479994087`*^9, 3.403529729804179*^9}, {3.4035298109874597`*^9, 
  3.403529827081156*^9}, {3.403588761194874*^9, 3.4035887620346637`*^9}, {
  3.40359196144024*^9, 3.4035920228083825`*^9}, {3.403629890725501*^9, 
  3.403629932041216*^9}}]
}, Open  ]],

Cell[CellGroupData[{

Cell["Getting started 5: creating and using simplification rules ", "Section",
 CellChangeTimes->{{3.4033384525984616`*^9, 3.4033384766691866`*^9}, {
  3.4033386741995125`*^9, 3.403338677967826*^9}, {3.4033388210729246`*^9, 
  3.403338835799521*^9}, {3.403460216154704*^9, 3.4034602378095684`*^9}, {
  3.403512348662737*^9, 3.4035123487584343`*^9}, {3.40359100980193*^9, 
  3.403591017272155*^9}, {3.4035910507084913`*^9, 3.4035910531800127`*^9}, {
  3.4035981372158637`*^9, 3.4035981532050743`*^9}, {3.403598264489944*^9, 
  3.4035982652086477`*^9}, {3.4035987635312457`*^9, 3.4035987702085524`*^9}, {
  3.4035988242265797`*^9, 3.4035988277058487`*^9}, {3.4035988820197554`*^9, 
  3.4035989212877502`*^9}, {3.403607782265533*^9, 3.4036077832967167`*^9}, {
  3.4036300713145285`*^9, 3.4036300809389124`*^9}}],

Cell[TextData[StyleBox["Simplification or change of notation are frequently \
required in symbolic processing. We can define patterns and pass them on to \
the FullSimplify command to be used in simplification of expressions. For \
example:",
 FontSize->16,
 FontSlant->"Italic"]], "Text",
 CellChangeTimes->{{3.403339183467698*^9, 3.4033392844085035`*^9}, {
  3.4033393996452684`*^9, 3.403339411131838*^9}, {3.4033394689425907`*^9, 
  3.4033394737733364`*^9}, {3.4034590645955963`*^9, 3.403459065147319*^9}, {
  3.4034593099002085`*^9, 3.4034594088128233`*^9}, {3.4034594530062838`*^9, 
  3.4034595138715286`*^9}, {3.4034598973626084`*^9, 3.4034599978327403`*^9}, {
  3.4034603019636655`*^9, 3.4034603667739706`*^9}, {3.4034605051928453`*^9, 
  3.403460533461544*^9}, {3.4034606662645674`*^9, 3.4034606827908535`*^9}, {
  3.403510788011648*^9, 3.4035108104887257`*^9}, {3.403511083995634*^9, 
  3.403511289284252*^9}, {3.403511331032557*^9, 3.403511367971599*^9}, {
  3.4035119249173975`*^9, 3.4035120132437754`*^9}, {3.4035122566510477`*^9, 
  3.4035122749750705`*^9}, {3.4035127511223297`*^9, 3.4035128513991146`*^9}, {
  3.403513482390955*^9, 3.4035135283850813`*^9}, {3.403513665178084*^9, 
  3.4035136711767235`*^9}, {3.403513738601119*^9, 3.4035137423450203`*^9}, {
  3.4035295479994087`*^9, 3.403529729804179*^9}, {3.4035298109874597`*^9, 
  3.403529827081156*^9}, {3.403588761194874*^9, 3.4035887620346637`*^9}, {
  3.40359196144024*^9, 3.4035920228083825`*^9}, {3.403598157799507*^9, 
  3.40359823502015*^9}, {3.403598773472992*^9, 3.4035988653977723`*^9}, {
  3.403598909185009*^9, 3.4035989095687733`*^9}}],

Cell[BoxData[{
 RowBox[{
  RowBox[{
   RowBox[{"Rule1", "[", 
    RowBox[{"A", "+", "B", "+", "C___"}], "]"}], ":=", 
   RowBox[{"X", "+", 
    RowBox[{"Plus", "[", "C", "]"}]}]}], ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{
   RowBox[{"Rule2", "[", 
    RowBox[{"Sin", "[", "M", "]"}], "]"}], ":=", "Y"}], 
  ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"CleanItUp", "=", 
   RowBox[{
    RowBox[{"FullSimplify", "[", 
     RowBox[{"#", ",", 
      RowBox[{"TransformationFunctions", "\[Rule]", 
       RowBox[{"{", 
        RowBox[{"Rule1", ",", "Rule2", ",", "Automatic"}], "}"}]}]}], "]"}], 
    "&"}]}], ";"}]}], "Input",
 CellChangeTimes->{{3.403599381732889*^9, 3.4035994142337384`*^9}, {
  3.4035994470695276`*^9, 3.403599594630396*^9}, {3.403599691859524*^9, 
  3.4035996961151114`*^9}, {3.4035997522687435`*^9, 3.40359977383377*^9}, {
  3.403599808158721*^9, 3.403599813181837*^9}, {3.4035998505544453`*^9, 
  3.403599863760631*^9}, {3.4035999188576903`*^9, 3.4035999396874123`*^9}, {
  3.403599994714164*^9, 3.4036000104963565`*^9}, {3.4036100389189496`*^9, 
  3.403610161441381*^9}}],

Cell[TextData[StyleBox["Triple underscore in the pattern specification \
indicates an optional element or sequence of elements. It is necessary \
because sums may contain more than two terms, and the other terms have to be \
kept as they are.",
 FontSize->16,
 FontSlant->"Italic"]], "Text",
 CellChangeTimes->{{3.403339183467698*^9, 3.4033392844085035`*^9}, {
   3.4033393996452684`*^9, 3.403339411131838*^9}, {3.4033394689425907`*^9, 
   3.4033394737733364`*^9}, {3.4034590645955963`*^9, 3.403459065147319*^9}, {
   3.4034593099002085`*^9, 3.4034594088128233`*^9}, {3.4034594530062838`*^9, 
   3.4034595138715286`*^9}, {3.4034598973626084`*^9, 
   3.4034599978327403`*^9}, {3.4034603019636655`*^9, 
   3.4034603667739706`*^9}, {3.4034605051928453`*^9, 3.403460533461544*^9}, {
   3.4034606662645674`*^9, 3.4034606827908535`*^9}, {3.403510788011648*^9, 
   3.4035108104887257`*^9}, {3.403511083995634*^9, 3.403511289284252*^9}, {
   3.403511331032557*^9, 3.403511367971599*^9}, {3.4035119249173975`*^9, 
   3.4035120132437754`*^9}, {3.4035122566510477`*^9, 
   3.4035122749750705`*^9}, {3.4035127511223297`*^9, 
   3.4035128513991146`*^9}, {3.403513482390955*^9, 3.4035135283850813`*^9}, {
   3.403513665178084*^9, 3.4035136711767235`*^9}, {3.403513738601119*^9, 
   3.4035137423450203`*^9}, {3.4035295479994087`*^9, 3.403529729804179*^9}, {
   3.4035298109874597`*^9, 3.403529827081156*^9}, {3.403588761194874*^9, 
   3.4035887620346637`*^9}, {3.40359196144024*^9, 3.4035920228083825`*^9}, {
   3.403598157799507*^9, 3.40359823502015*^9}, {3.403598773472992*^9, 
   3.4035988653977723`*^9}, {3.403598909185009*^9, 3.4035989095687733`*^9}, {
   3.403610187068647*^9, 3.4036102029953623`*^9}, {3.403610863513633*^9, 
   3.4036109049494576`*^9}, 3.4036109467690463`*^9}],

Cell[BoxData[
 RowBox[{"A", "+", "B", "+", 
  RowBox[{
   RowBox[{"Sin", "[", "M", "]"}], "^", "2"}]}]], "Input"],

Cell[BoxData[
 RowBox[{
  RowBox[{"A", "+", "B", "+", 
   RowBox[{
    RowBox[{"Sin", "[", "M", "]"}], "^", "2"}]}], "//", 
  "CleanItUp"}]], "Input"],

Cell[TextData[StyleBox["We will use these custom simplification patterns to \
reduce the relaxation theory expressions in the second tutorial.",
 FontSize->16,
 FontSlant->"Italic"]], "Text",
 CellChangeTimes->{{3.403339183467698*^9, 3.4033392844085035`*^9}, {
  3.4033393996452684`*^9, 3.403339411131838*^9}, {3.4033394689425907`*^9, 
  3.4033394737733364`*^9}, {3.4034590645955963`*^9, 3.403459065147319*^9}, {
  3.4034593099002085`*^9, 3.4034594088128233`*^9}, {3.4034594530062838`*^9, 
  3.4034595138715286`*^9}, {3.4034598973626084`*^9, 3.4034599978327403`*^9}, {
  3.4034603019636655`*^9, 3.4034603667739706`*^9}, {3.4034605051928453`*^9, 
  3.403460533461544*^9}, {3.4034606662645674`*^9, 3.4034606827908535`*^9}, {
  3.403510788011648*^9, 3.4035108104887257`*^9}, {3.403511083995634*^9, 
  3.403511289284252*^9}, {3.403511331032557*^9, 3.403511367971599*^9}, {
  3.4035119249173975`*^9, 3.4035120132437754`*^9}, {3.4035122566510477`*^9, 
  3.4035122749750705`*^9}, {3.4035127511223297`*^9, 3.4035128513991146`*^9}, {
  3.403513482390955*^9, 3.4035135283850813`*^9}, {3.403513665178084*^9, 
  3.4035136711767235`*^9}, {3.403513738601119*^9, 3.4035137423450203`*^9}, {
  3.4035295479994087`*^9, 3.403529729804179*^9}, {3.4035298109874597`*^9, 
  3.403529827081156*^9}, {3.403588761194874*^9, 3.4035887620346637`*^9}, {
  3.40359196144024*^9, 3.4035920228083825`*^9}, {3.403598157799507*^9, 
  3.40359833702534*^9}, {3.4036109504650993`*^9, 3.4036109785424037`*^9}}]
}, Open  ]],

Cell[CellGroupData[{

Cell["\<\
Getting started 6: Clebsch-Gordan expansions of spherical harmonic products\
\>", "Section",
 CellChangeTimes->{{3.4033384525984616`*^9, 3.4033384766691866`*^9}, {
  3.4033386741995125`*^9, 3.403338677967826*^9}, {3.4033388210729246`*^9, 
  3.403338835799521*^9}, {3.403460216154704*^9, 3.4034602378095684`*^9}, {
  3.403512348662737*^9, 3.4035123487584343`*^9}, {3.403591066947686*^9, 
  3.403591082633206*^9}, {3.403591378091834*^9, 3.4035913788847523`*^9}, {
  3.4035988890993805`*^9, 3.4035988892673388`*^9}, {3.403630093427371*^9, 
  3.4036301020508423`*^9}}],

Cell[TextData[StyleBox["A very time-consuming analytical problem in spin \
dynamics is expanding products of spherical harmonics into a sum of harmonics \
of a higer rank. Doing it by hand can be very slow. We will now program fully \
general matching patterns what will reduce any spherical harmonic product in \
milliseconds. Looking up the formulas in a textbook (Brink & Satchler in this \
case) and typing them in as replacement patterns results in",
 FontSize->16,
 FontSlant->"Italic"]], "Text",
 CellChangeTimes->{{3.403339183467698*^9, 3.4033392844085035`*^9}, {
  3.4033393996452684`*^9, 3.403339411131838*^9}, {3.4033394689425907`*^9, 
  3.4033394737733364`*^9}, {3.4034590645955963`*^9, 3.403459065147319*^9}, {
  3.4034593099002085`*^9, 3.4034594088128233`*^9}, {3.4034594530062838`*^9, 
  3.4034595138715286`*^9}, {3.4034598973626084`*^9, 3.4034599978327403`*^9}, {
  3.4034603019636655`*^9, 3.4034603667739706`*^9}, {3.4034605051928453`*^9, 
  3.403460533461544*^9}, {3.4034606662645674`*^9, 3.4034606827908535`*^9}, {
  3.403510788011648*^9, 3.4035108104887257`*^9}, {3.403511083995634*^9, 
  3.403511289284252*^9}, {3.403511331032557*^9, 3.403511367971599*^9}, {
  3.4035119249173975`*^9, 3.4035120132437754`*^9}, {3.4035122566510477`*^9, 
  3.4035122749750705`*^9}, {3.4035127511223297`*^9, 3.4035128513991146`*^9}, {
  3.403513482390955*^9, 3.4035135283850813`*^9}, {3.403513665178084*^9, 
  3.4035136711767235`*^9}, {3.403513738601119*^9, 3.4035137423450203`*^9}, {
  3.4035295479994087`*^9, 3.403529729804179*^9}, {3.4035298109874597`*^9, 
  3.403529827081156*^9}, {3.4035308264068437`*^9, 3.403531026826633*^9}, {
  3.4035886411673994`*^9, 3.403588684834526*^9}, {3.4035888188611045`*^9, 
  3.4035888221567917`*^9}, {3.403590674188645*^9, 3.403590780904471*^9}, {
  3.403591390314685*^9, 3.403591478896906*^9}, {3.4035916689091587`*^9, 
  3.4035917041695967`*^9}, {3.403591762851388*^9, 3.4035917646510773`*^9}, {
  3.403610999182684*^9, 3.403611030592783*^9}}],

Cell[BoxData[{
 RowBox[{
  RowBox[{"Off", "[", 
   RowBox[{"ClebschGordan", "::", "\"\<phy\>\""}], "]"}], 
  ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"Y", "/:", 
   RowBox[{
    RowBox[{"Y", "[", 
     RowBox[{"l1_", ",", "m1_"}], "]"}], " ", 
    RowBox[{"Y", "[", 
     RowBox[{"l2_", ",", "m2_"}], "]"}]}], ":=", 
   RowBox[{"Sum", "[", 
    RowBox[{
     RowBox[{"(", 
      RowBox[{
       SqrtBox[
        FractionBox[
         RowBox[{
          RowBox[{"(", 
           RowBox[{
            RowBox[{"2", " ", "l1"}], "+", "1"}], ")"}], " ", 
          RowBox[{"(", 
           RowBox[{
            RowBox[{"2", " ", "l2"}], "+", "1"}], ")"}]}], 
         RowBox[{"4", " ", "\[Pi]", " ", 
          RowBox[{"(", 
           RowBox[{
            RowBox[{"2", "L"}], "+", "1"}], ")"}]}]]], 
       RowBox[{"ClebschGordan", "[", 
        RowBox[{
         RowBox[{"{", 
          RowBox[{"l1", ",", "0"}], "}"}], ",", 
         RowBox[{"{", 
          RowBox[{"l2", ",", "0"}], "}"}], ",", 
         RowBox[{"{", 
          RowBox[{"L", ",", "0"}], "}"}]}], "]"}], " ", 
       RowBox[{"ClebschGordan", "[", 
        RowBox[{
         RowBox[{"{", 
          RowBox[{"l1", ",", "m1"}], "}"}], ",", 
         RowBox[{"{", 
          RowBox[{"l2", ",", "m2"}], "}"}], ",", 
         RowBox[{"{", 
          RowBox[{"L", ",", "M"}], "}"}]}], "]"}], " ", 
       RowBox[{"Y", "[", 
        RowBox[{"L", ",", "M"}], "]"}]}], ")"}], ",", 
     RowBox[{"{", 
      RowBox[{"L", ",", 
       RowBox[{"Abs", "[", 
        RowBox[{"l1", "-", "l2"}], "]"}], ",", 
       RowBox[{"l1", "+", "l2"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{"M", ",", 
       RowBox[{"-", "L"}], ",", "L"}], "}"}]}], "]"}]}], 
  ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"Y", "/:", 
   SuperscriptBox[
    RowBox[{"Y", "[", 
     RowBox[{"l_", ",", "m_"}], "]"}], "2"], ":=", 
   RowBox[{"Sum", "[", 
    RowBox[{
     RowBox[{"(", 
      RowBox[{
       SqrtBox[
        FractionBox[
         RowBox[{
          RowBox[{"(", 
           RowBox[{
            RowBox[{"2", " ", "l"}], "+", "1"}], ")"}], " ", 
          RowBox[{"(", 
           RowBox[{
            RowBox[{"2", " ", "l"}], "+", "1"}], ")"}]}], 
         RowBox[{"4", " ", "\[Pi]", " ", 
          RowBox[{"(", 
           RowBox[{
            RowBox[{"2", "L"}], "+", "1"}], ")"}]}]]], 
       RowBox[{"ClebschGordan", "[", 
        RowBox[{
         RowBox[{"{", 
          RowBox[{"l", ",", "0"}], "}"}], ",", 
         RowBox[{"{", 
          RowBox[{"l", ",", "0"}], "}"}], ",", 
         RowBox[{"{", 
          RowBox[{"L", ",", "0"}], "}"}]}], "]"}], " ", 
       RowBox[{"ClebschGordan", "[", 
        RowBox[{
         RowBox[{"{", 
          RowBox[{"l", ",", "m"}], "}"}], ",", 
         RowBox[{"{", 
          RowBox[{"l", ",", "m"}], "}"}], ",", 
         RowBox[{"{", 
          RowBox[{"L", ",", "M"}], "}"}]}], "]"}], " ", 
       RowBox[{"Y", "[", 
        RowBox[{"L", ",", "M"}], "]"}]}], ")"}], ",", 
     RowBox[{"{", 
      RowBox[{"L", ",", "0", ",", 
       RowBox[{"2", "l"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{"M", ",", 
       RowBox[{"-", "L"}], ",", "L"}], "}"}]}], "]"}]}], ";"}]}], "Input",
 CellChangeTimes->{{3.403591487887542*^9, 3.403591502305564*^9}, {
  3.403591551181342*^9, 3.4035916198800697`*^9}}],

Cell[TextData[StyleBox["The first line switches off some warnings, the second \
and the third specify the patterns. Because a square is different, in the \
pattern matching sense, from the product, it has to be specified separately. \
The power of these patterns is quite dramatic:",
 FontSize->16,
 FontSlant->"Italic"]], "Text",
 CellChangeTimes->{{3.403339183467698*^9, 3.4033392844085035`*^9}, {
   3.4033393996452684`*^9, 3.403339411131838*^9}, {3.4033394689425907`*^9, 
   3.4033394737733364`*^9}, {3.4034590645955963`*^9, 3.403459065147319*^9}, {
   3.4034593099002085`*^9, 3.4034594088128233`*^9}, {3.4034594530062838`*^9, 
   3.4034595138715286`*^9}, {3.4034598973626084`*^9, 
   3.4034599978327403`*^9}, {3.4034603019636655`*^9, 
   3.4034603667739706`*^9}, {3.4034605051928453`*^9, 3.403460533461544*^9}, {
   3.4034606662645674`*^9, 3.4034606827908535`*^9}, {3.403510788011648*^9, 
   3.4035108104887257`*^9}, {3.403511083995634*^9, 3.403511289284252*^9}, {
   3.403511331032557*^9, 3.403511367971599*^9}, {3.4035119249173975`*^9, 
   3.4035120132437754`*^9}, {3.4035122566510477`*^9, 
   3.4035122749750705`*^9}, {3.4035127511223297`*^9, 
   3.4035128513991146`*^9}, {3.403513482390955*^9, 3.4035135283850813`*^9}, {
   3.403513665178084*^9, 3.4035136711767235`*^9}, {3.403513738601119*^9, 
   3.4035137423450203`*^9}, {3.4035295479994087`*^9, 3.403529729804179*^9}, {
   3.4035298109874597`*^9, 3.403529827081156*^9}, {3.4035308264068437`*^9, 
   3.403531026826633*^9}, {3.4035886411673994`*^9, 3.403588684834526*^9}, {
   3.4035888188611045`*^9, 3.4035888221567917`*^9}, {3.403590674188645*^9, 
   3.403590780904471*^9}, {3.403591390314685*^9, 3.403591478896906*^9}, {
   3.4035915163896236`*^9, 3.403591520884453*^9}, {3.40359176768311*^9, 
   3.403591822277272*^9}, 3.403592098664856*^9, {3.4035921345004525`*^9, 
   3.4035921766247096`*^9}, {3.40359238295623*^9, 3.403592389145287*^9}, {
   3.403611045086972*^9, 3.403611053942851*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{
   RowBox[{"Y", "[", 
    RowBox[{"2", ",", "1"}], "]"}], " ", 
   RowBox[{"Y", "[", 
    RowBox[{"4", ",", 
     RowBox[{"-", "3"}]}], "]"}], " ", 
   RowBox[{"Y", "[", 
    RowBox[{"6", ",", "5"}], "]"}], " ", 
   RowBox[{"Y", "[", 
    RowBox[{"2", ",", "0"}], "]"}]}], "//", "ExpandAll"}]], "Input",
 CellChangeTimes->{{3.403591523384293*^9, 3.403591635097846*^9}, {
  3.403591826550436*^9, 3.4035918285415196`*^9}, {3.4035921134314885`*^9, 
  3.403592130677455*^9}, {3.40359219111304*^9, 3.40359219415972*^9}}],

Cell[TextData[StyleBox["The ExpandAll directive opens all brackets in the \
intermediate results, which helps the matching. It would be almost impossible \
for an average human not to make any arithmetical mistakes in the expansion \
above. We can even obtain some general cases and automatically prepare them \
for cut-and-paste into Fortran:",
 FontSize->16,
 FontSlant->"Italic"]], "Text",
 CellChangeTimes->{{3.403339183467698*^9, 3.4033392844085035`*^9}, {
   3.4033393996452684`*^9, 3.403339411131838*^9}, {3.4033394689425907`*^9, 
   3.4033394737733364`*^9}, {3.4034590645955963`*^9, 3.403459065147319*^9}, {
   3.4034593099002085`*^9, 3.4034594088128233`*^9}, {3.4034594530062838`*^9, 
   3.4034595138715286`*^9}, {3.4034598973626084`*^9, 
   3.4034599978327403`*^9}, {3.4034603019636655`*^9, 
   3.4034603667739706`*^9}, {3.4034605051928453`*^9, 3.403460533461544*^9}, {
   3.4034606662645674`*^9, 3.4034606827908535`*^9}, {3.403510788011648*^9, 
   3.4035108104887257`*^9}, {3.403511083995634*^9, 3.403511289284252*^9}, {
   3.403511331032557*^9, 3.403511367971599*^9}, {3.4035119249173975`*^9, 
   3.4035120132437754`*^9}, {3.4035122566510477`*^9, 
   3.4035122749750705`*^9}, {3.4035127511223297`*^9, 
   3.4035128513991146`*^9}, {3.403513482390955*^9, 3.4035135283850813`*^9}, {
   3.403513665178084*^9, 3.4035136711767235`*^9}, {3.403513738601119*^9, 
   3.4035137423450203`*^9}, {3.4035295479994087`*^9, 3.403529729804179*^9}, {
   3.4035298109874597`*^9, 3.403529827081156*^9}, {3.4035308264068437`*^9, 
   3.403531026826633*^9}, {3.4035886411673994`*^9, 3.403588684834526*^9}, {
   3.4035888188611045`*^9, 3.4035888221567917`*^9}, {3.403590674188645*^9, 
   3.403590780904471*^9}, {3.403591390314685*^9, 3.403591478896906*^9}, {
   3.4035915163896236`*^9, 3.403591520884453*^9}, {3.40359176768311*^9, 
   3.403591822277272*^9}, 3.403592098664856*^9, {3.4035921345004525`*^9, 
   3.4035921766247096`*^9}, {3.40359224124655*^9, 3.403592280653208*^9}, {
   3.403592391818944*^9, 3.403592400056698*^9}, {3.40359255360937*^9, 
   3.4035926720676084`*^9}}],

Cell[BoxData[{
 RowBox[{
  RowBox[{"$Assumptions", "=", 
   RowBox[{
    RowBox[{"m0", "\[Element]", "Integers"}], "&&", 
    RowBox[{"m1", "\[Element]", "Integers"}]}]}], 
  ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{
   RowBox[{
    RowBox[{"Y", "[", 
     RowBox[{"2", ",", "m0"}], "]"}], " ", 
    RowBox[{"Y", "[", 
     RowBox[{"2", ",", "m1"}], "]"}]}], "//", "Simplify"}], "//", 
  "FortranForm"}]}], "Input",
 CellChangeTimes->{{3.40359265468005*^9, 3.4035926571515713`*^9}}]
}, Open  ]],

Cell[CellGroupData[{

Cell["Exercise 2: scalar products of spherical harmonics", "Section",
 CellChangeTimes->{{3.4033384525984616`*^9, 3.4033384766691866`*^9}, {
   3.4033386741995125`*^9, 3.403338677967826*^9}, {3.4033388210729246`*^9, 
   3.403338835799521*^9}, {3.403460216154704*^9, 3.4034602378095684`*^9}, {
   3.403512348662737*^9, 3.4035123487584343`*^9}, {3.40359100980193*^9, 
   3.403591017272155*^9}, {3.4035910507084913`*^9, 3.4035910531800127`*^9}, {
   3.403592324368183*^9, 3.4035923340950994`*^9}, 3.4036301197997065`*^9}],

Cell[TextData[StyleBox["Being eigenfunctions of the angular momentum \
operator, spherical harmonics obey Hilbert space orthogonality relations:",
 FontSize->16,
 FontSlant->"Italic"]], "Text",
 CellChangeTimes->{{3.403339183467698*^9, 3.4033392844085035`*^9}, {
   3.4033393996452684`*^9, 3.403339411131838*^9}, {3.4033394689425907`*^9, 
   3.4033394737733364`*^9}, {3.4034590645955963`*^9, 3.403459065147319*^9}, {
   3.4034593099002085`*^9, 3.4034594088128233`*^9}, {3.4034594530062838`*^9, 
   3.4034595138715286`*^9}, {3.4034598973626084`*^9, 
   3.4034599978327403`*^9}, {3.4034603019636655`*^9, 
   3.4034603667739706`*^9}, {3.4034605051928453`*^9, 3.403460533461544*^9}, {
   3.4034606662645674`*^9, 3.4034606827908535`*^9}, {3.403510788011648*^9, 
   3.4035108104887257`*^9}, {3.403511083995634*^9, 3.403511289284252*^9}, {
   3.403511331032557*^9, 3.403511367971599*^9}, {3.4035119249173975`*^9, 
   3.4035120132437754`*^9}, {3.4035122566510477`*^9, 
   3.4035122749750705`*^9}, {3.4035127511223297`*^9, 
   3.4035128513991146`*^9}, {3.403513482390955*^9, 3.4035135283850813`*^9}, {
   3.403513665178084*^9, 3.4035136711767235`*^9}, {3.403513738601119*^9, 
   3.4035137423450203`*^9}, {3.4035295479994087`*^9, 3.403529729804179*^9}, {
   3.4035298109874597`*^9, 3.403529827081156*^9}, {3.4035308264068437`*^9, 
   3.403531026826633*^9}, {3.4035886411673994`*^9, 3.403588684834526*^9}, {
   3.4035888188611045`*^9, 3.4035888221567917`*^9}, {3.403590674188645*^9, 
   3.403590780904471*^9}, {3.403591390314685*^9, 3.403591478896906*^9}, {
   3.4035915163896236`*^9, 3.403591520884453*^9}, {3.40359176768311*^9, 
   3.403591822277272*^9}, 3.403592098664856*^9, {3.4035921345004525`*^9, 
   3.4035921766247096`*^9}, {3.40359238295623*^9, 3.403592389145287*^9}, {
   3.4035932411190777`*^9, 3.403593338477104*^9}, {3.4035934583043957`*^9, 
   3.4035934614721613`*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{
   RowBox[{"Y", "[", 
    RowBox[{"L", ",", "M"}], "]"}], ".", 
   RowBox[{"Y", "[", 
    RowBox[{"P", ",", "Q"}], "]"}]}], "=", 
  RowBox[{
   RowBox[{
    SubsuperscriptBox["\[Integral]", "0", 
     RowBox[{"2", " ", "\[Pi]"}]], 
    RowBox[{
     SubsuperscriptBox["\[Integral]", "0", "\[Pi]"], 
     RowBox[{
      RowBox[{"Y", "[", 
       RowBox[{"L", ",", "M"}], "]"}], " ", 
      RowBox[{"Y", "[", 
       RowBox[{"P", ",", 
        RowBox[{"-", "Q"}]}], "]"}], " ", 
      RowBox[{"Sin", "[", "\[Theta]", "]"}], 
      RowBox[{"\[DifferentialD]", "\[Theta]"}], 
      RowBox[{"\[DifferentialD]", "\[CurlyPhi]"}]}]}]}], "=", 
   FractionBox[
    RowBox[{
     SubscriptBox["\[Delta]", 
      RowBox[{"L", ",", "M"}]], " ", 
     SubscriptBox["\[Delta]", 
      RowBox[{"P", ",", "Q"}]]}], 
    RowBox[{
     RowBox[{"2", " ", "L"}], "+", "1"}]]}]}]], "DisplayFormula",
 CellChangeTimes->{{3.4035933468349676`*^9, 3.4035934422194877`*^9}, 
   3.4035935505602093`*^9, {3.40359363004731*^9, 3.40359364209439*^9}, {
   3.4035943457329483`*^9, 3.403594350009042*^9}},
 TextAlignment->Center,
 Magnification->1.25],

Cell[TextData[StyleBox["If taken explicitly, by human or machine, these \
double integrals take a very long time. Modify the definition of the 'Dot' \
operator to include patterns for scalar products of spherical harmonics. Use \
the result to find the coefficient before Y[41,-8] in the Clebsch-Gordan \
expansion of Y[22,5] Y[19,-13]. Hint: Mathematica's function for Kronecker \
symbol is KroneckerDelta[a,b]",
 FontSize->16,
 FontSlant->"Italic"]], "Text",
 CellChangeTimes->{{3.403339183467698*^9, 3.4033392844085035`*^9}, {
   3.4033393996452684`*^9, 3.403339411131838*^9}, {3.4033394689425907`*^9, 
   3.4033394737733364`*^9}, {3.4034590645955963`*^9, 3.403459065147319*^9}, {
   3.4034593099002085`*^9, 3.4034594088128233`*^9}, {3.4034594530062838`*^9, 
   3.4034595138715286`*^9}, {3.4034598973626084`*^9, 
   3.4034599978327403`*^9}, {3.4034603019636655`*^9, 
   3.4034603667739706`*^9}, {3.4034605051928453`*^9, 3.403460533461544*^9}, {
   3.4034606662645674`*^9, 3.4034606827908535`*^9}, {3.403510788011648*^9, 
   3.4035108104887257`*^9}, {3.403511083995634*^9, 3.403511289284252*^9}, {
   3.403511331032557*^9, 3.403511367971599*^9}, {3.4035119249173975`*^9, 
   3.4035120132437754`*^9}, {3.4035122566510477`*^9, 
   3.4035122749750705`*^9}, {3.4035127511223297`*^9, 
   3.4035128513991146`*^9}, {3.403513482390955*^9, 3.4035135283850813`*^9}, {
   3.403513665178084*^9, 3.4035136711767235`*^9}, {3.403513738601119*^9, 
   3.4035137423450203`*^9}, {3.4035295479994087`*^9, 3.403529729804179*^9}, {
   3.4035298109874597`*^9, 3.403529827081156*^9}, {3.4035308264068437`*^9, 
   3.403531026826633*^9}, {3.4035886411673994`*^9, 3.403588684834526*^9}, {
   3.4035888188611045`*^9, 3.4035888221567917`*^9}, {3.403590674188645*^9, 
   3.403590780904471*^9}, {3.403591390314685*^9, 3.403591478896906*^9}, {
   3.4035915163896236`*^9, 3.403591520884453*^9}, {3.40359176768311*^9, 
   3.403591822277272*^9}, 3.403592098664856*^9, {3.4035921345004525`*^9, 
   3.4035921766247096`*^9}, {3.40359238295623*^9, 3.403592389145287*^9}, {
   3.4035932411190777`*^9, 3.403593338477104*^9}, {3.4035934583043957`*^9, 
   3.4035934614721613`*^9}, {3.403593514026415*^9, 3.4035935633884897`*^9}, {
   3.403593766727038*^9, 3.403593872675335*^9}, {3.403593907960186*^9, 
   3.4035939288133435`*^9}, {3.403594070662663*^9, 3.403594078950218*^9}, {
   3.4035941754831023`*^9, 3.4035942204704576`*^9}, {3.4035944708431044`*^9, 
   3.4035944857620716`*^9}, {3.403594518505093*^9, 3.4035945188869047`*^9}}],

Cell[CellGroupData[{

Cell["Answer", "Text",
 CellGroupingRules->{GroupTogetherGrouping, 10000.},
 CellChangeTimes->{{3.403528988628961*^9, 3.403529017208186*^9}, {
   3.403529279737282*^9, 3.403529287722122*^9}, 3.403594678324942*^9, {
   3.40359470868921*^9, 3.403594721624905*^9}}],

Cell[BoxData[{
 RowBox[{
  RowBox[{"Unprotect", "[", "Dot", "]"}], ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"Dot", "/:", 
   RowBox[{"Dot", "[", 
    RowBox[{"A_", ",", 
     RowBox[{"B_", "+", "C_"}]}], "]"}], ":=", 
   RowBox[{
    RowBox[{"A", ".", "B"}], "+", 
    RowBox[{"A", ".", "C"}]}]}], ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"Dot", "/:", 
   RowBox[{"Dot", "[", 
    RowBox[{
     RowBox[{"A_", "+", "B_"}], ",", "C_"}], "]"}], ":=", 
   RowBox[{
    RowBox[{"A", ".", "C"}], "+", 
    RowBox[{"B", ".", "C"}]}]}], ";"}], "\[IndentingNewLine]", 
 RowBox[{"Dot", "/:", 
  RowBox[{"Dot", "[", 
   RowBox[{"A_", ",", 
    RowBox[{"B_", "*", "C_"}]}], "]"}], ":=", 
  RowBox[{
   RowBox[{"B", "*", 
    RowBox[{"A", ".", "C"}]}], "/;", 
   RowBox[{"B", "\[Element]", "Complexes"}]}]}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"Dot", "/:", 
   RowBox[{"Dot", "[", 
    RowBox[{
     RowBox[{"A_", "*", "B_"}], ",", "C_"}], "]"}], ":=", 
   RowBox[{
    RowBox[{"A", "*", 
     RowBox[{"B", ".", "C"}]}], "/;", 
    RowBox[{"A", "\[Element]", "Complexes"}]}]}], 
  "\[IndentingNewLine]"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"Dot", "/:", 
   RowBox[{
    RowBox[{"Y", "[", 
     RowBox[{"L_", ",", "M_"}], "]"}], ".", 
    RowBox[{"Y", "[", 
     RowBox[{"P_", ",", "Q_"}], "]"}]}], ":=", 
   RowBox[{
    RowBox[{"KroneckerDelta", "[", 
     RowBox[{"L", ",", "P"}], "]"}], " ", 
    RowBox[{
     RowBox[{"KroneckerDelta", "[", 
      RowBox[{"M", ",", "Q"}], "]"}], "/", 
     RowBox[{"(", 
      RowBox[{
       RowBox[{"2", " ", "L"}], "+", "1"}], ")"}]}]}]}], ";"}]}], "Input",
 CellGroupingRules->{GroupTogetherGrouping, 10000.},
 CellChangeTimes->{{3.4035947327218513`*^9, 3.403594753841593*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{
   RowBox[{"Y", "[", 
    RowBox[{"41", ",", 
     RowBox[{"-", "8"}]}], "]"}], ".", 
   RowBox[{"(", 
    RowBox[{
     RowBox[{"Y", "[", 
      RowBox[{"22", ",", "5"}], "]"}], " ", 
     RowBox[{"Y", "[", 
      RowBox[{"19", ",", 
       RowBox[{"-", "13"}]}], "]"}]}], ")"}]}], "/", 
  RowBox[{"(", 
   RowBox[{
    RowBox[{"Y", "[", 
     RowBox[{"41", ",", 
      RowBox[{"-", "8"}]}], "]"}], ".", 
    RowBox[{"Y", "[", 
     RowBox[{"41", ",", 
      RowBox[{"-", "8"}]}], "]"}]}], ")"}]}]], "Input",
 CellGroupingRules->{GroupTogetherGrouping, 10000.},
 CellChangeTimes->{{3.403594741706628*^9, 3.403594757749546*^9}}]
}, Closed]]
}, Open  ]],

Cell[CellGroupData[{

Cell["\<\
Getting started 7: using patterns in large-scale analytical transformations\
\>", "Section",
 CellChangeTimes->{{3.4033384525984616`*^9, 3.4033384766691866`*^9}, {
   3.4033386741995125`*^9, 3.403338677967826*^9}, {3.4033388210729246`*^9, 
   3.403338835799521*^9}, {3.403460216154704*^9, 3.4034602378095684`*^9}, {
   3.403512348662737*^9, 3.4035123487584343`*^9}, {3.403591066947686*^9, 
   3.403591082633206*^9}, {3.403591378091834*^9, 3.4035913788847523`*^9}, {
   3.403597752338154*^9, 3.403597773447154*^9}, 3.4036111003461313`*^9, {
   3.4036301464386263`*^9, 3.403630160803918*^9}}],

Cell[TextData[StyleBox["In many cases the equations are simply too big to be \
displayed. However, some terms may be known to be zero, others might need to \
be extracted, some terms may have nice Taylor expansions, or may be replaced \
with an asymptotic expansion, or be secular, etc. In other words, a \
complicated transformation has to be applied to something we cannot even \
print (in extreme cases, these may be gigabytes of analytical formulae). This \
can be done in a variety of ways, as demonstrated inthe examples below.",
 FontSize->16,
 FontSlant->"Italic"]], "Text",
 CellChangeTimes->{{3.403339183467698*^9, 3.4033392844085035`*^9}, {
   3.4033393996452684`*^9, 3.403339411131838*^9}, {3.4033394689425907`*^9, 
   3.4033394737733364`*^9}, {3.4034590645955963`*^9, 3.403459065147319*^9}, {
   3.4034593099002085`*^9, 3.4034594088128233`*^9}, {3.4034594530062838`*^9, 
   3.4034595138715286`*^9}, {3.4034598973626084`*^9, 
   3.4034599978327403`*^9}, {3.4034603019636655`*^9, 
   3.4034603667739706`*^9}, {3.4034605051928453`*^9, 3.403460533461544*^9}, {
   3.4034606662645674`*^9, 3.4034606827908535`*^9}, {3.403510788011648*^9, 
   3.4035108104887257`*^9}, {3.403511083995634*^9, 3.403511289284252*^9}, {
   3.403511331032557*^9, 3.403511367971599*^9}, {3.4035119249173975`*^9, 
   3.4035120132437754`*^9}, {3.4035122566510477`*^9, 
   3.4035122749750705`*^9}, {3.4035127511223297`*^9, 
   3.4035128513991146`*^9}, {3.403513482390955*^9, 3.4035135283850813`*^9}, {
   3.403513665178084*^9, 3.4035136711767235`*^9}, {3.403513738601119*^9, 
   3.4035137423450203`*^9}, {3.4035295479994087`*^9, 3.403529729804179*^9}, {
   3.4035298109874597`*^9, 3.403529827081156*^9}, {3.4035308264068437`*^9, 
   3.403531026826633*^9}, {3.4035886411673994`*^9, 3.403588684834526*^9}, {
   3.4035888188611045`*^9, 3.4035888221567917`*^9}, {3.403590674188645*^9, 
   3.403590780904471*^9}, {3.403591390314685*^9, 3.403591478896906*^9}, {
   3.4035915163896236`*^9, 3.403591520884453*^9}, {3.40359176768311*^9, 
   3.403591822277272*^9}, 3.403592098664856*^9, {3.4035921345004525`*^9, 
   3.4035921766247096`*^9}, {3.40359238295623*^9, 3.403592389145287*^9}, {
   3.4035932411190777`*^9, 3.403593338477104*^9}, {3.4035934583043957`*^9, 
   3.4035934614721613`*^9}, {3.403614175215733*^9, 3.40361418230317*^9}, 
   3.40362516902185*^9, {3.403625467481263*^9, 3.4036255024301977`*^9}, {
   3.4036296364917727`*^9, 3.403629799118083*^9}, {3.4041343978302*^9, 
   3.4041344352702*^9}, {3.4041366800165997`*^9, 3.4041366807497997`*^9}}],

Cell[TextData[StyleBox["Find out positions at which x to any power appears:",
 FontSize->16,
 FontSlant->"Italic"]], "Text",
 CellChangeTimes->{{3.403339183467698*^9, 3.4033392844085035`*^9}, {
   3.4033393996452684`*^9, 3.403339411131838*^9}, {3.4033394689425907`*^9, 
   3.4033394737733364`*^9}, {3.4034590645955963`*^9, 3.403459065147319*^9}, {
   3.4034593099002085`*^9, 3.4034594088128233`*^9}, {3.4034594530062838`*^9, 
   3.4034595138715286`*^9}, {3.4034598973626084`*^9, 
   3.4034599978327403`*^9}, {3.4034603019636655`*^9, 
   3.4034603667739706`*^9}, {3.4034605051928453`*^9, 3.403460533461544*^9}, {
   3.4034606662645674`*^9, 3.4034606827908535`*^9}, {3.403510788011648*^9, 
   3.4035108104887257`*^9}, {3.403511083995634*^9, 3.403511289284252*^9}, {
   3.403511331032557*^9, 3.403511367971599*^9}, {3.4035119249173975`*^9, 
   3.4035120132437754`*^9}, {3.4035122566510477`*^9, 
   3.4035122749750705`*^9}, {3.4035127511223297`*^9, 
   3.4035128513991146`*^9}, {3.403513482390955*^9, 3.4035135283850813`*^9}, {
   3.403513665178084*^9, 3.4035136711767235`*^9}, {3.403513738601119*^9, 
   3.4035137423450203`*^9}, {3.4035295479994087`*^9, 3.403529729804179*^9}, {
   3.4035298109874597`*^9, 3.403529827081156*^9}, {3.4035308264068437`*^9, 
   3.403531026826633*^9}, {3.4035886411673994`*^9, 3.403588684834526*^9}, {
   3.4035888188611045`*^9, 3.4035888221567917`*^9}, {3.403590674188645*^9, 
   3.403590780904471*^9}, {3.403591390314685*^9, 3.403591478896906*^9}, {
   3.4035915163896236`*^9, 3.403591520884453*^9}, {3.40359176768311*^9, 
   3.403591822277272*^9}, 3.403592098664856*^9, {3.4035921345004525`*^9, 
   3.4035921766247096`*^9}, {3.40359238295623*^9, 3.403592389145287*^9}, {
   3.4035932411190777`*^9, 3.403593338477104*^9}, {3.4035934583043957`*^9, 
   3.4035934614721613`*^9}, {3.403614158914042*^9, 3.4036141614812603`*^9}, {
   3.4041343352586*^9, 3.4041343556478*^9}, {3.4041345110394*^9, 
   3.4041345312726*^9}}],

Cell[BoxData[
 RowBox[{"Position", "[", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{
     RowBox[{"1", "+", 
      RowBox[{"x", "^", "2"}]}], ",", "5", ",", 
     RowBox[{"x", "^", "4"}], ",", 
     RowBox[{"a", "+", 
      RowBox[{
       RowBox[{"(", 
        RowBox[{"1", "+", 
         RowBox[{"x", "^", "2"}]}], ")"}], "^", "2"}]}]}], "}"}], ",", 
   RowBox[{"x", "^", "_"}]}], "]"}]], "Input",
 CellChangeTimes->{{3.403625114665954*^9, 3.403625149114921*^9}, {
   3.403625307095044*^9, 3.403625315971429*^9}, {3.4036254515770073`*^9, 
   3.4036254578539495`*^9}, {3.4041341620362*^9, 3.4041341663106003`*^9}, {
   3.4041342017382*^9, 3.4041342118626003`*^9}, {3.404134302561*^9, 
   3.4041343231373997`*^9}, 3.4041344914458*^9, {3.4041346707366*^9, 
   3.4041346888482*^9}}],

Cell[TextData[StyleBox["Return x from all cases of f[x] found:",
 FontSize->16,
 FontSlant->"Italic"]], "Text",
 CellChangeTimes->{{3.403339183467698*^9, 3.4033392844085035`*^9}, {
   3.4033393996452684`*^9, 3.403339411131838*^9}, {3.4033394689425907`*^9, 
   3.4033394737733364`*^9}, {3.4034590645955963`*^9, 3.403459065147319*^9}, {
   3.4034593099002085`*^9, 3.4034594088128233`*^9}, {3.4034594530062838`*^9, 
   3.4034595138715286`*^9}, {3.4034598973626084`*^9, 
   3.4034599978327403`*^9}, {3.4034603019636655`*^9, 
   3.4034603667739706`*^9}, {3.4034605051928453`*^9, 3.403460533461544*^9}, {
   3.4034606662645674`*^9, 3.4034606827908535`*^9}, {3.403510788011648*^9, 
   3.4035108104887257`*^9}, {3.403511083995634*^9, 3.403511289284252*^9}, {
   3.403511331032557*^9, 3.403511367971599*^9}, {3.4035119249173975`*^9, 
   3.4035120132437754`*^9}, {3.4035122566510477`*^9, 
   3.4035122749750705`*^9}, {3.4035127511223297`*^9, 
   3.4035128513991146`*^9}, {3.403513482390955*^9, 3.4035135283850813`*^9}, {
   3.403513665178084*^9, 3.4035136711767235`*^9}, {3.403513738601119*^9, 
   3.4035137423450203`*^9}, {3.4035295479994087`*^9, 3.403529729804179*^9}, {
   3.4035298109874597`*^9, 3.403529827081156*^9}, {3.4035308264068437`*^9, 
   3.403531026826633*^9}, {3.4035886411673994`*^9, 3.403588684834526*^9}, {
   3.4035888188611045`*^9, 3.4035888221567917`*^9}, {3.403590674188645*^9, 
   3.403590780904471*^9}, {3.403591390314685*^9, 3.403591478896906*^9}, {
   3.4035915163896236`*^9, 3.403591520884453*^9}, {3.40359176768311*^9, 
   3.403591822277272*^9}, 3.403592098664856*^9, {3.4035921345004525`*^9, 
   3.4035921766247096`*^9}, {3.40359238295623*^9, 3.403592389145287*^9}, {
   3.4035932411190777`*^9, 3.403593338477104*^9}, {3.4035934583043957`*^9, 
   3.4035934614721613`*^9}, {3.403614158914042*^9, 3.4036141614812603`*^9}, {
   3.4041343352586*^9, 3.4041343556478*^9}, {3.4041345110394*^9, 
   3.4041345312726*^9}, {3.4041346957434*^9, 3.4041347109533997`*^9}}],

Cell[BoxData[
 RowBox[{"Cases", "[", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{"1", ",", "1", ",", 
     RowBox[{"f", "[", "a", "]"}], ",", "2", ",", "3", ",", "y", ",", 
     RowBox[{"f", "[", "8", "]"}], ",", "9", ",", 
     RowBox[{"f", "[", "10", "]"}]}], "}"}], ",", 
   RowBox[{
    RowBox[{"f", "[", "x_", "]"}], "\[Rule]", "x"}]}], "]"}]], "Input",
 CellChangeTimes->{{3.403625114665954*^9, 3.403625149114921*^9}, {
   3.403625307095044*^9, 3.403625315971429*^9}, {3.4036254515770073`*^9, 
   3.4036254578539495`*^9}, {3.4041341620362*^9, 3.4041341663106003`*^9}, {
   3.4041342017382*^9, 3.4041342118626003`*^9}, {3.404134302561*^9, 
   3.4041343231373997`*^9}, 3.4041344914458*^9, {3.4041346707366*^9, 
   3.4041346888482*^9}, 3.4041347367869997`*^9}],

Cell[TextData[StyleBox["Count the number of elements in a sum that depend on \
z ('infinity' means to look at all levels of the expression):",
 FontSize->16,
 FontSlant->"Italic"]], "Text",
 CellChangeTimes->{{3.403339183467698*^9, 3.4033392844085035`*^9}, {
   3.4033393996452684`*^9, 3.403339411131838*^9}, {3.4033394689425907`*^9, 
   3.4033394737733364`*^9}, {3.4034590645955963`*^9, 3.403459065147319*^9}, {
   3.4034593099002085`*^9, 3.4034594088128233`*^9}, {3.4034594530062838`*^9, 
   3.4034595138715286`*^9}, {3.4034598973626084`*^9, 
   3.4034599978327403`*^9}, {3.4034603019636655`*^9, 
   3.4034603667739706`*^9}, {3.4034605051928453`*^9, 3.403460533461544*^9}, {
   3.4034606662645674`*^9, 3.4034606827908535`*^9}, {3.403510788011648*^9, 
   3.4035108104887257`*^9}, {3.403511083995634*^9, 3.403511289284252*^9}, {
   3.403511331032557*^9, 3.403511367971599*^9}, {3.4035119249173975`*^9, 
   3.4035120132437754`*^9}, {3.4035122566510477`*^9, 
   3.4035122749750705`*^9}, {3.4035127511223297`*^9, 
   3.4035128513991146`*^9}, {3.403513482390955*^9, 3.4035135283850813`*^9}, {
   3.403513665178084*^9, 3.4035136711767235`*^9}, {3.403513738601119*^9, 
   3.4035137423450203`*^9}, {3.4035295479994087`*^9, 3.403529729804179*^9}, {
   3.4035298109874597`*^9, 3.403529827081156*^9}, {3.4035308264068437`*^9, 
   3.403531026826633*^9}, {3.4035886411673994`*^9, 3.403588684834526*^9}, {
   3.4035888188611045`*^9, 3.4035888221567917`*^9}, {3.403590674188645*^9, 
   3.403590780904471*^9}, {3.403591390314685*^9, 3.403591478896906*^9}, {
   3.4035915163896236`*^9, 3.403591520884453*^9}, {3.40359176768311*^9, 
   3.403591822277272*^9}, 3.403592098664856*^9, {3.4035921345004525`*^9, 
   3.4035921766247096`*^9}, {3.40359238295623*^9, 3.403592389145287*^9}, {
   3.4035932411190777`*^9, 3.403593338477104*^9}, {3.4035934583043957`*^9, 
   3.4035934614721613`*^9}, {3.403614158914042*^9, 3.4036141614812603`*^9}, {
   3.4041343352586*^9, 3.4041343556478*^9}, {3.4041345110394*^9, 
   3.4041345312726*^9}, {3.4041346957434*^9, 3.4041347109533997`*^9}, {
   3.404134841697*^9, 3.4041348807438*^9}, {3.4041350770698*^9, 
   3.404135114541*^9}}],

Cell[BoxData[
 RowBox[{"Count", "[", 
  RowBox[{
   RowBox[{
    RowBox[{"2", " ", "a"}], "+", "b", "+", "c", "+", "q", "+", "z", "+", 
    RowBox[{"Cos", "[", "z", "]"}], "+", 
    RowBox[{"f", "[", "a", "]"}], "+", 
    RowBox[{"Sin", "[", "b", "]"}]}], ",", "z", ",", "Infinity"}], 
  "]"}]], "Input",
 CellChangeTimes->{{3.403625114665954*^9, 3.403625149114921*^9}, {
   3.403625307095044*^9, 3.403625315971429*^9}, {3.4036254515770073`*^9, 
   3.4036254578539495`*^9}, {3.4041341620362*^9, 3.4041341663106003`*^9}, {
   3.4041342017382*^9, 3.4041342118626003`*^9}, {3.404134302561*^9, 
   3.4041343231373997`*^9}, 3.4041344914458*^9, {3.4041346707366*^9, 
   3.4041346888482*^9}, 3.4041347367869997`*^9, {3.4041348180318003`*^9, 
   3.4041348212609997`*^9}, {3.4041348854393997`*^9, 
   3.4041350657130003`*^9}, {3.4041351678462*^9, 3.4041351712158003`*^9}}],

Cell[TextData[StyleBox["Pick out numeric quantities from a product:",
 FontSize->16,
 FontSlant->"Italic"]], "Text",
 CellChangeTimes->{{3.403339183467698*^9, 3.4033392844085035`*^9}, {
   3.4033393996452684`*^9, 3.403339411131838*^9}, {3.4033394689425907`*^9, 
   3.4033394737733364`*^9}, {3.4034590645955963`*^9, 3.403459065147319*^9}, {
   3.4034593099002085`*^9, 3.4034594088128233`*^9}, {3.4034594530062838`*^9, 
   3.4034595138715286`*^9}, {3.4034598973626084`*^9, 
   3.4034599978327403`*^9}, {3.4034603019636655`*^9, 
   3.4034603667739706`*^9}, {3.4034605051928453`*^9, 3.403460533461544*^9}, {
   3.4034606662645674`*^9, 3.4034606827908535`*^9}, {3.403510788011648*^9, 
   3.4035108104887257`*^9}, {3.403511083995634*^9, 3.403511289284252*^9}, {
   3.403511331032557*^9, 3.403511367971599*^9}, {3.4035119249173975`*^9, 
   3.4035120132437754`*^9}, {3.4035122566510477`*^9, 
   3.4035122749750705`*^9}, {3.4035127511223297`*^9, 
   3.4035128513991146`*^9}, {3.403513482390955*^9, 3.4035135283850813`*^9}, {
   3.403513665178084*^9, 3.4035136711767235`*^9}, {3.403513738601119*^9, 
   3.4035137423450203`*^9}, {3.4035295479994087`*^9, 3.403529729804179*^9}, {
   3.4035298109874597`*^9, 3.403529827081156*^9}, {3.4035308264068437`*^9, 
   3.403531026826633*^9}, {3.4035886411673994`*^9, 3.403588684834526*^9}, {
   3.4035888188611045`*^9, 3.4035888221567917`*^9}, {3.403590674188645*^9, 
   3.403590780904471*^9}, {3.403591390314685*^9, 3.403591478896906*^9}, {
   3.4035915163896236`*^9, 3.403591520884453*^9}, {3.40359176768311*^9, 
   3.403591822277272*^9}, 3.403592098664856*^9, {3.4035921345004525`*^9, 
   3.4035921766247096`*^9}, {3.40359238295623*^9, 3.403592389145287*^9}, {
   3.4035932411190777`*^9, 3.403593338477104*^9}, {3.4035934583043957`*^9, 
   3.4035934614721613`*^9}, {3.403614158914042*^9, 3.4036141614812603`*^9}, {
   3.4041343352586*^9, 3.4041343556478*^9}}],

Cell[BoxData[
 RowBox[{"Select", "[", 
  RowBox[{
   RowBox[{"7", " ", 
    SuperscriptBox["\[Pi]", "2"], " ", 
    SuperscriptBox["x", "2"], " ", 
    SuperscriptBox["y", "2"]}], ",", "NumericQ"}], "]"}]], "Input",
 CellChangeTimes->{{3.403625114665954*^9, 3.403625149114921*^9}, {
   3.403625307095044*^9, 3.403625315971429*^9}, {3.4036254515770073`*^9, 
   3.4036254578539495`*^9}, {3.4041341620362*^9, 3.4041341663106003`*^9}, {
   3.4041342017382*^9, 3.4041342118626003`*^9}, {3.404134302561*^9, 
   3.4041343231373997`*^9}, 3.4041352214477997`*^9}],

Cell[TextData[StyleBox["Delete occurrences of a certain patern:",
 FontSize->16,
 FontSlant->"Italic"]], "Text",
 CellChangeTimes->{{3.403339183467698*^9, 3.4033392844085035`*^9}, {
   3.4033393996452684`*^9, 3.403339411131838*^9}, {3.4033394689425907`*^9, 
   3.4033394737733364`*^9}, {3.4034590645955963`*^9, 3.403459065147319*^9}, {
   3.4034593099002085`*^9, 3.4034594088128233`*^9}, {3.4034594530062838`*^9, 
   3.4034595138715286`*^9}, {3.4034598973626084`*^9, 
   3.4034599978327403`*^9}, {3.4034603019636655`*^9, 
   3.4034603667739706`*^9}, {3.4034605051928453`*^9, 3.403460533461544*^9}, {
   3.4034606662645674`*^9, 3.4034606827908535`*^9}, {3.403510788011648*^9, 
   3.4035108104887257`*^9}, {3.403511083995634*^9, 3.403511289284252*^9}, {
   3.403511331032557*^9, 3.403511367971599*^9}, {3.4035119249173975`*^9, 
   3.4035120132437754`*^9}, {3.4035122566510477`*^9, 
   3.4035122749750705`*^9}, {3.4035127511223297`*^9, 
   3.4035128513991146`*^9}, {3.403513482390955*^9, 3.4035135283850813`*^9}, {
   3.403513665178084*^9, 3.4035136711767235`*^9}, {3.403513738601119*^9, 
   3.4035137423450203`*^9}, {3.4035295479994087`*^9, 3.403529729804179*^9}, {
   3.4035298109874597`*^9, 3.403529827081156*^9}, {3.4035308264068437`*^9, 
   3.403531026826633*^9}, {3.4035886411673994`*^9, 3.403588684834526*^9}, {
   3.4035888188611045`*^9, 3.4035888221567917`*^9}, {3.403590674188645*^9, 
   3.403590780904471*^9}, {3.403591390314685*^9, 3.403591478896906*^9}, {
   3.4035915163896236`*^9, 3.403591520884453*^9}, {3.40359176768311*^9, 
   3.403591822277272*^9}, 3.403592098664856*^9, {3.4035921345004525`*^9, 
   3.4035921766247096`*^9}, {3.40359238295623*^9, 3.403592389145287*^9}, {
   3.4035932411190777`*^9, 3.403593338477104*^9}, {3.4035934583043957`*^9, 
   3.4035934614721613`*^9}, {3.403614158914042*^9, 3.4036141614812603`*^9}, {
   3.4041343352586*^9, 3.4041343556478*^9}, {3.4041352276566*^9, 
   3.4041352419462*^9}}],

Cell[BoxData[
 RowBox[{"DeleteCases", "[", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{
     RowBox[{"f", "[", "a", "]"}], ",", 
     RowBox[{"g", "[", "b", "]"}], ",", 
     RowBox[{"f", "[", "c", "]"}]}], "}"}], ",", 
   RowBox[{"f", "[", "_", "]"}]}], "]"}]], "Input",
 CellChangeTimes->{{3.403625114665954*^9, 3.403625149114921*^9}, {
   3.403625307095044*^9, 3.403625315971429*^9}, {3.4036254515770073`*^9, 
   3.4036254578539495`*^9}, {3.4041341620362*^9, 3.4041341663106003`*^9}, {
   3.4041342017382*^9, 3.4041342118626003`*^9}, {3.404134302561*^9, 
   3.4041343231373997`*^9}, 3.4041352214477997`*^9, {3.4041353005554*^9, 
   3.4041353171694*^9}}],

Cell[TextData[StyleBox["Find all elements flanked by the same element:",
 FontSize->16,
 FontSlant->"Italic"]], "Text",
 CellChangeTimes->{{3.403339183467698*^9, 3.4033392844085035`*^9}, {
   3.4033393996452684`*^9, 3.403339411131838*^9}, {3.4033394689425907`*^9, 
   3.4033394737733364`*^9}, {3.4034590645955963`*^9, 3.403459065147319*^9}, {
   3.4034593099002085`*^9, 3.4034594088128233`*^9}, {3.4034594530062838`*^9, 
   3.4034595138715286`*^9}, {3.4034598973626084`*^9, 
   3.4034599978327403`*^9}, {3.4034603019636655`*^9, 
   3.4034603667739706`*^9}, {3.4034605051928453`*^9, 3.403460533461544*^9}, {
   3.4034606662645674`*^9, 3.4034606827908535`*^9}, {3.403510788011648*^9, 
   3.4035108104887257`*^9}, {3.403511083995634*^9, 3.403511289284252*^9}, {
   3.403511331032557*^9, 3.403511367971599*^9}, {3.4035119249173975`*^9, 
   3.4035120132437754`*^9}, {3.4035122566510477`*^9, 
   3.4035122749750705`*^9}, {3.4035127511223297`*^9, 
   3.4035128513991146`*^9}, {3.403513482390955*^9, 3.4035135283850813`*^9}, {
   3.403513665178084*^9, 3.4035136711767235`*^9}, {3.403513738601119*^9, 
   3.4035137423450203`*^9}, {3.4035295479994087`*^9, 3.403529729804179*^9}, {
   3.4035298109874597`*^9, 3.403529827081156*^9}, {3.4035308264068437`*^9, 
   3.403531026826633*^9}, {3.4035886411673994`*^9, 3.403588684834526*^9}, {
   3.4035888188611045`*^9, 3.4035888221567917`*^9}, {3.403590674188645*^9, 
   3.403590780904471*^9}, {3.403591390314685*^9, 3.403591478896906*^9}, {
   3.4035915163896236`*^9, 3.403591520884453*^9}, {3.40359176768311*^9, 
   3.403591822277272*^9}, 3.403592098664856*^9, {3.4035921345004525`*^9, 
   3.4035921766247096`*^9}, {3.40359238295623*^9, 3.403592389145287*^9}, {
   3.4035932411190777`*^9, 3.403593338477104*^9}, {3.4035934583043957`*^9, 
   3.4035934614721613`*^9}, {3.403614158914042*^9, 3.4036141614812603`*^9}, {
   3.4041343352586*^9, 3.4041343556478*^9}, {3.4041352276566*^9, 
   3.4041352419462*^9}, {3.4041353225514*^9, 3.404135333019*^9}}],

Cell[BoxData[
 RowBox[{"ReplaceList", "[", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{"a", ",", "b", ",", "c", ",", "a", ",", "d", ",", "b", ",", "d"}],
     "}"}], ",", 
   RowBox[{
    RowBox[{"{", 
     RowBox[{"___", ",", "x_", ",", "y__", ",", "x_", ",", "___"}], "}"}], 
    "\[Rule]", 
    RowBox[{"{", 
     RowBox[{"x", ",", 
      RowBox[{"{", "y", "}"}]}], "}"}]}]}], "]"}]], "Input",
 CellChangeTimes->{{3.403625114665954*^9, 3.403625149114921*^9}, {
   3.403625307095044*^9, 3.403625315971429*^9}, {3.4036254515770073`*^9, 
   3.4036254578539495`*^9}, {3.4041341620362*^9, 3.4041341663106003`*^9}, {
   3.4041342017382*^9, 3.4041342118626003`*^9}, {3.404134302561*^9, 
   3.4041343231373997`*^9}, 3.4041352214477997`*^9, {3.4041353005554*^9, 
   3.4041353171694*^9}}]
}, Open  ]],

Cell[CellGroupData[{

Cell["Large-scale transformations: weeding a lengthy Taylor series", "Section",
 CellChangeTimes->{{3.4033384525984616`*^9, 3.4033384766691866`*^9}, {
   3.4033386741995125`*^9, 3.403338677967826*^9}, {3.4033388210729246`*^9, 
   3.403338835799521*^9}, {3.403460216154704*^9, 3.4034602378095684`*^9}, {
   3.403512348662737*^9, 3.4035123487584343`*^9}, {3.403591066947686*^9, 
   3.403591082633206*^9}, {3.403591378091834*^9, 3.4035913788847523`*^9}, {
   3.403597752338154*^9, 3.403597773447154*^9}, 3.4036111003461313`*^9, {
   3.4036301464386263`*^9, 3.403630160803918*^9}, {3.4041354113934*^9, 
   3.4041354939486*^9}, {3.4041365644674*^9, 3.4041365757306004`*^9}}],

Cell[TextData[StyleBox["In this example we will extract all tems with total \
power of 22 from a three-dimensional Taylor series that takes up several \
screens",
 FontSize->16,
 FontSlant->"Italic"]], "Text",
 CellChangeTimes->{{3.403339183467698*^9, 3.4033392844085035`*^9}, {
   3.4033393996452684`*^9, 3.403339411131838*^9}, {3.4033394689425907`*^9, 
   3.4033394737733364`*^9}, {3.4034590645955963`*^9, 3.403459065147319*^9}, {
   3.4034593099002085`*^9, 3.4034594088128233`*^9}, {3.4034594530062838`*^9, 
   3.4034595138715286`*^9}, {3.4034598973626084`*^9, 
   3.4034599978327403`*^9}, {3.4034603019636655`*^9, 
   3.4034603667739706`*^9}, {3.4034605051928453`*^9, 3.403460533461544*^9}, {
   3.4034606662645674`*^9, 3.4034606827908535`*^9}, {3.403510788011648*^9, 
   3.4035108104887257`*^9}, {3.403511083995634*^9, 3.403511289284252*^9}, {
   3.403511331032557*^9, 3.403511367971599*^9}, {3.4035119249173975`*^9, 
   3.4035120132437754`*^9}, {3.4035122566510477`*^9, 
   3.4035122749750705`*^9}, {3.4035127511223297`*^9, 
   3.4035128513991146`*^9}, {3.403513482390955*^9, 3.4035135283850813`*^9}, {
   3.403513665178084*^9, 3.4035136711767235`*^9}, {3.403513738601119*^9, 
   3.4035137423450203`*^9}, {3.4035295479994087`*^9, 3.403529729804179*^9}, {
   3.4035298109874597`*^9, 3.403529827081156*^9}, {3.4035308264068437`*^9, 
   3.403531026826633*^9}, {3.4035886411673994`*^9, 3.403588684834526*^9}, {
   3.4035888188611045`*^9, 3.4035888221567917`*^9}, {3.403590674188645*^9, 
   3.403590780904471*^9}, {3.403591390314685*^9, 3.403591478896906*^9}, {
   3.4035915163896236`*^9, 3.403591520884453*^9}, {3.40359176768311*^9, 
   3.403591822277272*^9}, 3.403592098664856*^9, {3.4035921345004525`*^9, 
   3.4035921766247096`*^9}, {3.40359238295623*^9, 3.403592389145287*^9}, {
   3.4035932411190777`*^9, 3.403593338477104*^9}, {3.4035934583043957`*^9, 
   3.4035934614721613`*^9}, {3.403614158914042*^9, 3.4036141614812603`*^9}, {
   3.4041343352586*^9, 3.4041343556478*^9}, {3.4041352276566*^9, 
   3.4041352419462*^9}, {3.4041353225514*^9, 3.404135333019*^9}, {
   3.4041355245713997`*^9, 3.4041355700766*^9}, {3.4041357546558*^9, 
   3.404135792127*^9}, {3.4041363435246*^9, 3.404136343743*^9}, {
   3.4041363837882*^9, 3.4041364170006*^9}}],

Cell[BoxData[{
 RowBox[{"A", "=", 
  RowBox[{
   RowBox[{
    RowBox[{"Series", "[", 
     RowBox[{
      RowBox[{"Sin", "[", 
       RowBox[{
        RowBox[{"x", " ", "y", " ", "z"}], "+", 
        RowBox[{"3", "x"}], "-", 
        SuperscriptBox["z", "2"]}], "]"}], ",", 
      RowBox[{"{", 
       RowBox[{"x", ",", "0", ",", "23"}], "}"}], ",", 
      RowBox[{"{", 
       RowBox[{"y", ",", "0", ",", "23"}], "}"}], ",", 
      RowBox[{"{", 
       RowBox[{"z", ",", "0", ",", "23"}], "}"}]}], "]"}], "//", "Normal"}], "//",
    "ExpandAll"}]}], "\[IndentingNewLine]", 
 RowBox[{"Cases", "[", 
  RowBox[{"A", ",", 
   RowBox[{
    RowBox[{"B_", " ", 
     RowBox[{"x", "^", "n_"}], " ", 
     RowBox[{"y", "^", "k_"}], " ", 
     RowBox[{"z", "^", "m_"}]}], "/;", 
    RowBox[{
     RowBox[{"n", "+", "k", "+", "m"}], "\[Equal]", "22"}]}]}], 
  "]"}]}], "Input",
 CellChangeTimes->{{3.4041355744602003`*^9, 3.4041357159365997`*^9}, {
  3.4041358069782*^9, 3.4041358777398*^9}, {3.4041360141462*^9, 
  3.4041361624086*^9}, {3.4041362083506002`*^9, 3.4041363621198*^9}}],

Cell[TextData[StyleBox["Question: which terms were not matched by the pattern \
above? Which additional patterns are necessary?",
 FontSize->16,
 FontSlant->"Italic"]], "Text",
 CellChangeTimes->{{3.403339183467698*^9, 3.4033392844085035`*^9}, {
   3.4033393996452684`*^9, 3.403339411131838*^9}, {3.4033394689425907`*^9, 
   3.4033394737733364`*^9}, {3.4034590645955963`*^9, 3.403459065147319*^9}, {
   3.4034593099002085`*^9, 3.4034594088128233`*^9}, {3.4034594530062838`*^9, 
   3.4034595138715286`*^9}, {3.4034598973626084`*^9, 
   3.4034599978327403`*^9}, {3.4034603019636655`*^9, 
   3.4034603667739706`*^9}, {3.4034605051928453`*^9, 3.403460533461544*^9}, {
   3.4034606662645674`*^9, 3.4034606827908535`*^9}, {3.403510788011648*^9, 
   3.4035108104887257`*^9}, {3.403511083995634*^9, 3.403511289284252*^9}, {
   3.403511331032557*^9, 3.403511367971599*^9}, {3.4035119249173975`*^9, 
   3.4035120132437754`*^9}, {3.4035122566510477`*^9, 
   3.4035122749750705`*^9}, {3.4035127511223297`*^9, 
   3.4035128513991146`*^9}, {3.403513482390955*^9, 3.4035135283850813`*^9}, {
   3.403513665178084*^9, 3.4035136711767235`*^9}, {3.403513738601119*^9, 
   3.4035137423450203`*^9}, {3.4035295479994087`*^9, 3.403529729804179*^9}, {
   3.4035298109874597`*^9, 3.403529827081156*^9}, {3.4035308264068437`*^9, 
   3.403531026826633*^9}, {3.4035886411673994`*^9, 3.403588684834526*^9}, {
   3.4035888188611045`*^9, 3.4035888221567917`*^9}, {3.403590674188645*^9, 
   3.403590780904471*^9}, {3.403591390314685*^9, 3.403591478896906*^9}, {
   3.4035915163896236`*^9, 3.403591520884453*^9}, {3.40359176768311*^9, 
   3.403591822277272*^9}, 3.403592098664856*^9, {3.4035921345004525`*^9, 
   3.4035921766247096`*^9}, {3.40359238295623*^9, 3.403592389145287*^9}, {
   3.4035932411190777`*^9, 3.403593338477104*^9}, {3.4035934583043957`*^9, 
   3.4035934614721613`*^9}, {3.403614158914042*^9, 3.4036141614812603`*^9}, {
   3.4041343352586*^9, 3.4041343556478*^9}, {3.4041352276566*^9, 
   3.4041352419462*^9}, {3.4041353225514*^9, 3.404135333019*^9}, {
   3.4041355245713997`*^9, 3.4041355700766*^9}, {3.4041357546558*^9, 
   3.404135792127*^9}, {3.4041363435246*^9, 3.404136343743*^9}, {
   3.4041363837882*^9, 3.4041364789482*^9}}]
}, Open  ]],

Cell[CellGroupData[{

Cell["\<\
Large-scale transformations: collecting terms around pre-defined symbols\
\>", "Section",
 CellChangeTimes->{{3.4033384525984616`*^9, 3.4033384766691866`*^9}, {
   3.4033386741995125`*^9, 3.403338677967826*^9}, {3.4033388210729246`*^9, 
   3.403338835799521*^9}, {3.403460216154704*^9, 3.4034602378095684`*^9}, {
   3.403512348662737*^9, 3.4035123487584343`*^9}, {3.403591066947686*^9, 
   3.403591082633206*^9}, {3.403591378091834*^9, 3.4035913788847523`*^9}, {
   3.403597752338154*^9, 3.403597773447154*^9}, 3.4036111003461313`*^9, {
   3.4036301464386263`*^9, 3.403630160803918*^9}, {3.4041354113934*^9, 
   3.4041354939486*^9}, {3.4041365644674*^9, 3.4041366192390003`*^9}}],

Cell[TextData[StyleBox["Collecting terms frequently makes the expressions \
easier to analyze. An example from one of the relaxation theory worksheets:",
 FontSize->16,
 FontSlant->"Italic"]], "Text",
 CellChangeTimes->{{3.403339183467698*^9, 3.4033392844085035`*^9}, {
   3.4033393996452684`*^9, 3.403339411131838*^9}, {3.4033394689425907`*^9, 
   3.4033394737733364`*^9}, {3.4034590645955963`*^9, 3.403459065147319*^9}, {
   3.4034593099002085`*^9, 3.4034594088128233`*^9}, {3.4034594530062838`*^9, 
   3.4034595138715286`*^9}, {3.4034598973626084`*^9, 
   3.4034599978327403`*^9}, {3.4034603019636655`*^9, 
   3.4034603667739706`*^9}, {3.4034605051928453`*^9, 3.403460533461544*^9}, {
   3.4034606662645674`*^9, 3.4034606827908535`*^9}, {3.403510788011648*^9, 
   3.4035108104887257`*^9}, {3.403511083995634*^9, 3.403511289284252*^9}, {
   3.403511331032557*^9, 3.403511367971599*^9}, {3.4035119249173975`*^9, 
   3.4035120132437754`*^9}, {3.4035122566510477`*^9, 
   3.4035122749750705`*^9}, {3.4035127511223297`*^9, 
   3.4035128513991146`*^9}, {3.403513482390955*^9, 3.4035135283850813`*^9}, {
   3.403513665178084*^9, 3.4035136711767235`*^9}, {3.403513738601119*^9, 
   3.4035137423450203`*^9}, {3.4035295479994087`*^9, 3.403529729804179*^9}, {
   3.4035298109874597`*^9, 3.403529827081156*^9}, {3.4035308264068437`*^9, 
   3.403531026826633*^9}, {3.4035886411673994`*^9, 3.403588684834526*^9}, {
   3.4035888188611045`*^9, 3.4035888221567917`*^9}, {3.403590674188645*^9, 
   3.403590780904471*^9}, {3.403591390314685*^9, 3.403591478896906*^9}, {
   3.4035915163896236`*^9, 3.403591520884453*^9}, {3.40359176768311*^9, 
   3.403591822277272*^9}, 3.403592098664856*^9, {3.4035921345004525`*^9, 
   3.4035921766247096`*^9}, {3.40359238295623*^9, 3.403592389145287*^9}, {
   3.4035932411190777`*^9, 3.403593338477104*^9}, {3.4035934583043957`*^9, 
   3.4035934614721613`*^9}, {3.403614158914042*^9, 3.4036141614812603`*^9}, {
   3.404136760965*^9, 3.4041367862214003`*^9}}],

Cell[BoxData[{
 RowBox[{"A", "=", 
  RowBox[{
   RowBox[{
    RowBox[{"-", 
     FractionBox["1", "45"]}], " ", 
    RowBox[{"J", "[", 
     RowBox[{
      SubscriptBox["\[Tau]", "\"\<c\>\""], ",", "0"}], "]"}], " ", 
    SubsuperscriptBox["\[CapitalDelta]", "\"\<G1\>\"", "2"]}], "-", 
   RowBox[{
    FractionBox["1", "60"], " ", 
    RowBox[{"J", "[", 
     RowBox[{
      SubscriptBox["\[Tau]", "\"\<c\>\""], ",", 
      SubscriptBox["\[Omega]", "L"]}], "]"}], " ", 
    SubsuperscriptBox["\[CapitalDelta]", "\"\<G1\>\"", "2"]}], "-", 
   RowBox[{
    FractionBox["1", "180"], " ", 
    RowBox[{"J", "[", 
     RowBox[{
      SubscriptBox["\[Tau]", "\"\<c\>\""], ",", "0"}], "]"}], " ", 
    SubsuperscriptBox["\[CapitalDelta]", "\"\<HFC\>\"", "2"]}], "-", 
   RowBox[{
    FractionBox["1", "240"], " ", 
    RowBox[{"J", "[", 
     RowBox[{
      SubscriptBox["\[Tau]", "\"\<c\>\""], ",", 
      SubscriptBox["\[Omega]", "F"]}], "]"}], " ", 
    SubsuperscriptBox["\[CapitalDelta]", "\"\<HFC\>\"", "2"]}], "-", 
   RowBox[{
    FractionBox["1", "80"], " ", 
    RowBox[{"J", "[", 
     RowBox[{
      SubscriptBox["\[Tau]", "\"\<c\>\""], ",", 
      SubscriptBox["\[Omega]", "L"]}], "]"}], " ", 
    SubsuperscriptBox["\[CapitalDelta]", "\"\<HFC\>\"", "2"]}], "-", 
   RowBox[{
    FractionBox["1", "720"], " ", 
    RowBox[{"J", "[", 
     RowBox[{
      SubscriptBox["\[Tau]", "\"\<c\>\""], ",", 
      RowBox[{
       RowBox[{"-", 
        SubscriptBox["\[Omega]", "F"]}], "+", 
       SubscriptBox["\[Omega]", "L"]}]}], "]"}], " ", 
    SubsuperscriptBox["\[CapitalDelta]", "\"\<HFC\>\"", "2"]}], "-", 
   RowBox[{
    FractionBox["2", "15"], " ", 
    RowBox[{"J", "[", 
     RowBox[{
      SubscriptBox["\[Tau]", "\"\<c\>\""], ",", "0"}], "]"}], " ", 
    SubscriptBox["X", 
     RowBox[{"\"\<G1\>\"", ",", "\"\<HFC\>\""}]]}], "-", 
   RowBox[{
    FractionBox["1", "10"], " ", 
    RowBox[{"J", "[", 
     RowBox[{
      SubscriptBox["\[Tau]", "\"\<c\>\""], ",", 
      SubscriptBox["\[Omega]", "L"]}], "]"}], " ", 
    SubscriptBox["X", 
     RowBox[{
     "\"\<G1\>\"", ",", "\"\<HFC\>\""}]]}]}]}], "\[IndentingNewLine]", 
 RowBox[{"Collect", "[", 
  RowBox[{"A", ",", 
   RowBox[{"{", 
    RowBox[{
     SubsuperscriptBox["\[CapitalDelta]", "\"\<G1\>\"", "2"], ",", 
     SubsuperscriptBox["\[CapitalDelta]", "\"\<HFC\>\"", "2"], ",", 
     RowBox[{"X", "[", 
      RowBox[{"\"\<G1\>\"", ",", "\"\<HFC\>\""}], "]"}]}], "}"}], ",", 
   "FullSimplify"}], "]"}]}], "Input",
 CellChangeTimes->{{3.4041368410866003`*^9, 3.4041368671074*^9}, {
  3.4041369116766*^9, 3.404136912441*^9}, {3.4041369517061996`*^9, 
  3.4041370625754004`*^9}}],

Cell[TextData[StyleBox["The rearranged form has the spectral densities neatly \
collected by their respective relaxation mechanism (g anisotropy, HFC \
anisotropy and their cross-correlation) . The 'FullSimplify' operation is \
applied to each of the collected groups.",
 FontSize->16,
 FontSlant->"Italic"]], "Text",
 CellChangeTimes->{{3.403339183467698*^9, 3.4033392844085035`*^9}, {
   3.4033393996452684`*^9, 3.403339411131838*^9}, {3.4033394689425907`*^9, 
   3.4033394737733364`*^9}, {3.4034590645955963`*^9, 3.403459065147319*^9}, {
   3.4034593099002085`*^9, 3.4034594088128233`*^9}, {3.4034594530062838`*^9, 
   3.4034595138715286`*^9}, {3.4034598973626084`*^9, 
   3.4034599978327403`*^9}, {3.4034603019636655`*^9, 
   3.4034603667739706`*^9}, {3.4034605051928453`*^9, 3.403460533461544*^9}, {
   3.4034606662645674`*^9, 3.4034606827908535`*^9}, {3.403510788011648*^9, 
   3.4035108104887257`*^9}, {3.403511083995634*^9, 3.403511289284252*^9}, {
   3.403511331032557*^9, 3.403511367971599*^9}, {3.4035119249173975`*^9, 
   3.4035120132437754`*^9}, {3.4035122566510477`*^9, 
   3.4035122749750705`*^9}, {3.4035127511223297`*^9, 
   3.4035128513991146`*^9}, {3.403513482390955*^9, 3.4035135283850813`*^9}, {
   3.403513665178084*^9, 3.4035136711767235`*^9}, {3.403513738601119*^9, 
   3.4035137423450203`*^9}, {3.4035295479994087`*^9, 3.403529729804179*^9}, {
   3.4035298109874597`*^9, 3.403529827081156*^9}, {3.4035308264068437`*^9, 
   3.403531026826633*^9}, {3.4035886411673994`*^9, 3.403588684834526*^9}, {
   3.4035888188611045`*^9, 3.4035888221567917`*^9}, {3.403590674188645*^9, 
   3.403590780904471*^9}, {3.403591390314685*^9, 3.403591478896906*^9}, {
   3.4035915163896236`*^9, 3.403591520884453*^9}, {3.40359176768311*^9, 
   3.403591822277272*^9}, 3.403592098664856*^9, {3.4035921345004525`*^9, 
   3.4035921766247096`*^9}, {3.40359238295623*^9, 3.403592389145287*^9}, {
   3.4035932411190777`*^9, 3.403593338477104*^9}, {3.4035934583043957`*^9, 
   3.4035934614721613`*^9}, {3.403614158914042*^9, 3.4036141614812603`*^9}, {
   3.404136760965*^9, 3.4041367862214003`*^9}, {3.4041370953354*^9, 
   3.4041372235206003`*^9}}]
}, Open  ]],

Cell[CellGroupData[{

Cell["Getting started 9: playing with patterns", "Section",
 CellChangeTimes->{{3.4033384525984616`*^9, 3.4033384766691866`*^9}, {
   3.4033386741995125`*^9, 3.403338677967826*^9}, {3.4033388210729246`*^9, 
   3.403338835799521*^9}, {3.403460216154704*^9, 3.4034602378095684`*^9}, {
   3.403512348662737*^9, 3.4035123487584343`*^9}, {3.403591066947686*^9, 
   3.403591082633206*^9}, {3.403591378091834*^9, 3.4035913788847523`*^9}, {
   3.403597752338154*^9, 3.403597773447154*^9}, 3.4036111003461313`*^9, {
   3.4036301464386263`*^9, 3.403630160803918*^9}, {3.403701701932208*^9, 
   3.4037017054827623`*^9}, 3.4037018649969664`*^9, {3.4041354168534*^9, 
   3.4041354171342*^9}}],

Cell[TextData[StyleBox["Think of a recent analytical problem that you had \
encountered. Which functions needed to be transformed? What is the minimal \
set of rules that is sufficient to transform them? Try creating a coherent \
pattern set for those functions and applying it to the problem. Remember that \
Mathematica has a large number of built-in functions which may be used or \
extended as you see fit. The help system is fairly extensive -- just \
highlight what you want and press F1.",
 FontSize->16,
 FontSlant->"Italic"]], "Text",
 CellChangeTimes->{{3.403339183467698*^9, 3.4033392844085035`*^9}, {
   3.4033393996452684`*^9, 3.403339411131838*^9}, {3.4033394689425907`*^9, 
   3.4033394737733364`*^9}, {3.4034590645955963`*^9, 3.403459065147319*^9}, {
   3.4034593099002085`*^9, 3.4034594088128233`*^9}, {3.4034594530062838`*^9, 
   3.4034595138715286`*^9}, {3.4034598973626084`*^9, 
   3.4034599978327403`*^9}, {3.4034603019636655`*^9, 
   3.4034603667739706`*^9}, {3.4034605051928453`*^9, 3.403460533461544*^9}, {
   3.4034606662645674`*^9, 3.4034606827908535`*^9}, {3.403510788011648*^9, 
   3.4035108104887257`*^9}, {3.403511083995634*^9, 3.403511289284252*^9}, {
   3.403511331032557*^9, 3.403511367971599*^9}, {3.4035119249173975`*^9, 
   3.4035120132437754`*^9}, {3.4035122566510477`*^9, 
   3.4035122749750705`*^9}, {3.4035127511223297`*^9, 
   3.4035128513991146`*^9}, {3.403513482390955*^9, 3.4035135283850813`*^9}, {
   3.403513665178084*^9, 3.4035136711767235`*^9}, {3.403513738601119*^9, 
   3.4035137423450203`*^9}, {3.4035295479994087`*^9, 3.403529729804179*^9}, {
   3.4035298109874597`*^9, 3.403529827081156*^9}, {3.4035308264068437`*^9, 
   3.403531026826633*^9}, {3.4035886411673994`*^9, 3.403588684834526*^9}, {
   3.4035888188611045`*^9, 3.4035888221567917`*^9}, {3.403590674188645*^9, 
   3.403590780904471*^9}, {3.403591390314685*^9, 3.403591478896906*^9}, {
   3.4035915163896236`*^9, 3.403591520884453*^9}, {3.40359176768311*^9, 
   3.403591822277272*^9}, 3.403592098664856*^9, {3.4035921345004525`*^9, 
   3.4035921766247096`*^9}, {3.40359238295623*^9, 3.403592389145287*^9}, {
   3.4035932411190777`*^9, 3.403593338477104*^9}, {3.4035934583043957`*^9, 
   3.4035934614721613`*^9}, {3.403614175215733*^9, 3.40361418230317*^9}, 
   3.40362516902185*^9, {3.403625467481263*^9, 3.4036255024301977`*^9}, {
   3.4036296364917727`*^9, 3.403629799118083*^9}, {3.4037017133875294`*^9, 
   3.403701803505785*^9}, {3.4037018789784937`*^9, 3.4037019476996813`*^9}}]
}, Open  ]]
}, Open  ]]
},
WindowSize->{1117, 764},
WindowMargins->{{20, Automatic}, {Automatic, 30}},
ShowSelection->True,
FrontEndVersion->"6.0 for Microsoft Windows (64-bit) (April 28, 2007)",
StyleDefinitions->Notebook[{
   Cell[
    StyleData[StyleDefinitions -> "Default.nb"]], 
   Cell[
    StyleData["Text"], CellDingbat -> None, TextJustification -> 1., FontSize -> 
    16, FontSlant -> "Italic", FontColor -> GrayLevel[0], Background -> 
    RGBColor[0.9019607843137255, 0.9411764705882353, 0.8509803921568627]]}, 
  Visible -> False, FrontEndVersion -> 
  "6.0 for Microsoft Windows (64-bit) (April 28, 2007)", StyleDefinitions -> 
  "PrivateStylesheetFormatting.nb"]
]
(* End of Notebook Content *)

(* Internal cache information *)
(*CellTagsOutline
CellTagsIndex->{}
*)
(*CellTagsIndex
CellTagsIndex->{}
*)
(*NotebookFileOutline
Notebook[{
Cell[CellGroupData[{
Cell[590, 23, 329, 4, 83, "Title"],
Cell[922, 29, 919, 14, 93, "Text"],
Cell[CellGroupData[{
Cell[1866, 47, 484, 8, 71, "Section"],
Cell[2353, 57, 598, 13, 71, "Text"],
Cell[2954, 72, 404, 9, 31, "Input"],
Cell[3361, 83, 597, 9, 71, "Text"],
Cell[3961, 94, 598, 13, 31, "Input"],
Cell[4562, 109, 788, 18, 71, "Text"],
Cell[5353, 129, 406, 9, 31, "Input"],
Cell[5762, 140, 565, 11, 93, "Text"]
}, Open  ]],
Cell[CellGroupData[{
Cell[6364, 156, 439, 5, 71, "Section"],
Cell[6806, 163, 564, 9, 49, "Text"],
Cell[7373, 174, 613, 13, 31, "Input"],
Cell[7989, 189, 769, 16, 71, "Text"],
Cell[8761, 207, 277, 8, 31, "Input"],
Cell[9041, 217, 1011, 15, 71, "Text"],
Cell[10055, 234, 1453, 39, 112, "Input"],
Cell[11511, 275, 1181, 17, 93, "Text"],
Cell[12695, 294, 486, 11, 31, "Input"]
}, Open  ]],
Cell[CellGroupData[{
Cell[13218, 310, 386, 5, 71, "Section"],
Cell[13607, 317, 1413, 20, 93, "Text"],
Cell[15023, 339, 639, 17, 52, "Input"],
Cell[15665, 358, 1332, 19, 71, "Text"],
Cell[17000, 379, 1860, 58, 268, "Input"],
Cell[18863, 439, 813, 13, 49, "Text"],
Cell[19679, 454, 1670, 41, 31, "Input"],
Cell[21352, 497, 1180, 18, 71, "Text"]
}, Open  ]],
Cell[CellGroupData[{
Cell[22569, 520, 481, 6, 71, "Section"],
Cell[23053, 528, 2009, 28, 137, "Text"],
Cell[CellGroupData[{
Cell[25087, 560, 190, 3, 49, "Text",
 CellGroupingRules->{GroupTogetherGrouping, 10000.}],
Cell[25280, 565, 4347, 130, 292, "Input",
 CellGroupingRules->{GroupTogetherGrouping, 10000.}],
Cell[29630, 697, 1595, 40, 31, "Input",
 CellGroupingRules->{GroupTogetherGrouping, 10000.}],
Cell[31228, 739, 1678, 25, 71, "Text",
 CellGroupingRules->{GroupTogetherGrouping, 10000.}]
}, Closed]]
}, Open  ]],
Cell[CellGroupData[{
Cell[32955, 770, 430, 5, 71, "Section"],
Cell[33388, 777, 1578, 22, 93, "Text"],
Cell[34969, 801, 386, 9, 44, "Input"],
Cell[35358, 812, 1955, 31, 115, "Text"],
Cell[37316, 845, 1324, 39, 115, "Input"],
Cell[38643, 886, 1973, 27, 115, "Text"],
Cell[40619, 915, 1076, 29, 35, "Input"],
Cell[41698, 946, 1412, 20, 49, "Text"]
}, Open  ]],
Cell[CellGroupData[{
Cell[43147, 971, 808, 10, 71, "Section"],
Cell[43958, 983, 1618, 23, 71, "Text"],
Cell[45579, 1008, 1109, 27, 72, "Input"],
Cell[46691, 1037, 1769, 26, 71, "Text"],
Cell[48463, 1065, 113, 3, 31, "Input"],
Cell[48579, 1070, 150, 5, 31, "Input"],
Cell[48732, 1077, 1471, 20, 49, "Text"]
}, Open  ]],
Cell[CellGroupData[{
Cell[50240, 1102, 574, 9, 71, "Section"],
Cell[50817, 1113, 1983, 27, 93, "Text"],
Cell[52803, 1142, 3320, 103, 201, "Input"],
Cell[56126, 1247, 1954, 28, 71, "Text"],
Cell[58083, 1277, 550, 14, 31, "Input"],
Cell[58636, 1293, 2067, 30, 93, "Text"],
Cell[60706, 1325, 492, 15, 52, "Input"]
}, Open  ]],
Cell[CellGroupData[{
Cell[61235, 1345, 518, 6, 71, "Section"],
Cell[61756, 1353, 1873, 27, 49, "Text"],
Cell[63632, 1382, 1152, 34, 48, "DisplayFormula"],
Cell[64787, 1418, 2492, 35, 93, "Text"],
Cell[CellGroupData[{
Cell[67304, 1457, 262, 4, 49, "Text",
 CellGroupingRules->{GroupTogetherGrouping, 10000.}],
Cell[67569, 1463, 1745, 54, 152, "Input",
 CellGroupingRules->{GroupTogetherGrouping, 10000.}],
Cell[69317, 1519, 663, 22, 31, "Input",
 CellGroupingRules->{GroupTogetherGrouping, 10000.}]
}, Closed]]
}, Open  ]],
Cell[CellGroupData[{
Cell[70029, 1547, 600, 9, 71, "Section"],
Cell[70632, 1558, 2528, 35, 115, "Text"],
Cell[73163, 1595, 1944, 28, 49, "Text"],
Cell[75110, 1625, 777, 19, 31, "Input"],
Cell[75890, 1646, 1977, 28, 49, "Text"],
Cell[77870, 1676, 760, 15, 31, "Input"],
Cell[78633, 1693, 2145, 31, 49, "Text"],
Cell[80781, 1726, 863, 16, 31, "Input"],
Cell[81647, 1744, 1890, 27, 49, "Text"],
Cell[83540, 1773, 554, 11, 33, "Input"],
Cell[84097, 1786, 1932, 28, 49, "Text"],
Cell[86032, 1816, 653, 14, 31, "Input"],
Cell[86688, 1832, 1980, 28, 49, "Text"],
Cell[88671, 1862, 778, 18, 31, "Input"]
}, Open  ]],
Cell[CellGroupData[{
Cell[89486, 1885, 669, 8, 71, "Section"],
Cell[90158, 1895, 2252, 33, 49, "Text"],
Cell[92413, 1930, 1073, 30, 54, "Input"],
Cell[93489, 1962, 2219, 32, 49, "Text"]
}, Open  ]],
Cell[CellGroupData[{
Cell[95745, 1999, 689, 10, 71, "Section"],
Cell[96437, 2011, 1977, 28, 49, "Text"],
Cell[98417, 2041, 2648, 74, 102, "Input"],
Cell[101068, 2117, 2144, 31, 71, "Text"]
}, Open  ]],
Cell[CellGroupData[{
Cell[103249, 2153, 679, 9, 71, "Section"],
Cell[103931, 2164, 2495, 35, 115, "Text"]
}, Open  ]]
}, Open  ]]
}
]
*)

(* End of internal cache information *)