(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 4.2' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 100348, 3934]*) (*NotebookOutlinePosition[ 136181, 5162]*) (* CellTagsIndexPosition[ 136137, 5158]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Gamma & Beta Functions", "Title", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ R.G. Palmer 7/8/04\ \>", "Subsubtitle", Evaluatable->False, TextAlignment->Center, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Background and Copyright", "Subsection"], Cell[TextData[{ "This ", StyleBox["Mathematica", FontSlant->"Italic"], " Notebook was written by Richard G. Palmer (Physics Department, Duke \ University) for use in a course he taught (from Phy 230, \"Mathematical \ Method in Physics\"). As of 2004, it has been made available for general \ non-profit use under the following copyright provision." }], "Text"], Cell[TextData[{ StyleBox["This Mathematica Notebook is Copyright Richard G. Palmer, 2004", FontWeight->"Bold"], ". It may be freely used by individuals, and by classes at academic \ institutions, provided:\n1. Credit is given to Richard Palmer as the original \ author; and\n2. It is not bought or sold or exchanged for profit, or \ incorporated into material that is bought or sold or exchanged for profit.\n\ Any other use requires the written permission of Richard Palmer, Dept. of \ Physics, Box 90305, Duke University, Durham, NC 27708, USA. See ", StyleBox["http://www.phy.duke.edu/~palmer", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " for the email address." }], "Text"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " Version: 4.2 (and OK for 5)" }], "Text"], Cell[TextData[{ "See ", StyleBox["http://www.phy.duke.edu/~palmer/notebooks/", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " for other ", StyleBox["Mathematica", FontSlant->"Italic"], " noteboks by Richard Palmer." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Preface"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["This Notebook concerns Gamma and Beta functions. It also \ introduces and exemplifies some ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" principles", Evaluatable->False, AspectRatioFixed->True], StyleBox[".", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" ", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["There are 3 problems embedded in the Notebook.", "Text", Evaluatable->False, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["The Euler integral for ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[CapitalGamma]", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Plain"], StyleBox["(x)", Evaluatable->False, AspectRatioFixed->True] }], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "[", StyleBox["Mathematica", FontSlant->"Italic"], " Book: 1.7.1, 1.1.6, 2.1.3]" }], "Text"], Cell[TextData[StyleBox["Mathematically the gamma function \[CapitalGamma](x) \ may be defined in several ways, the most generally useful being the Euler \ integral:", Evaluatable->False, AspectRatioFixed->True]], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ gamma[x_] := Integrate[t^(x-1) Exp[-t], {t,0,\[Infinity]}]\ \>", \ "Input", TextAlignment->Left, AspectRatioFixed->True], Cell["\<\ (It won't produce any output, but ignore the spelling \ warning.)\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell["Notes", "Text", CellFrame->True, CellDingbat->None, TextAlignment->Center, FontWeight->"Bold", Background->RGBColor[1, 1, 0]], Cell[TextData[{ "1. You should have evaluated the above function. It doesn't produce any \ output (besides the spelling warning) because it's just a definition of a \ function. The syntax\n\n", StyleBox["function", FontSize->14, FontSlant->"Italic"], StyleBox["[", FontSize->14], StyleBox["argument", FontSize->14, FontSlant->"Italic"], StyleBox["_] := ", FontSize->14], StyleBox["expression", FontSize->14, FontSlant->"Italic"], StyleBox["\n", FontSlant->"Italic"], "\n is a very important ", StyleBox["Mathematica", FontSlant->"Italic"], " idiom for defining a function. See section 1.7.1 of the book. Remember \ the \"_\" and the \":=\"." }], "Text", CellFrame->True, Evaluatable->False, TextAlignment->Left, AspectRatioFixed->True, Background->RGBColor[1, 1, 0]], Cell[TextData[{ StyleBox["2. You should have ", Evaluatable->False, AspectRatioFixed->True], StyleBox["read", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" the expression you evaluated, and made sure you understood it. \ It's sometimes easy to go through a Notebook evaluating things but not \ understanding them. Try to practice critical reading. In the present case: \ Did you recognize the expression as the standard definition of the gamma \ function? Did you wonder whether ", Evaluatable->False, AspectRatioFixed->True], StyleBox["t", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" needed defining? (no, but there'd be trouble if ", Evaluatable->False, AspectRatioFixed->True], StyleBox["t", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" already had a value.) Did you notice the syntax for specifying \ a ", Evaluatable->False, AspectRatioFixed->True], StyleBox["definite", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" integral? Did you enjoy the ability to use \"\[Infinity]\" (or \ ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Infinity", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"], StyleBox[") as a numerical constant? [And did you wonder what ", Evaluatable->False, AspectRatioFixed->True], StyleBox["2*Infinity", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" or ", Evaluatable->False, AspectRatioFixed->True], StyleBox["-Infinity", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" would give? Try them if you wish.]", Evaluatable->False, AspectRatioFixed->True] }], "Text", CellFrame->True, Evaluatable->False, AspectRatioFixed->True, Background->RGBColor[1, 1, 0]], Cell[TextData[ "OK, let's try out our new definition. Try to anticipate (i.e. guess!) what \ each of these will give before evaluating it:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["gamma[2]", "Input", AspectRatioFixed->True], Cell["gamma[5]", "Input", AspectRatioFixed->True], Cell["gamma[1/2]", "Input", AspectRatioFixed->True], Cell["gamma[7/2]", "Input", AspectRatioFixed->True], Cell["gamma[0]", "Input", AspectRatioFixed->True], Cell["gamma[1/4]", "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["The last example give the game away. ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" recognizes our integral as a gamma function, and substitutes its \ own internal function ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Gamma[x]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" as the result. Note carefully the difference between ", Evaluatable->False, AspectRatioFixed->True], StyleBox["gamma[x]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" (our very own integral), and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Gamma[x]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" (", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["'s built-in function). Internal ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" functions always begin with an upper-case letter (or a $ sign); \ ", Evaluatable->False, AspectRatioFixed->True], StyleBox["you", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" should normally use lower-case to avoid any possible conflict.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Observe that ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" did not get an explicit answer for ", Evaluatable->False, AspectRatioFixed->True], StyleBox["gamma[1/4]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[", because there isn't one. \[CapitalGamma](x) only only has an \ explicit value in terms of elementary functions when x is an integer or \ half-integer. Though many people would call \[CapitalGamma](x) itself pretty \ elementary, like sin(x) or log(x). That's the game of the name; you give a \ name to an integral you can't do, then suddenly it's do-able, and pretty soon \ people are calling it elementary!", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Of course we can get a ", Evaluatable->False, AspectRatioFixed->True], StyleBox["numerical", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " answer for \[CapitalGamma](1/4) if we want it. Here are three different \ ways to do that:", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["gamma[0.25]", "Input", AspectRatioFixed->True], Cell["gamma[1/4] //N", "Input", AspectRatioFixed->True], Cell["N[gamma[1/4]]", "Input", AspectRatioFixed->True], Cell[TextData[{ "In the first case we gave ", StyleBox["Mathematica", FontSlant->"Italic"], " an approximate value (a floating-point number) for the argument, so it \ assumed that an approximate result would be OK, and did a numerical \ calculation. This is a general principle whenever ", StyleBox["Mathematica", FontSlant->"Italic"], " encounters numbers with decimal points -- it assumes that they're \ approximate and does an approximate calculation from there on." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["The second and third cases are equivalent ways of applying the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["N[]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" function to the expression ", Evaluatable->False, AspectRatioFixed->True], StyleBox["gamma[1/4]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[". The ", Evaluatable->False, AspectRatioFixed->True], StyleBox["N[]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[ " function forces its argument to be evaluated numerically. In fact any \ one-argument function ", Evaluatable->False, AspectRatioFixed->True], StyleBox["f[]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" can be applied to an argument ", Evaluatable->False, AspectRatioFixed->True], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" as", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["f[x]", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"], StyleBox[" or as ", Evaluatable->False, AspectRatioFixed->True], StyleBox["x //f", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"], StyleBox[" or as ", Evaluatable->False, AspectRatioFixed->True], StyleBox["f @ x", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"] }], "Print", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "These are referred to the standard form, postfix form, and prefix form \ respectively, and can be used pretty much interchangeably. [Not quite \ identically, because their precedences are different; see section 2.1.3 if \ you care.] Thus we could also have written "], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["1/4 //gamma //N", "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["although that seems a little wierd. I rarely use the prefix form \ ", Evaluatable->False, AspectRatioFixed->True], StyleBox["f @ x", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[", but I often find the postfix form ", Evaluatable->False, AspectRatioFixed->True], StyleBox["x //f ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" natural.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["[Optional] Mathematical uNease", "Subsubsection", FontColor->RGBColor[1, 0, 1]], Cell[TextData[{ StyleBox["Does the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["N[]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" function bother you? In a way it should if you've come from a \ strict mathematical background. Because it doesn't just take the value of \ its argument and do something to it, like all well-behaved mathematical \ functions. Instead it changes the way its argument is evaluated.", Evaluatable->False, AspectRatioFixed->True] }], "Text", CellDingbat->None, CellMargins->{{53, Inherited}, {Inherited, Inherited}}, Evaluatable->False, TextAlignment->Left, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Actually ", Evaluatable->False, AspectRatioFixed->True], StyleBox["most", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" functions in ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" behave in the ordinary mathematical way; first they evaluate \ their arguments, and then they do something to the results. But there are a \ number of important exceptions, including ", Evaluatable->False, AspectRatioFixed->True], StyleBox["N[]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[".", Evaluatable->False, AspectRatioFixed->True] }], "Text", CellDingbat->None, CellMargins->{{53, Inherited}, {Inherited, Inherited}}, Evaluatable->False, TextAlignment->Left, AspectRatioFixed->True], Cell[TextData[{ StyleBox["From now on we'll mainly use ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["'s internal ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Gamma[]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" function instead of our own ", Evaluatable->False, AspectRatioFixed->True], StyleBox["gamma[]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[". Actually it doesn't make much difference, and it's pretty hard \ to find an argument for which they don't give the same value (challenge!).", Evaluatable->False, AspectRatioFixed->True] }], "Text", CellMargins->{{12, Inherited}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Factorials and double factorials", "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "[", StyleBox["Mathematica", FontSlant->"Italic"], " Book: 1.2.1, 1.7.3, 1.8.2, 3.7.4]" }], "Text"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " also understand the usual factorial notation:" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["4!", "Input", AspectRatioFixed->True], Cell["100!", "Input", AspectRatioFixed->True], Cell["(1/2)!", "Input", AspectRatioFixed->True], Cell["(2/3)!", "Input", AspectRatioFixed->True], Cell["0.66666666667!", "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["It seems to treat ", Evaluatable->False, AspectRatioFixed->True], StyleBox["x!", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" as exactly the same as ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Gamma[x+1]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[", whether or not ", Evaluatable->False, AspectRatioFixed->True], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" is an integer.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ It is often more useful to have a list of results, which you can \ then manipulate further:\ \>", "Text"], Cell[BoxData[ \(Table[\(n!\), \ {n, 10}]\)], "Input"], Cell["or:", "Text"], Cell[BoxData[ \(\(\(nten\)\(\ \)\(=\)\(Table[\(n!\), \ {n, 10}]\)\(\ \)\)\)], "Input"], Cell[BoxData[ \(nten\)], "Input"], Cell[BoxData[ \(Sqrt[nten]\)], "Input"], Cell[TextData[{ StyleBox["Table[f,{i,imax}]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " givea list of the values of ", StyleBox["f", FontSlant->"Italic"], " as ", StyleBox["i", FontSlant->"Italic"], " runs from 1 to ", StyleBox["imax", FontSlant->"Italic"], ".\n", StyleBox["Table[f,{i,imin,imax}]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " givea list of values with ", StyleBox["i", FontSlant->"Italic"], " running from ", StyleBox["imin", FontSlant->"Italic"], " to ", StyleBox["imax", FontSlant->"Italic"], ".\n", StyleBox["Table[f,{i,imin,imax,di}]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " use steps of ", StyleBox["di", FontSlant->"Italic"], "." }], "Text"], Cell[BoxData[ \(Table[Gamma[z], {z, 1, 2, 0.1}]\)], "Input"], Cell[TextData["The \"double factorial\" function"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[" n!! = n(n-2)(n-4)..."], "Print", Evaluatable->False, LineSpacing->{1, 0}, AspectRatioFixed->True], Cell[TextData[StyleBox["is also recognized as long as n is an integer.", Evaluatable->False, AspectRatioFixed->True]], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["5!!", "Input", AspectRatioFixed->True], Cell["6!!", "Input", AspectRatioFixed->True], Cell[TextData[StyleBox["If it is not an integer, then, as usual, it just \ gives you back what you asked.", Evaluatable->False, AspectRatioFixed->True]], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["(1/2)!!", "Input", AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell["Plots for real x", "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "[", StyleBox["Mathematica", FontSlant->"Italic"], " Book: 1.9.1]" }], "Text"], Cell[TextData[{ "Now let's look at some pictures. Here's a", StyleBox[" positive real axis of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Gamma[x]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[":", Evaluatable->False, AspectRatioFixed->True] }], "Text"], Cell[BoxData[ \(Plot[Gamma[x], {x, 0.1, 5}]\)], "Input"], Cell[TextData[StyleBox["Let's have a look when you include the negative \ axis:", Evaluatable->False, AspectRatioFixed->True]], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["Plot[Gamma[x],{x,-3,5}]", "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " plots the right thing despite coming from the poles at x = -3, -2, -1, \ and 0." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Note how flat ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[CapitalGamma]", Evaluatable->False, AspectRatioFixed->True, FontSize->12], StyleBox["(x) is near its minimum between x=1 and x=2. We'll come back to \ that. ", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell["Plots for complex z", "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "[ ", StyleBox["Mathematica", FontSlant->"Italic"], " Book: 1.1.5, 1.9.3, 1.9.7]" }], "Text"], Cell[TextData[{ StyleBox["Now let's look at ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[CapitalGamma]", Evaluatable->False, AspectRatioFixed->True, FontSize->12], StyleBox["(z) for complex z. The most natural thing to try is a 3D plot of \ the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["absolute value", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" (or ", Evaluatable->False, AspectRatioFixed->True], StyleBox["modulus", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[", or ", Evaluatable->False, AspectRatioFixed->True], StyleBox["magnitude", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[") in the x-y plane (for z = x+iy).", Evaluatable->False, AspectRatioFixed->True], " ", StyleBox["You can get real and imaginary parts, and absolute value with the \ functions ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Re[]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[", ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Im[]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[", ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Abs[]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[". For example (first defining ", Evaluatable->False, AspectRatioFixed->True], StyleBox["z", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["): ", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["z = 2 - 3\[ImaginaryI]", "Input", AspectRatioFixed->True], Cell["Re[z]", "Input", AspectRatioFixed->True], Cell["Im[z]", "Input", AspectRatioFixed->True], Cell["Abs[z]", "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["To write a complex number in ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" you must use ", Evaluatable->False, AspectRatioFixed->True], StyleBox["I", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" or ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[ImaginaryI]", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontWeight->"Bold"], StyleBox[" for i, and be sure not to write things like ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Iy", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" (no space), which would be taken as a single variable called ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Iy", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[".", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["Now let's look at a picture.", "Text"], Cell["\<\ Plot3D[Abs[Gamma[x + \[ImaginaryI] y]], {x,-3,3}, {y,-1,1}]\ \>", \ "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Well, that's not very satisfying, it it? The errors came from \ evaluating ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Gamma[-3]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" (where there's a pole), and led to a gap there in the plot. \ More importantly, we don't get a very good sense of the surface, and in \ particular don't really see the poles at -3, -2, -1, and 0.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ To get a better plot requires a good bit of experimentation (as is \ often the case with 3D plots). After playing for a while my best effort was \ the following::\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ Plot3D[Abs[Gamma[x + \[ImaginaryI] y]], {x,-3.5,3.5}, {y,-1,1}, \tPlotPoints->60, \tPlotRange->{0,5}, \tBoxRatios->{1,1,1}, \tViewPoint->{-1.4, -2.3, 2.000}, \tAxesLabel->{\"x\",\"y\",\"|Gamma|\"}, \tClipFill->{GrayLevel[0],GrayLevel[0]} ]\ \>", "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Note that I used ", Evaluatable->False, AspectRatioFixed->True], StyleBox["PlotPoints->60", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" to get a 60 x 60 grid instead of the default 15 x 15 used above. \ I also changed the box shape and the viewpoint to improve the overall \ appearance.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["We could also look at the real or imaginary part of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[CapitalGamma]", Evaluatable->False, AspectRatioFixed->True, FontSize->12], StyleBox["(z), but I'll leave that for now.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Problem 1", FontColor->RGBColor[1, 0, 1]]], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Try defining a function ", Evaluatable->False, AspectRatioFixed->True], StyleBox["ngamma[x]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" that works like our ", Evaluatable->False, AspectRatioFixed->True], StyleBox["gamma[x]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" but uses the numerical integration function ", Evaluatable->False, AspectRatioFixed->True], StyleBox["NIntegrate[]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" instead of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Integrate[]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[". Try these values of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[": 3, 0.5, 10, 0.2, -0.5, 2+3\[ImaginaryI], and compare with ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Gamma[x]//N", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[". See how well the two values agree by subtracting them. \ Discuss the results.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell["Gamma function identities", "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" does ", Evaluatable->False, AspectRatioFixed->True], StyleBox["not", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" have any built-in rules corresponding to the most famous \ properties of the gamma function:", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox["1. \[CapitalGamma](x+1) = x \[CapitalGamma](x)\t\t \ (recursion relation)", Evaluatable->False, AspectRatioFixed->True]], "Print", Evaluatable->False, GeneratedCell->False, CellAutoOverwrite->False, AspectRatioFixed->True], Cell[TextData[StyleBox["2. \[CapitalGamma](x) \[CapitalGamma](1-x) = \[Pi] \ / sin \[Pi]x (reflection formula)", Evaluatable->False, AspectRatioFixed->True]], "Print", Evaluatable->False, GeneratedCell->False, CellAutoOverwrite->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["3. x! (x-1/2)! = ", Evaluatable->False, AspectRatioFixed->True], StyleBox["(2x)! ", Evaluatable->False, AspectRatioFixed->True], StyleBox["sqrt(\[Pi])/2", Evaluatable->False, AspectRatioFixed->True], StyleBox["2x", Evaluatable->False, AspectRatioFixed->True, FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox[" (duplication formula)", Evaluatable->False, AspectRatioFixed->True] }], "Print", Evaluatable->False, GeneratedCell->False, CellAutoOverwrite->False, AspectRatioFixed->True], Cell[TextData[ "For example, you or I could use the above relations to simplify the \ following:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["Gamma[5/4] / Gamma[1/4]", "Input", AspectRatioFixed->True], Cell["(7/3)! / (4/3)!", "Input", AspectRatioFixed->True], Cell["Gamma[1/4] Gamma[3/4]", "Input", AspectRatioFixed->True], Cell["(-3/4)! (-1/4)!", "Input", AspectRatioFixed->True], Cell["2! (3/2)!", "Input", AspectRatioFixed->True], Cell[TextData[ "Of course the last one worked because I used integer and half-integer \ values."], "Text", Evaluatable->False, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell["Finding a minimum", "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "[", StyleBox["Mathematica", FontSlant->"Italic"], " Book: 1.6.5, 3.9.8]" }], "Text"], Cell[TextData[{ StyleBox["Finding the minimum of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[CapitalGamma]", Evaluatable->False, AspectRatioFixed->True, FontSize->12], StyleBox["(z) between z=1 and z=2 is a famous numerical problem. It's \ tricky to get an accurate answer on an ordinary computer because ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[CapitalGamma]", Evaluatable->False, AspectRatioFixed->True, FontSize->12], StyleBox["(z) is very flat near the minimum, as we saw in the plots. But ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" seems to spoil us with a good answer without any fuss:", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["FindMinimum[Gamma[t], {t, 1.5}]", "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Here I asked for a minimum near t=1.5. The result was a list of \ two elements, first the value of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Gamma[t]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" at the minimum, and then the location of t for the minimum. The \ location is written as a rule, ", Evaluatable->False, AspectRatioFixed->True], StyleBox["t -> value", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[", for reasons that we'll come back to that.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["We can check the result by evaluating ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Gamma[t]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" at 1.46163:", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["Gamma[1.46163]", "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Experiment to find out how far from 1.46163 you have to go to see \ a change in this 6 digit value", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[". Because the function is so flat, you'll have to go quite a \ way.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["You should find that we could only determine the location of the \ minimum to about 3 decimal places if we could only evaluate ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Gamma[t]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" to 6 decimal places. Based on such evaluations, we can only say \ it's between t=1.461 and 1.462. More generally, if we could evaluate ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Gamma[t]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" to n decimal places, we could get ", Evaluatable->False, AspectRatioFixed->True], StyleBox["t", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" to about n/2 places. (Why?)", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Seeing this problem, we should be somewhat suspicious of ", StyleBox["Mathematica", FontSlant->"Italic"], "'s result. Is it really 1.46163? Is the last digit meaningful? How can \ we check?" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["One way to check is to evaluate ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Gamma[t]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" to more decimal places using ", Evaluatable->False, AspectRatioFixed->True], StyleBox["N[", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["expression", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold", FontSlant->"Italic"], StyleBox[", ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["digits", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold", FontSlant->"Italic"], StyleBox["]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[":", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["N[Gamma[1.46163], 12]", "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Now you can (if you wish) try neighboring values and verify that \ 1.46163 ", Evaluatable->False, AspectRatioFixed->True], StyleBox["is", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" correct as far as it goes. In fact ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" used much higher internal precision in evaluating ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Gamma[]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" while locating the minimum, even though by default it only \ printed 6 decimal places. But in general you should practice suspicion in \ circumstances like this, and not trust results blindly.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell["Removing definitions: =. and Clear[] and Remove[]", "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "[", StyleBox["Mathematica", FontSlant->"Italic"], " Book: 1.2.2, 1.4.2]" }], "Text"], Cell[TextData[{ StyleBox["Note", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"], StyleBox[": This section and the next wander off from the topic of gamma \ functions. They discuss some important ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" operators that we'll need to understand. Roughly the same \ material is discussed in Section 1.4.2 of the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" book.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Before we go on, we're going to remove the definition that we \ made earlier for ", Evaluatable->False, AspectRatioFixed->True], StyleBox["z", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[". Otherwise every ", Evaluatable->False, AspectRatioFixed->True], StyleBox["z", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" from now on would get replaced by ", Evaluatable->False, AspectRatioFixed->True], StyleBox["2 - 3\[ImaginaryI]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[". It's ", Evaluatable->False, AspectRatioFixed->True], StyleBox["very", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" important to remember to remove old definitions when you want to \ reuse a name; not doing so is probably the most common beginner's error in ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[". For example, the following innocent-looking command fails \ because ", Evaluatable->False, AspectRatioFixed->True], StyleBox["z", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" has a fixed value already:", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["Integrate[1/(1+z^2), {z,0,1}]", "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["You can remove a definition with the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["unset", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" operator \"", Evaluatable->False, AspectRatioFixed->True], StyleBox["=.", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["\", so here we need:", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["z =.", "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Now the above ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Integrate[]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" will work; try it again.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["You're allowed to leave a space between the \"", Evaluatable->False, AspectRatioFixed->True], StyleBox["=", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["\" and the \"", Evaluatable->False, AspectRatioFixed->True], StyleBox[".", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["\", so I could have written \"", Evaluatable->False, AspectRatioFixed->True], StyleBox["z = .", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["\", but it's still really only one operator. You should think of \ it as unsetting or undefining the left hand side, not as setting it equal to \ \".\".", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Another ways to remove a value for ", Evaluatable->False, AspectRatioFixed->True], StyleBox["z", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" are ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Clear[z]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Remove[z]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[", but there are not quite the same in general, though here any \ would do OK.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Clear[", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["symbol", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold", FontSlant->"Italic"], StyleBox["]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" or ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Remove[", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["symbol", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold", FontSlant->"Italic"], StyleBox["]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" only works on ", Evaluatable->False, AspectRatioFixed->True], StyleBox["symbols", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[", and clears ", Evaluatable->False, AspectRatioFixed->True], StyleBox["all", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" the associated ", Evaluatable->False, AspectRatioFixed->True], StyleBox["definition", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["s. ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Remove[", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["symbol", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold", FontSlant->"Italic"], StyleBox["]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" will also clears the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["attributes, options and messages", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[", but we won't discuss them here. ", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["\"", Evaluatable->False, AspectRatioFixed->True], StyleBox["lhs", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" =.", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["\" removes any definitions for ", Evaluatable->False, AspectRatioFixed->True], StyleBox["lhs", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[", which could be anything that could appear on the left-hand-side \ of an ", Evaluatable->False, AspectRatioFixed->True], StyleBox["=", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" or a ", Evaluatable->False, AspectRatioFixed->True], StyleBox[":=", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" operator.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["For example, let's define some rules for a symbol ", Evaluatable->False, AspectRatioFixed->True], StyleBox["a", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[":", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ a[1] := 0; a[n_] := n^2;\ \>", "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Note that the first rule would be used for ", Evaluatable->False, AspectRatioFixed->True], StyleBox["a[1]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[", but the second rule would be used for ", Evaluatable->False, AspectRatioFixed->True], StyleBox["a[2]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[". The general idea is that the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["most specific", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" applicable rule is used. Try it out:", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["{a[1], a[2], a[3], a[4]}", "Input", AspectRatioFixed->True], Cell[TextData[{ "(A compact way to try several things at once is to put them in a list like \ this.) ", StyleBox["We can see the rules for ", Evaluatable->False, AspectRatioFixed->True], StyleBox["a", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" with:", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(\(?a\)\)], "Input"], Cell[TextData[{ StyleBox["Suppose that we now wanted to remove the first rule. ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Clear[a[1]]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" does ", Evaluatable->False, AspectRatioFixed->True], StyleBox["not", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" work, because it only applies to ", Evaluatable->False, AspectRatioFixed->True], StyleBox["symbols", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[", and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["a[1]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" is not a plain symbol: ", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Clear[a[1]]\)], "Input"], Cell[TextData[{ StyleBox["But ", Evaluatable->False, AspectRatioFixed->True], StyleBox["lhs", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" =.", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" does:", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["a[1] =.", "Input", AspectRatioFixed->True], Cell[TextData[{ "Now the ", StyleBox["second", FontSlant->"Italic"], " rule applies:" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["{a[1], a[2], a[3], a[4]}", "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["We can see a rule remaining for ", Evaluatable->False, AspectRatioFixed->True], StyleBox["a", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" with:", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["?a", "Input", AspectRatioFixed->True], Cell["Good. Of course we can always use", "Text", Evaluatable->False, AspectRatioFixed->True], Cell["Clear[a]", "Input", AspectRatioFixed->True], Cell["?a", "Input", AspectRatioFixed->True], Cell["or", "Text", Evaluatable->False, AspectRatioFixed->True], Cell["Remove[a]", "Input", AspectRatioFixed->True], Cell[BoxData[ \(\(?a\)\)], "Input"], Cell[TextData[{ StyleBox["to clear all the rules associated with ", Evaluatable->False, AspectRatioFixed->True], StyleBox["a", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" if that's what we want. But remember that \"", Evaluatable->False, AspectRatioFixed->True], StyleBox["lhs", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" =.", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["\" is a lot more flexible, and can be used for selective editing \ of rules.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell["Rules(->) and Replacements(/.) ", "Section"], Cell[TextData[{ "[", StyleBox["Mathematica", FontSlant->"Italic"], " Book: 1.4.2]" }], "Text"], Cell[TextData[{ StyleBox["One approach to avoiding trouble after making definitions like ", Evaluatable->False, AspectRatioFixed->True], StyleBox["z = 2 - 3\[ImaginaryI] ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["(and forgetting to remove them) is not to make them at all! Some \ ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" purists avoid such definitions almost entirely. The idea is to \ work only with symbols (like ", Evaluatable->False, AspectRatioFixed->True], StyleBox["z", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["), and to do all your calculations symbolically, only \ substituting in numbers at the end (if at all). I don't advocate such a \ purist approach, but it is useful to avoid definitions in some cases.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["But how do you substitute in numbers without defining a value for \ ", Evaluatable->False, AspectRatioFixed->True], StyleBox["z", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["? I'm glad you asked. That's where the replacement operator ", Evaluatable->False, AspectRatioFixed->True], StyleBox["/.", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" comes in. ", Evaluatable->False, AspectRatioFixed->True], StyleBox["/.", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" is one of the most important operators in ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" after ", Evaluatable->False, AspectRatioFixed->True], StyleBox["=", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True], StyleBox[":=", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[". It specifies the replacement of a symbol (or subexpression) in \ an expression using a ", Evaluatable->False, AspectRatioFixed->True], StyleBox["rule", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" specified with \[Rule] (or ", Evaluatable->False, AspectRatioFixed->True], StyleBox["->", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["). You've already seen that options to functions like ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Plot[]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" are specified by such ", Evaluatable->False, AspectRatioFixed->True], StyleBox["rules", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" (e.g. ", Evaluatable->False, AspectRatioFixed->True], StyleBox["PlotPoints\[Rule]60", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["). The usual syntax for a ", Evaluatable->False, AspectRatioFixed->True], StyleBox["/.", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" replacement is: ", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["expression", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic"], StyleBox[" /. ", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox["symbol", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic"], StyleBox[" \[Rule] ", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox["replacement", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic"] }], "Print", Evaluatable->False, GeneratedCell->False, CellAutoOverwrite->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Every occurrence of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["symbol", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" in ", Evaluatable->False, AspectRatioFixed->True], StyleBox["expression", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" is replaced by ", Evaluatable->False, AspectRatioFixed->True], StyleBox["replacement", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold", FontSlant->"Italic"], StyleBox[". This is usually read as \"", Evaluatable->False, AspectRatioFixed->True], StyleBox["expression", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" at the point where ", Evaluatable->False, AspectRatioFixed->True], StyleBox["symbol", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" goes to ", Evaluatable->False, AspectRatioFixed->True], StyleBox["replacement", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["\". Thinking of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["/.", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" as \"", Evaluatable->False, AspectRatioFixed->True], StyleBox["at the point where", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["\"", Evaluatable->False, AspectRatioFixed->True, FontSize->12], StyleBox[" makes it easy to remember. For example:", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(x^2 /. x \[Rule] 3\)], "Input"], Cell["Sin[5 t] /. t \[Rule] 2.1", "Input", AspectRatioFixed->True], Cell["x^2 Log[x + y] /. x \[Rule] a", "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["This is very different from using a definition (assignment) like \ ", Evaluatable->False, AspectRatioFixed->True], StyleBox["x = 3", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" or ", Evaluatable->False, AspectRatioFixed->True], StyleBox[" ", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"], StyleBox["t = 2.1", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" which would ", Evaluatable->False, AspectRatioFixed->True], StyleBox["permanently", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" change ", Evaluatable->False, AspectRatioFixed->True], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" or ", Evaluatable->False, AspectRatioFixed->True], StyleBox["t", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" (until changed again). ", Evaluatable->False, AspectRatioFixed->True], StyleBox["/.", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" only makes the replacement for the expression at its left.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "M", StyleBox["ore than one replacement rule may be specified after ", Evaluatable->False, AspectRatioFixed->True], StyleBox["/.", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"], StyleBox[" ", Evaluatable->False, AspectRatioFixed->True, FontSize->12], StyleBox["by putting them in a list. For example:", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ x^2 + 3 x y /. {x \[Rule] r Cos[theta], y \[Rule] r \ Sin[theta]}\ \>", "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Besides making substitutions to specialize or simplify an \ equation, you also need to use the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["/.", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" replacement operator to extract actual values when a ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" function returns a rule. For example, we found the location of \ the minimum of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[CapitalGamma]", Evaluatable->False, AspectRatioFixed->True, FontSize->12], StyleBox["(x) between 1 and 2 with:", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["FindMinimum[Gamma[t], {t,1,2}]", "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["The result is returned as a list, of which the second element is \ a ", Evaluatable->False, AspectRatioFixed->True], StyleBox["rule", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" for t. Suppose you now want to calculate t-1 at that point; you \ need to extract a value for t itself. The following is the standard way:", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(%[\([2]\)]\)], "Input"], Cell["t-1 /. %", "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Make sure you understand that; it's a very common idiom. In the \ expression ", Evaluatable->False, AspectRatioFixed->True], StyleBox["%", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" means the result of the previous expression. More precisely it \ means the result of the last expression ", Evaluatable->False, AspectRatioFixed->True], StyleBox["evaluated", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[", which is not necessarily the one immediately before it in the \ Notebook. Be careful with evaluation order when there are ", Evaluatable->False, AspectRatioFixed->True], StyleBox["%", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox["'s around.", Evaluatable->False, AspectRatioFixed->True], " ", StyleBox["The ", Evaluatable->False, AspectRatioFixed->True], StyleBox["[[2]]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" takes the second element of the list referred to as ", Evaluatable->False, AspectRatioFixed->True], StyleBox["%", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[". That still leaves an unneccessary extra pair of braces ", Evaluatable->False, AspectRatioFixed->True], StyleBox["{...}", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[", but that doesn't matter because ", Evaluatable->False, AspectRatioFixed->True], StyleBox["/.", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" is willing to accept a list (even with only one argument) for \ its right-hand-side.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["Here's the whole problem as one evaluation:", "Text"], Cell["t-1 /. FindMinimum[Gamma[t], {t,1,2}][[2]]", "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Make sure you understand that; it's a very common idiom. If you \ still feel hazy please read section 1.4.2 of the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" book.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Problem 2", FontColor->RGBColor[1, 0, 1]]], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Locate the minima and maxima of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[CapitalGamma]", Evaluatable->False, AspectRatioFixed->True, FontSize->12], StyleBox["(x) between -10 and 0. Y", Evaluatable->False, AspectRatioFixed->True], "ou may find ", StyleBox["Table[...,{n,-9,0}]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " helpful. You might want to read section 3.9.8 of the ", StyleBox["Mathematica", FontSlant->"Italic"], " book." }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell["Digamma, PolyGamma, and Euler's constant", "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "[", StyleBox["Mathematica", FontSlant->"Italic"], " Book: 3.2.10, 3.9.4]" }], "Text"], Cell[TextData[{ StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" can evaluate the derivative of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Gamma[x]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" symbolically:", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["D[Gamma[x], x]", "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["What ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" calls ", Evaluatable->False, AspectRatioFixed->True], StyleBox["PolyGamma[0, x]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" is normally called the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["digamma function", Evaluatable->False, AspectRatioFixed->True, FontVariations->{"Underline"->True}], StyleBox[" \[Psi](x), and is the logarithmic derivative of the gamma \ function:", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(\[Psi] \((x)\)\ = \ \ \(\(dlog\ \[CapitalGamma] \((x)\)\)\/dx = \ \(\(\ \[CapitalGamma]\^'\) \((x)\)\)\/\(\[CapitalGamma] \((x)\)\)\)\)], "Text", FontSize->14], Cell[TextData[{ StyleBox["The n", Evaluatable->False, AspectRatioFixed->True], StyleBox["th", Evaluatable->False, AspectRatioFixed->True, FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox[" derivative of \[Psi](x) -- i.e. the (n+1)", Evaluatable->False, AspectRatioFixed->True], StyleBox["th", Evaluatable->False, AspectRatioFixed->True, FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox[" derivative of log \[CapitalGamma][x] -- is what is commonly \ called the polygamma function, written as \[Psi]", Evaluatable->False, AspectRatioFixed->True], StyleBox["n", Evaluatable->False, AspectRatioFixed->True, FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["(x):", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(\(\[Psi]\_n\) \((x)\)\ = \ \(\(\(d\^n\) \[Psi] \((x)\)\)\/dx\^n = \ \(\ \(d\^\(n + 1\)\) log\ \[CapitalGamma] \((x)\)\)\/dx\^\(n + 1\)\)\)], "Text", FontSize->14], Cell[TextData[{ StyleBox["The ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" notation for this is ", Evaluatable->False, AspectRatioFixed->True], StyleBox["PolyGamma[n, x]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[". Thus the digamma function \[Psi](x) is \[Psi]", Evaluatable->False, AspectRatioFixed->True], StyleBox["0", Evaluatable->False, AspectRatioFixed->True, FontSize->12, FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["(x), or ", Evaluatable->False, AspectRatioFixed->True], StyleBox["PolyGamma[0,x]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[", which may also be abbreviated to ", Evaluatable->False, AspectRatioFixed->True], StyleBox["PolyGamma[x]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[".", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Check that Mathematica knows all this by evaluating one of the \ higher derivatives of ", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["Log[Gamma[x]]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[". (If you don't know how to do multiple derivatives, ", Evaluatable->False, AspectRatioFixed->True], "you can choose the ", StyleBox["?", FontWeight->"Bold"], StyleBox["symbol", FontWeight->"Bold", FontSlant->"Italic"], " command", StyleBox[".)", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ Here are some interesting values of the digamma and polygamma \ functions: \ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell["PolyGamma[1.46163]", "Input", AspectRatioFixed->True], Cell["PolyGamma[1]", "Input", AspectRatioFixed->True], Cell["% // N", "Input", AspectRatioFixed->True], Cell["PolyGamma[2]", "Input", AspectRatioFixed->True], Cell[BoxData[ \(PolyGamma[1, \ 1]\)], "Input"], Cell[TextData[StyleBox["The first value is of course the minimum point for \ \[CapitalGamma](z), where the digamma function should go through 0. And so \ it does, more or less.", Evaluatable->False, AspectRatioFixed->True]], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["The constant ", Evaluatable->False, AspectRatioFixed->True], StyleBox["EulerGamma", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" is called Euler's constant, with value 0.577216 as you saw \ above. It's usually written as ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Gamma]", Evaluatable->False, AspectRatioFixed->True, FontSize->12], StyleBox[" (gamma). It comes up all over the place, especially in \ connection with infinite series, gamma functions, and the exponential \ integral. Its standard definition is as the limit of", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["\t1 + 1/2 + 1/3 + 1/4 +... + 1/n - log n"], "Print", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "as n goes to infinity. The series part alone isn't convergent, but the \ subtraction of log n makes it finite in the limit."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "Summing this series is definitely not the best way to calculate Euler's \ constant -- it's extremely slow to converge. But let's try it anyway, first \ defining ", Evaluatable->False, AspectRatioFixed->True], StyleBox["euler[n]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[ " as the approximation given above (using n terms of the series). ", Evaluatable->False, AspectRatioFixed->True], StyleBox["NSum[", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["expr", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold", FontSlant->"Italic"], StyleBox[", {k,n}]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" computes a numerical sum of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["expr", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" with ", Evaluatable->False, AspectRatioFixed->True], StyleBox["k", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" going from 1 to ", Evaluatable->False, AspectRatioFixed->True], StyleBox["n", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[".", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[" euler[n_] := NSum[1/k,{k,n}] - N[Log[n]]", "Input", AspectRatioFixed->True], Cell[" euler[10]", "Input", AspectRatioFixed->True], Cell[" euler[100]", "Input", AspectRatioFixed->True], Cell[" euler[1000]", "Input", AspectRatioFixed->True], Cell[" euler[10000]", "Input", AspectRatioFixed->True], Cell[" euler[100000]", "Input", AspectRatioFixed->True], Cell[" euler[1000000]", "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "Were you surprised at how fast it was to sum a million terms? You should \ be! In fact ", Evaluatable->False, AspectRatioFixed->True], StyleBox["NSum[]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[ " is clever and constructs an approximation to the sum; it didn't really \ add up all those terms. If you really wanted to add up every term you could \ use ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Sum[...]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" instead of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["NSum[...]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[". But don't try that for a million terms (10000 is OK).", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Problem 3", FontColor->RGBColor[1, 0, 1]]], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox["The digamma function \[Psi](x) approaches ln x for \ large x. The difference is of order 1/x, so we can write", Evaluatable->False, AspectRatioFixed->True]], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["\[Psi]", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontColor->GrayLevel[0]], StyleBox["(x) = ln x + C/x + O(1/x^2)", Evaluatable->False, AspectRatioFixed->True, FontColor->GrayLevel[0]] }], "Message", Evaluatable->False, GeneratedCell->False, CellAutoOverwrite->False, AspectRatioFixed->True, FontSize->14], Cell[TextData[{ "(where the last term reads \"plus terms of order 1 over x squared\"). \ Find the value of the coefficient C ", StyleBox["numerically", FontSlant->"Italic"], " by making a suitable plot from which you can read off the value." }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell["Beta function", "Section", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ The Beta function often comes up in integrals of trigonometric \ functions and of powers. For example:\ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell["Integrate[x^m (1-x)^n, {x,0,1}]", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "Of course, we want ", StyleBox["m", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " and ", StyleBox["n", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " are \"nice\" (i.e. ", StyleBox["Re[m]>-1 && Re[n]>-1", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], "). If we like to see at first:" }], "Text"], Cell[TextData[{ "Integrate[x^m (1-x)^n, {x,0,1},\nAssumptions\[Rule]{", StyleBox["Re[m]>-1 && Re[n]>-1", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], "}]" }], "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "That's normally written as B(m+1, n+1) (where B is an uppercase ", StyleBox["\[Beta]", FontSize->12], ", which is indistinguisable from an uppercase b). In general" }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ \[CapitalGamma](p) \[CapitalGamma](q) B(p,q) = ---------- (*) \[CapitalGamma](p+q)\ \>", "Print", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " uses ", StyleBox["Beta[p,q]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " for this, and can often give you an exact value. For example:" }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell["Beta[2/3,3]", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell["Beta[2+3\[ImaginaryI],4]", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ Of course we could always use the above relation (*) to evaluate \ things like this, but we typically won't get such a nice rational answer. \ For the preceding examples, for instance:\ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell["Gamma[2/3] Gamma[3] / Gamma[3+2/3]", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ Gamma[2+3\[ImaginaryI]] Gamma[4]/ Gamma[6+3\[ImaginaryI]]\ \>", \ "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "You can see that it tried to evaluate each gamma function individually, \ only succeeding for integer and half-integer values. In the second case, for \ example, the ", StyleBox["ratio", FontSlant->"Italic"], " of ", StyleBox["Gamma[2/3]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " and ", StyleBox["Gamma[11/3]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " has a nice rational value (using the recursion relation) even though \ neither ", StyleBox["Gamma[]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " does individually." }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "Of course you can check ", StyleBox["numerically", FontSlant->"Italic"], " that (for example) ", StyleBox["Beta[2/3,3]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " and ", StyleBox["Gamma[2/3]Gamma[3]/Gamma[3+2/3]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " are equal:" }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell["Beta[2/3,3] - Gamma[2/3]Gamma[3]/Gamma[3+2/3] //N", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "That's close enough, isn't it? The ", StyleBox["//N", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " (or ", StyleBox["N[", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["expr", FontFamily->"Courier", FontSize->12, FontWeight->"Bold", FontSlant->"Italic"], StyleBox["]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], ") made ", StyleBox["Mathematica", FontSlant->"Italic"], " evaluate the two terms individually using the default machine precision \ (probably about 16 decimal places), and then subtract them. They weren't \ quite equal to the last digit due to \"rounding errors\" so we didn't quite \ get zero when we subtracted. You can check the normal machine precision:" }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \($MachinePrecision\)], "Input"], Cell[TextData[{ "What happens if we use ", StyleBox["N[", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["expr,digits", FontFamily->"Courier", FontSize->12, FontWeight->"Bold", FontSlant->"Italic"], StyleBox["]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], "? Well it depends whether you are using ", StyleBox["Mathematica", FontSlant->"Italic"], " Version 5 or before. It is much better in Version 5. You can check your \ version and then read the right subsection" }], "Text"], Cell[BoxData[ \($Version\)], "Input"], Cell[CellGroupData[{ Cell["Version 1-4", "Subsubsection"], Cell[TextData[{ StyleBox["N[", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["expr", FontSize->12, FontSlant->"Italic"], StyleBox[",", FontFamily->"Courier", FontSize->12, FontWeight->"Bold", FontSlant->"Italic"], StyleBox["digits", FontSize->12, FontSlant->"Italic"], StyleBox["]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " if ", StyleBox["digits", FontSlant->"Italic"], " \[LessEqual] ", StyleBox["$MachinePrecision", FontFamily->"Courier", FontWeight->"Bold"], ", then it just uses the machine precision internally. This does ", StyleBox["not", FontSlant->"Italic"], " specify the precision of the final result (i.e. ", StyleBox["expr", FontSlant->"Italic"], "); we get the same results:" }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell["N[Beta[2/3,3] - Gamma[2/3]Gamma[3]/Gamma[3+2/3], 4]", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell["N[Beta[2/3,3] - Gamma[2/3]Gamma[3]/Gamma[3+2/3], 10]", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "This is not very good, but ", StyleBox["Mathematica", FontSlant->"Italic"], " made it better in Version 5." }], "Text"], Cell[TextData[{ " ", StyleBox["N[", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["expr", FontSize->12, FontSlant->"Italic"], StyleBox[",", FontFamily->"Courier", FontSize->12, FontWeight->"Bold", FontSlant->"Italic"], StyleBox["digits", FontSize->12, FontSlant->"Italic"], StyleBox["]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " if ", StyleBox["digits", FontSlant->"Italic"], " > ", StyleBox["$MachinePrecision", FontFamily->"Courier", FontWeight->"Bold"], ", then it uses more intermediate calculations to attempt to give a final \ result (i.e. ", StyleBox["expr", FontSlant->"Italic"], ") with a ", StyleBox["digi", FontSlant->"Italic"], "t-precision. But it uses up to ", StyleBox["$MaxExtraPrecision", "MR", FontWeight->"Bold"], " gives the maximum number of extra digits of precision for intermediate \ calculations, and if that's not enough we get a error:" }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell["N[Beta[2/3,3] - Gamma[2/3]Gamma[3]/Gamma[3+2/3], 20]", "Input", ImageRegion->{{0, 1}, {0, 1}}] }, Closed]], Cell[CellGroupData[{ Cell["Version 5", "Subsubsection"], Cell[TextData[{ " ", StyleBox["N[", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["expr", FontSize->12, FontSlant->"Italic"], StyleBox["]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " uses machine-precision numbers." }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ " ", StyleBox["N[", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["expr", FontSize->12, FontSlant->"Italic"], StyleBox[",", FontFamily->"Courier", FontSize->12, FontWeight->"Bold", FontSlant->"Italic"], StyleBox["digits", FontSize->12, FontSlant->"Italic"], StyleBox["]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " uses arbitrary-precision numbers, using more intermediate calculations to \ attempt to give a final result (i.e. ", StyleBox["expr", FontSlant->"Italic"], ") with a ", StyleBox["digi", FontSlant->"Italic"], "t-precision. But it uses up to ", StyleBox["$MaxExtraPrecision", "MR", FontWeight->"Bold"], " gives the maximum number of extra digits of precision for intermediate \ calculations, and if that's not enough we get a error:" }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell["N[Beta[2/3,3] - Gamma[2/3]Gamma[3]/Gamma[3+2/3], 4]", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell["N[Beta[2/3,3] - Gamma[2/3]Gamma[3]/Gamma[3+2/3], 20]", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "You can read more from the \"", StyleBox["More...", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}], "\" help", ", or read ", StyleBox["The Mathematica Book", FontSlant->"Italic"], " in section 3.1.4-6." }], "Text"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Beta plots", "Section", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "Here is the obvious plot of ", StyleBox["Beta[n, m]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " in the n-m plane. It's infinite when either n or m is zero (why?), so \ we'll start at 0.5:" }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell["Plot3D[Beta[n,m], {n,0.5,3},{m,0.5,3}]", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ A nice baby ski slope, huh? You can make it a bit more challenging \ by starting closer to 0 (i.e. reduce the 0.5's).\ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "If you want to see something really weird, try plotting it in a range \ including negative n's or m's. You may get some error messages as it hits \ the poles (whenever n and/or m is a non-positive integer), and may want to \ add an option like ", StyleBox["PlotPoints\[Rule]30", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], ". Not recommended for skiing--stay out of the back-country!" }], "Text", ImageRegion->{{0, 1}, {0, 1}}] }, Closed]], Cell[CellGroupData[{ Cell["Summary of Mathematica Functions/Idioms", "Section"], Cell[TextData[{ StyleBox["function", FontSlant->"Italic"], StyleBox["[", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["argument", FontSlant->"Italic"], StyleBox["_", FontFamily->"Courier", FontSize->12, FontWeight->"Bold", FontSlant->"Italic"], StyleBox["] := ", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["expr", FontSlant->"Italic"] }], "Text"], Cell[TextData[{ StyleBox["expr ", FontSlant->"Italic"], StyleBox["/.", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" symbol ", FontSlant->"Italic"], StyleBox["\[Rule]", FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" replacement", FontSlant->"Italic"] }], "Text"], Cell[TextData[{ StyleBox["expr", FontSlant->"Italic"], StyleBox["[[", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["i", FontSlant->"Italic"], StyleBox["]]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " (gives the ", Cell[BoxData[ SuperscriptBox[ StyleBox["i", FontSlant->"Italic"], "th"]]], " part of expression)" }], "Text"], Cell[TextData[{ StyleBox["lhs", FontSlant->"Italic"], " ", StyleBox["=.\nClear[", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["symbol", FontSlant->"Italic"], StyleBox["]\nRemove[", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["symbol", FontSlant->"Italic"], StyleBox["]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"] }], "Text"], Cell[TextData[{ StyleBox["expr", FontSlant->"Italic"], StyleBox["//N\nN[", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["expr", FontSlant->"Italic"], StyleBox["]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " (uses machine-precision numbers)", StyleBox["\nN[", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["expr,digits", FontSlant->"Italic"], StyleBox["]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " (uses arbitrary-precision numbers for Version 5)" }], "Text"], Cell[TextData[{ StyleBox["Table[", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["expr", FontSlant->"Italic"], StyleBox[",{", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["i,imax", FontSlant->"Italic"], StyleBox["}", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], "\n", StyleBox["Table[", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["expr", FontSlant->"Italic"], StyleBox[",{", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["i,imin,imax", FontSlant->"Italic"], StyleBox["}", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], "\n", StyleBox["Table[", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["expr", FontSlant->"Italic"], StyleBox[",{", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["i,imin,imax,di", FontSlant->"Italic"], StyleBox["}", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"] }], "Text"], Cell[TextData[{ StyleBox["Re[", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["expr", FontSlant->"Italic"], StyleBox["]\nIm[", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["expr", FontSlant->"Italic"], StyleBox["]\nAbs[", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["expr", FontSlant->"Italic"], StyleBox["]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"] }], "Text", Evaluatable->False], Cell[TextData[{ StyleBox["Plot[", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["expr", FontSlant->"Italic"], StyleBox[",{", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["x,xmin,xmax", FontSlant->"Italic"], StyleBox["}]\nPlot3D[", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["expr", FontSlant->"Italic"], StyleBox[",{", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["x,xmin,xmax", FontSlant->"Italic"], StyleBox["},{", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["y,ymin,ymax", FontSlant->"Italic"], StyleBox["}]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"] }], "Text"], Cell[TextData[{ StyleBox["FindMinimum[", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["expr", FontSlant->"Italic"], StyleBox[",{", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["x,x0", FontSlant->"Italic"], StyleBox["}]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " (starting from ", StyleBox["x0", FontSlant->"Italic"], ")\n", StyleBox["FindMinimum[", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["expr", FontSlant->"Italic"], StyleBox[",{", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["x,x1,x2", FontSlant->"Italic"], StyleBox["}]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " (starting between ", StyleBox["x1", FontSlant->"Italic"], " to ", StyleBox["x2", FontSlant->"Italic"], ")" }], "Text"], Cell[TextData[{ StyleBox["D[", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["expr", FontSlant->"Italic"], StyleBox[",x", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["]\n", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["Integrate[", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["expr", FontSlant->"Italic"], StyleBox[",{x,x0,x1}", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["]\n", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["Integrate[", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["expr", FontSlant->"Italic"], StyleBox[",{x,x0,x1}", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[",Assumptions\[Rule]{", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["assumptions", FontSlant->"Italic"], StyleBox["}]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"] }], "Text"], Cell[TextData[{ StyleBox["NSum[", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["expr", FontSlant->"Italic"], StyleBox[",{", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["k,n", FontSlant->"Italic"], StyleBox["}", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["]", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " (k=1, 2, ...,n)" }], "Text"] }, Closed]] }, Open ]] }, FrontEndVersion->"4.2 for X", ScreenRectangle->{{0, 1280}, {0, 1024}}, WindowToolbars->"RulerBar", WindowSize->{520, 627}, WindowMargins->{{47, Automatic}, {66, Automatic}}, PrintingPageRange->{Automatic, Automatic}, PrintingOptions->{"PaperSize"->{612, 792}, "PaperOrientation"->"Portrait", "Magnification"->1}, PrivateNotebookOptions->{"ColorPalette"->{RGBColor, -1}}, ShowCellLabel->True, ShowCellTags->False, RenderingOptions->{"ObjectDithering"->True, "RasterDithering"->False}, StyleDefinitions -> Notebook[{ Cell[CellGroupData[{ Cell["Style Definitions", "Subtitle"], Cell["\<\ Modify the definitions below to change the default appearance of \ all cells in a given style. Make modifications to any definition using \ commands in the Format menu.\ \>", "Text"], Cell[CellGroupData[{ Cell["Style Environment Names", "Section"], Cell[StyleData[All, "Working"], PageWidth->WindowWidth, CellLabelMargins->{{12, Inherited}, {Inherited, Inherited}}, ScriptMinSize->9], Cell[StyleData[All, "Presentation"], PageWidth->WindowWidth, CellLabelMargins->{{24, Inherited}, {Inherited, Inherited}}, ScriptMinSize->12], Cell[StyleData[All, "Condensed"], PageWidth->WindowWidth, CellLabelMargins->{{8, Inherited}, {Inherited, Inherited}}, ScriptMinSize->8], Cell[StyleData[All, "Printout"], PageWidth->PaperWidth, CellLabelMargins->{{2, Inherited}, {Inherited, Inherited}}, ScriptMinSize->5, PrivateFontOptions->{"FontType"->"Outline"}] }, Closed]], Cell[CellGroupData[{ Cell["Notebook Options", "Section"], Cell["\<\ The options defined for the style below will be used at the \ Notebook level.\ \>", "Text"], Cell[StyleData["Notebook"], PageHeaders->{{Cell[ TextData[ { CounterBox[ "Page"]}], "PageNumber"], None, Cell[ TextData[ { ValueBox[ "FileName"]}], "Header"]}, {Cell[ TextData[ { ValueBox[ "FileName"]}], "Header"], None, Cell[ TextData[ { CounterBox[ "Page"]}], "PageNumber"]}}, CellFrameLabelMargins->6, StyleMenuListing->None] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Headings", "Section", FontColor->RGBColor[0, 0, 1]], Cell[CellGroupData[{ Cell[StyleData["Title"], CellMargins->{{12, Inherited}, {20, 40}}, CellGroupingRules->{"TitleGrouping", 0}, PageBreakBelow->False, DefaultNewInlineCellStyle->"None", InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-0.36, -0.1}, {0, -0}}, BoxBaselineShift -> -0.2], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica"}, LanguageCategory->"NaturalLanguage", CounterIncrements->"Title", CounterAssignments->{{"Section", 0}, {"Equation", 0}, {"Figure", 0}, { "Subtitle", 0}, {"Subsubtitle", 0}}, FontFamily->"Helvetica", FontSize->36, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], Cell[StyleData["Title", "Presentation"], CellMargins->{{24, 10}, {20, 40}}, LineSpacing->{1, 0}, FontSize->44], Cell[StyleData["Title", "Condensed"], CellMargins->{{8, 10}, {4, 8}}, FontSize->20], Cell[StyleData["Title", "Printout"], CellMargins->{{2, 10}, {12, 30}}, FontSize->24] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subtitle"], CellMargins->{{12, Inherited}, {20, 15}}, CellGroupingRules->{"TitleGrouping", 10}, PageBreakBelow->False, DefaultNewInlineCellStyle->"None", InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-0.36, -0.1}, {0, -0}}, BoxBaselineShift -> -0.2], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica"}, LanguageCategory->"NaturalLanguage", CounterIncrements->"Subtitle", CounterAssignments->{{"Section", 0}, {"Equation", 0}, {"Figure", 0}, { "Subsubtitle", 0}}, FontFamily->"Helvetica", FontSize->24, FontColor->RGBColor[0, 0, 1]], Cell[StyleData["Subtitle", "Presentation"], CellMargins->{{24, 10}, {20, 20}}, LineSpacing->{1, 0}, FontSize->36], Cell[StyleData["Subtitle", "Condensed"], CellMargins->{{8, 10}, {4, 4}}, FontSize->14], Cell[StyleData["Subtitle", "Printout"], CellMargins->{{2, 10}, {12, 8}}, FontSize->18] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subsubtitle"], CellMargins->{{12, Inherited}, {20, 15}}, CellGroupingRules->{"TitleGrouping", 20}, PageBreakBelow->False, DefaultNewInlineCellStyle->"None", InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-0.36, -0.1}, {0, -0}}, BoxBaselineShift -> -0.2], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica"}, LanguageCategory->"NaturalLanguage", CounterIncrements->"Subsubtitle", CounterAssignments->{{"Section", 0}, {"Equation", 0}, {"Figure", 0}}, FontFamily->"Helvetica", FontSize->14, FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], Cell[StyleData["Subsubtitle", "Presentation"], CellMargins->{{24, 10}, {20, 20}}, LineSpacing->{1, 0}, FontSize->24], Cell[StyleData["Subsubtitle", "Condensed"], CellMargins->{{8, 10}, {8, 8}}, FontSize->12], Cell[StyleData["Subsubtitle", "Printout"], CellMargins->{{2, 10}, {12, 8}}, FontSize->14] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Section"], CellDingbat->"\[FilledSquare]", CellMargins->{{25, Inherited}, {8, 24}}, CellGroupingRules->{"SectionGrouping", 30}, PageBreakBelow->False, DefaultNewInlineCellStyle->"None", InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-0.36, -0.1}, {0, -0}}, BoxBaselineShift -> -0.2], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica"}, LanguageCategory->"NaturalLanguage", CounterIncrements->"Section", CounterAssignments->{{"Subsection", 0}, {"Subsubsection", 0}}, FontFamily->"Helvetica", FontSize->16, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], Cell[StyleData["Section", "Presentation"], CellMargins->{{40, 10}, {11, 32}}, LineSpacing->{1, 0}, FontSize->24], Cell[StyleData["Section", "Condensed"], CellMargins->{{18, Inherited}, {6, 12}}, FontSize->12], Cell[StyleData["Section", "Printout"], CellMargins->{{13, 0}, {7, 22}}, FontSize->14] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subsection"], CellDingbat->"\[FilledSmallSquare]", CellMargins->{{22, Inherited}, {8, 20}}, CellGroupingRules->{"SectionGrouping", 40}, PageBreakBelow->False, DefaultNewInlineCellStyle->"None", InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-0.36, -0.1}, {0, -0}}, BoxBaselineShift -> -0.2], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica"}, LanguageCategory->"NaturalLanguage", CounterIncrements->"Subsection", CounterAssignments->{{"Subsubsection", 0}}, FontFamily->"Times", FontSize->14, FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], Cell[StyleData["Subsection", "Presentation"], CellMargins->{{36, 10}, {11, 32}}, LineSpacing->{1, 0}, FontSize->22], Cell[StyleData["Subsection", "Condensed"], CellMargins->{{16, Inherited}, {6, 12}}, FontSize->12], Cell[StyleData["Subsection", "Printout"], CellMargins->{{9, 0}, {7, 22}}, FontSize->12] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subsubsection"], CellDingbat->"\[FilledSmallSquare]", CellMargins->{{22, Inherited}, {8, 18}}, CellGroupingRules->{"SectionGrouping", 50}, PageBreakBelow->False, DefaultNewInlineCellStyle->"None", InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-0.36, -0.1}, {0, -0}}, BoxBaselineShift -> -0.2], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica"}, LanguageCategory->"NaturalLanguage", CounterIncrements->"Subsubsection", FontFamily->"Times", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]], Cell[StyleData["Subsubsection", "Presentation"], CellMargins->{{34, 10}, {11, 26}}, LineSpacing->{1, 0}, FontSize->18], Cell[StyleData["Subsubsection", "Condensed"], CellMargins->{{17, Inherited}, {6, 12}}, FontSize->10], Cell[StyleData["Subsubsection", "Printout"], CellMargins->{{9, 0}, {7, 14}}, FontSize->11] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Body Text", "Section"], Cell[CellGroupData[{ Cell[StyleData["Text"], CellMargins->{{12, 10}, {7, 7}}, InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-0.36, -0.1}, {0, -0}}, BoxBaselineShift -> -0.2], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica"}, Hyphenation->True, LineSpacing->{1, 3}, CounterIncrements->"Text", FontSize->12], Cell[StyleData["Text", "Presentation"], CellMargins->{{24, 10}, {10, 10}}, LineSpacing->{1, 5}, FontSize->16], Cell[StyleData["Text", "Condensed"], CellMargins->{{8, 10}, {6, 6}}, LineSpacing->{1, 1}, FontSize->11], Cell[StyleData["Text", "Printout"], CellMargins->{{2, 2}, {6, 6}}, TextJustification->0.5, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["SmallText"], CellMargins->{{12, 10}, {6, 6}}, DefaultNewInlineCellStyle->"None", Hyphenation->True, LineSpacing->{1, 3}, LanguageCategory->"NaturalLanguage", CounterIncrements->"SmallText", FontFamily->"Helvetica", FontSize->9], Cell[StyleData["SmallText", "Presentation"], CellMargins->{{24, 10}, {8, 8}}, LineSpacing->{1, 5}, FontSize->12], Cell[StyleData["SmallText", "Condensed"], CellMargins->{{8, 10}, {5, 5}}, LineSpacing->{1, 2}, FontSize->9], Cell[StyleData["SmallText", "Printout"], CellMargins->{{2, 2}, {5, 5}}, TextJustification->0.5, FontSize->7] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Input/Output", "Section"], Cell["\<\ The cells in this section define styles used for input and output \ to the kernel. Be careful when modifying, renaming, or removing these \ styles, because the front end associates special meanings with these style \ names. Some attributes for these styles are actually set in FormatType Styles \ (in the last section of this stylesheet). \ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Input"], CellMargins->{{45, 10}, {5, 7}}, Evaluatable->True, CellGroupingRules->"InputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, DefaultFormatType->DefaultInputFormatType, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, LanguageCategory->"Formula", FormatType->InputForm, ShowStringCharacters->True, NumberMarks->True, LinebreakAdjustments->{0.85, 2, 10, 0, 1}, CounterIncrements->"Input", FontWeight->"Bold"], Cell[StyleData["Input", "Presentation"], CellMargins->{{72, Inherited}, {8, 10}}, LineSpacing->{1, 0}, FontSize->16], Cell[StyleData["Input", "Condensed"], CellMargins->{{40, 10}, {2, 3}}, FontSize->11], Cell[StyleData["Input", "Printout"], CellMargins->{{39, 0}, {4, 6}}, LinebreakAdjustments->{0.85, 2, 10, 1, 1}, FontSize->9] }, Closed]], Cell[StyleData["InputOnly"], Evaluatable->True, CellGroupingRules->"InputGrouping", CellHorizontalScrolling->True, DefaultFormatType->DefaultInputFormatType, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, LanguageCategory->"Formula", FormatType->InputForm, ShowStringCharacters->True, NumberMarks->True, LinebreakAdjustments->{0.85, 2, 10, 0, 1}, CounterIncrements->"Input", StyleMenuListing->None, FontWeight->"Bold"], Cell[CellGroupData[{ Cell[StyleData["Output"], CellMargins->{{47, 10}, {7, 5}}, CellEditDuplicate->True, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, DefaultFormatType->DefaultOutputFormatType, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, LanguageCategory->"Formula", FormatType->InputForm, CounterIncrements->"Output"], Cell[StyleData["Output", "Presentation"], CellMargins->{{72, Inherited}, {10, 8}}, LineSpacing->{1, 0}, FontSize->16], Cell[StyleData["Output", "Condensed"], CellMargins->{{41, Inherited}, {3, 2}}, FontSize->11], Cell[StyleData["Output", "Printout"], CellMargins->{{39, 0}, {6, 4}}, FontSize->9] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Message"], CellMargins->{{45, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"OutputGrouping", PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, DefaultFormatType->DefaultOutputFormatType, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, FormatType->InputForm, CounterIncrements->"Message", StyleMenuListing->None, FontSize->11, FontColor->RGBColor[0, 0, 1]], Cell[StyleData["Message", "Presentation"], CellMargins->{{72, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}, FontSize->16], Cell[StyleData["Message", "Condensed"], CellMargins->{{41, Inherited}, {Inherited, Inherited}}, FontSize->11], Cell[StyleData["Message", "Printout"], CellMargins->{{39, Inherited}, {Inherited, Inherited}}, FontSize->7, FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Print"], CellMargins->{{45, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, DefaultFormatType->DefaultOutputFormatType, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, FormatType->InputForm, CounterIncrements->"Print", StyleMenuListing->None], Cell[StyleData["Print", "Presentation"], CellMargins->{{72, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}, FontSize->16], Cell[StyleData["Print", "Condensed"], CellMargins->{{41, Inherited}, {Inherited, Inherited}}, FontSize->11], Cell[StyleData["Print", "Printout"], CellMargins->{{39, Inherited}, {Inherited, Inherited}}, FontSize->8] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Graphics"], CellMargins->{{4, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"GraphicsGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, DefaultFormatType->DefaultOutputFormatType, LanguageCategory->None, FormatType->InputForm, CounterIncrements->"Graphics", ImageMargins->{{43, Inherited}, {Inherited, 0}}, StyleMenuListing->None, FontFamily->"Courier", FontSize->10], Cell[StyleData["Graphics", "Presentation"], ImageMargins->{{62, Inherited}, {Inherited, 0}}], Cell[StyleData["Graphics", "Condensed"], ImageMargins->{{38, Inherited}, {Inherited, 0}}, Magnification->0.6], Cell[StyleData["Graphics", "Printout"], ImageMargins->{{30, Inherited}, {Inherited, 0}}, Magnification->0.8] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["CellLabel"], StyleMenuListing->None, FontFamily->"Helvetica", FontSize->9, FontColor->RGBColor[0, 0, 1]], Cell[StyleData["CellLabel", "Presentation"], FontSize->12], Cell[StyleData["CellLabel", "Condensed"], FontSize->9], Cell[StyleData["CellLabel", "Printout"], FontFamily->"Courier", FontSize->8, FontSlant->"Italic", FontColor->GrayLevel[0]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Inline Formatting", "Section"], Cell["\<\ These styles are for modifying individual words or letters in a \ cell exclusive of the cell tag.\ \>", "Text"], Cell[StyleData["RM"], StyleMenuListing->None, FontWeight->"Plain", FontSlant->"Plain"], Cell[StyleData["BF"], StyleMenuListing->None, FontWeight->"Bold"], Cell[StyleData["IT"], StyleMenuListing->None, FontSlant->"Italic"], Cell[StyleData["TR"], StyleMenuListing->None, FontFamily->"Times", FontWeight->"Plain", FontSlant->"Plain"], Cell[StyleData["TI"], StyleMenuListing->None, FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], Cell[StyleData["TB"], StyleMenuListing->None, FontFamily->"Times", FontWeight->"Bold", FontSlant->"Plain"], Cell[StyleData["TBI"], StyleMenuListing->None, FontFamily->"Times", FontWeight->"Bold", FontSlant->"Italic"], Cell[StyleData["MR"], StyleMenuListing->None, FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], Cell[StyleData["MO"], StyleMenuListing->None, FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Italic"], Cell[StyleData["MB"], StyleMenuListing->None, FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], Cell[StyleData["MBO"], StyleMenuListing->None, FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Italic"], Cell[StyleData["SR"], StyleMenuListing->None, FontFamily->"Helvetica", FontWeight->"Plain", FontSlant->"Plain"], Cell[StyleData["SO"], StyleMenuListing->None, FontFamily->"Helvetica", FontWeight->"Plain", FontSlant->"Italic"], Cell[StyleData["SB"], StyleMenuListing->None, FontFamily->"Helvetica", FontWeight->"Bold", FontSlant->"Plain"], Cell[StyleData["SBO"], StyleMenuListing->None, FontFamily->"Helvetica", FontWeight->"Bold", FontSlant->"Italic"], Cell[CellGroupData[{ Cell[StyleData["SO10"], StyleMenuListing->None, FontFamily->"Helvetica", FontSize->10, FontWeight->"Plain", FontSlant->"Italic"], Cell[StyleData["SO10", "Printout"], StyleMenuListing->None, FontFamily->"Helvetica", FontSize->7, FontWeight->"Plain", FontSlant->"Italic"], Cell[StyleData["SO10", "EnhancedPrintout"], StyleMenuListing->None, FontFamily->"Futura", FontSize->7, FontWeight->"Plain", FontSlant->"Italic"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Formulas and Programming", "Section"], Cell[CellGroupData[{ Cell[StyleData["InlineFormula"], CellMargins->{{10, 4}, {0, 8}}, CellHorizontalScrolling->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, LanguageCategory->"Formula", ScriptLevel->1, SingleLetterItalics->True], Cell[StyleData["InlineFormula", "Presentation"], CellMargins->{{24, 10}, {10, 10}}, LineSpacing->{1, 5}, FontSize->16], Cell[StyleData["InlineFormula", "Condensed"], CellMargins->{{8, 10}, {6, 6}}, LineSpacing->{1, 1}, FontSize->11], Cell[StyleData["InlineFormula", "Printout"], CellMargins->{{2, 0}, {6, 6}}, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["DisplayFormula"], CellMargins->{{42, Inherited}, {Inherited, Inherited}}, CellHorizontalScrolling->True, DefaultFormatType->DefaultInputFormatType, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, LanguageCategory->"Formula", ScriptLevel->0, SingleLetterItalics->True, UnderoverscriptBoxOptions->{LimitsPositioning->True}], Cell[StyleData["DisplayFormula", "Presentation"], LineSpacing->{1, 5}, FontSize->16], Cell[StyleData["DisplayFormula", "Condensed"], LineSpacing->{1, 1}, FontSize->11], Cell[StyleData["DisplayFormula", "Printout"], FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Program"], CellFrame->{{0, 0}, {0.5, 0.5}}, CellMargins->{{10, 4}, {0, 8}}, CellHorizontalScrolling->True, Hyphenation->False, LanguageCategory->"Formula", ScriptLevel->1, FontFamily->"Courier"], Cell[StyleData["Program", "Presentation"], CellMargins->{{24, 10}, {10, 10}}, LineSpacing->{1, 5}, FontSize->16], Cell[StyleData["Program", "Condensed"], CellMargins->{{8, 10}, {6, 6}}, LineSpacing->{1, 1}, FontSize->11], Cell[StyleData["Program", "Printout"], CellMargins->{{2, 0}, {6, 6}}, FontSize->9] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Hyperlink Styles", "Section"], Cell["\<\ The cells below define styles useful for making hypertext \ ButtonBoxes. The \"Hyperlink\" style is for links within the same Notebook, \ or between Notebooks.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Hyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2]}]&), Active->True, ButtonNote->ButtonData}], Cell[StyleData["Hyperlink", "Presentation"], FontSize->16], Cell[StyleData["Hyperlink", "Condensed"], FontSize->11], Cell[StyleData["Hyperlink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell["\<\ The following styles are for linking automatically to the on-line \ help system.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["MainBookLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "MainBook", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["MainBookLink", "Presentation"], FontSize->16], Cell[StyleData["MainBookLink", "Condensed"], FontSize->11], Cell[StyleData["MainBookLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["AddOnsLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontFamily->"Courier", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "AddOns", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["AddOnsLink", "Presentation"], FontSize->16], Cell[StyleData["AddOnsLink", "Condensed"], FontSize->11], Cell[StyleData["AddOnsLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["RefGuideLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontFamily->"Courier", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "RefGuide", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["RefGuideLink", "Presentation"], FontSize->16], Cell[StyleData["RefGuideLink", "Condensed"], FontSize->11], Cell[StyleData["RefGuideLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["GettingStartedLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "GettingStarted", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["GettingStartedLink", "Presentation"], FontSize->16], Cell[StyleData["GettingStartedLink", "Condensed"], FontSize->11], Cell[StyleData["GettingStartedLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["OtherInformationLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "OtherInformation", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["OtherInformationLink", "Presentation"], FontSize->16], Cell[StyleData["OtherInformationLink", "Condensed"], FontSize->11], Cell[StyleData["OtherInformationLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Headers and Footers", "Section"], Cell[StyleData["Header"], CellMargins->{{0, 0}, {4, 1}}, DefaultNewInlineCellStyle->"None", LanguageCategory->"NaturalLanguage", StyleMenuListing->None, FontSize->10, FontSlant->"Italic"], Cell[StyleData["Footer"], CellMargins->{{0, 0}, {0, 4}}, DefaultNewInlineCellStyle->"None", LanguageCategory->"NaturalLanguage", StyleMenuListing->None, FontSize->9, FontSlant->"Italic"], Cell[StyleData["PageNumber"], CellMargins->{{0, 0}, {4, 1}}, StyleMenuListing->None, FontFamily->"Times", FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell["Palette Styles", "Section"], Cell["\<\ The cells below define styles that define standard \ ButtonFunctions, for use in palette buttons.\ \>", "Text"], Cell[StyleData["Paste"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, After]}]&)}], Cell[StyleData["Evaluate"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], SelectionEvaluate[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["EvaluateCell"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionMove[ FrontEnd`InputNotebook[ ], All, Cell, 1], FrontEnd`SelectionEvaluateCreateCell[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["CopyEvaluate"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`SelectionCreateCell[ FrontEnd`InputNotebook[ ], All], FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionEvaluate[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["CopyEvaluateCell"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`SelectionCreateCell[ FrontEnd`InputNotebook[ ], All], FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionEvaluateCreateCell[ FrontEnd`InputNotebook[ ], All]}]&)}] }, Closed]], Cell[CellGroupData[{ Cell["Placeholder Styles", "Section"], Cell["\<\ The cells below define styles useful for making placeholder \ objects in palette templates.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Placeholder"], Placeholder->True, StyleMenuListing->None, FontSlant->"Italic", FontColor->RGBColor[0.890623, 0.864698, 0.384756], TagBoxOptions->{Editable->False, Selectable->False, StripWrapperBoxes->False}], Cell[StyleData["Placeholder", "Presentation"]], Cell[StyleData["Placeholder", "Condensed"]], Cell[StyleData["Placeholder", "Printout"]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["PrimaryPlaceholder"], StyleMenuListing->None, DrawHighlighted->True, FontSlant->"Italic", Background->RGBColor[0.912505, 0.891798, 0.507774], TagBoxOptions->{Editable->False, Selectable->False, StripWrapperBoxes->False}], Cell[StyleData["PrimaryPlaceholder", "Presentation"]], Cell[StyleData["PrimaryPlaceholder", "Condensed"]], Cell[StyleData["PrimaryPlaceholder", "Printout"]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["FormatType Styles", "Section"], Cell["\<\ The cells below define styles that are mixed in with the styles \ of most cells. If a cell's FormatType matches the name of one of the styles \ defined below, then that style is applied between the cell's style and its \ own options. This is particularly true of Input and Output.\ \>", "Text"], Cell[StyleData["CellExpression"], PageWidth->Infinity, CellMargins->{{6, Inherited}, {Inherited, Inherited}}, ShowCellLabel->False, ShowSpecialCharacters->False, AllowInlineCells->False, Hyphenation->False, AutoItalicWords->{}, StyleMenuListing->None, FontFamily->"Courier", FontSize->12, Background->GrayLevel[1]], Cell[StyleData["InputForm"], InputAutoReplacements->{}, AllowInlineCells->False, Hyphenation->False, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["OutputForm"], PageWidth->Infinity, TextAlignment->Left, LineSpacing->{0.6, 1}, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["StandardForm"], InputAutoReplacements->{ "->"->"\[Rule]", ":>"->"\[RuleDelayed]", "<="->"\[LessEqual]", ">="->"\[GreaterEqual]", "!="->"\[NotEqual]", "=="->"\[Equal]", Inherited}, LineSpacing->{1.25, 0}, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["TraditionalForm"], InputAutoReplacements->{ "->"->"\[Rule]", ":>"->"\[RuleDelayed]", "<="->"\[LessEqual]", ">="->"\[GreaterEqual]", "!="->"\[NotEqual]", "=="->"\[Equal]", Inherited}, LineSpacing->{1.25, 0}, SingleLetterItalics->True, TraditionalFunctionNotation->True, DelimiterMatching->None, StyleMenuListing->None], Cell["\<\ The style defined below is mixed in to any cell that is in an \ inline cell within another.\ \>", "Text"], Cell[StyleData["InlineCell"], TextAlignment->Left, ScriptLevel->1, StyleMenuListing->None], Cell[StyleData["InlineCellEditing"], StyleMenuListing->None, Background->RGBColor[1, 0.749996, 0.8]] }, Closed]], Cell[CellGroupData[{ Cell["Automatic Styles", "Section"], Cell["\<\ The cells below define styles that are used to affect the display \ of certain types of objects in typeset expressions. For example, \ \"UnmatchedBracket\" style defines how unmatched bracket, curly bracket, and \ parenthesis characters are displayed (typically by coloring them to make them \ stand out).\ \>", "Text"], Cell[StyleData["UnmatchedBracket"], StyleMenuListing->None, FontColor->RGBColor[0.760006, 0.330007, 0.8]] }, Closed]] }, Open ]] }] ] (******************************************************************* Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. The cache data will then be recreated when you save this file from within Mathematica. *******************************************************************) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[1776, 53, 87, 2, 110, "Title", Evaluatable->False], Cell[1866, 57, 122, 6, 71, "Subsubtitle", Evaluatable->False], Cell[CellGroupData[{ Cell[2013, 67, 46, 0, 45, "Subsection"], Cell[2062, 69, 371, 8, 86, "Text"], Cell[2436, 79, 719, 14, 158, "Text"], Cell[3158, 95, 112, 4, 32, "Text"], Cell[3273, 101, 266, 10, 50, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[3576, 116, 87, 2, 29, "Subsection", Evaluatable->False], Cell[3666, 120, 580, 21, 50, "Text", Evaluatable->False], Cell[4249, 143, 110, 2, 32, "Text", Evaluatable->False] }, Closed]], Cell[CellGroupData[{ Cell[4396, 150, 347, 13, 35, "Section", Evaluatable->False], Cell[4746, 165, 118, 5, 32, "Text"], Cell[4867, 172, 272, 6, 50, "Text", Evaluatable->False], Cell[5142, 180, 134, 5, 27, "Input"], Cell[5279, 187, 137, 5, 32, "Text", Evaluatable->False], Cell[5419, 194, 141, 5, 48, "Text"], Cell[5563, 201, 840, 29, 157, "Text", Evaluatable->False], Cell[6406, 232, 2162, 71, 174, "Text", Evaluatable->False], Cell[8571, 305, 198, 4, 50, "Text", Evaluatable->False], Cell[8772, 311, 51, 1, 27, "Input"], Cell[8826, 314, 51, 1, 27, "Input"], Cell[8880, 317, 53, 1, 27, "Input"], Cell[8936, 320, 53, 1, 27, "Input"], Cell[8992, 323, 51, 1, 27, "Input"], Cell[9046, 326, 53, 1, 27, "Input"], Cell[9102, 329, 1839, 63, 122, "Text", Evaluatable->False], Cell[10944, 394, 989, 27, 122, "Text", Evaluatable->False], Cell[11936, 423, 430, 15, 50, "Text", Evaluatable->False], Cell[12369, 440, 54, 1, 27, "Input"], Cell[12426, 443, 57, 1, 27, "Input"], Cell[12486, 446, 56, 1, 27, "Input"], Cell[12545, 449, 549, 13, 104, "Text", Evaluatable->False], Cell[13097, 464, 1418, 53, 68, "Text", Evaluatable->False], Cell[14518, 519, 529, 21, 23, "Print", Evaluatable->False], Cell[15050, 542, 342, 6, 68, "Text", Evaluatable->False], Cell[15395, 550, 58, 1, 27, "Input"], Cell[15456, 553, 677, 25, 50, "Text", Evaluatable->False], Cell[CellGroupData[{ Cell[16158, 582, 87, 1, 42, "Subsubsection"], Cell[16248, 585, 743, 21, 109, "Text", Evaluatable->False], Cell[16994, 608, 998, 34, 90, "Text", Evaluatable->False], Cell[17995, 644, 1008, 33, 90, "Text", Evaluatable->False] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[19052, 683, 99, 2, 35, "Section", Evaluatable->False], Cell[19154, 687, 125, 5, 33, "Text"], Cell[19282, 694, 178, 6, 33, "Text", Evaluatable->False], Cell[19463, 702, 45, 1, 27, "Input"], Cell[19511, 705, 47, 1, 27, "Input"], Cell[19561, 708, 49, 1, 27, "Input"], Cell[19613, 711, 49, 1, 27, "Input"], Cell[19665, 714, 57, 1, 27, "Input"], Cell[19725, 717, 844, 33, 52, "Text", Evaluatable->False], Cell[20572, 752, 115, 3, 52, "Text"], Cell[20690, 757, 57, 1, 27, "Input"], Cell[20750, 760, 19, 0, 33, "Text"], Cell[20772, 762, 90, 1, 27, "Input"], Cell[20865, 765, 37, 1, 27, "Input"], Cell[20905, 768, 43, 1, 27, "Input"], Cell[20951, 771, 824, 37, 71, "Text"], Cell[21778, 810, 64, 1, 27, "Input"], Cell[21845, 813, 107, 2, 33, "Text", Evaluatable->False], Cell[21955, 817, 121, 3, 21, "Print", Evaluatable->False], Cell[22079, 822, 178, 4, 33, "Text", Evaluatable->False], Cell[22260, 828, 46, 1, 27, "Input"], Cell[22309, 831, 46, 1, 27, "Input"], Cell[22358, 834, 213, 5, 33, "Text", Evaluatable->False], Cell[22574, 841, 50, 1, 27, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[22661, 847, 83, 2, 35, "Section", Evaluatable->False], Cell[22747, 851, 104, 5, 33, "Text"], Cell[22854, 858, 379, 14, 33, "Text"], Cell[23236, 874, 60, 1, 27, "Input"], Cell[23299, 877, 187, 5, 33, "Text", Evaluatable->False], Cell[23489, 884, 66, 1, 27, "Input"], Cell[23558, 887, 212, 7, 52, "Text", Evaluatable->False], Cell[23773, 896, 398, 14, 52, "Text", Evaluatable->False] }, Closed]], Cell[CellGroupData[{ Cell[24208, 915, 86, 2, 35, "Section", Evaluatable->False], Cell[24297, 919, 119, 5, 33, "Text"], Cell[24419, 926, 1984, 76, 90, "Text", Evaluatable->False], Cell[26406, 1004, 65, 1, 27, "Input"], Cell[26474, 1007, 48, 1, 27, "Input"], Cell[26525, 1010, 48, 1, 27, "Input"], Cell[26576, 1013, 49, 1, 27, "Input"], Cell[26628, 1016, 1280, 48, 71, "Text", Evaluatable->False], Cell[27911, 1066, 44, 0, 33, "Text"], Cell[27958, 1068, 114, 4, 27, "Input"], Cell[28075, 1074, 629, 18, 90, "Text", Evaluatable->False], Cell[28707, 1094, 235, 6, 52, "Text", Evaluatable->False], Cell[28945, 1102, 290, 10, 132, "Input"], Cell[29238, 1114, 530, 17, 71, "Text", Evaluatable->False], Cell[29771, 1133, 396, 13, 52, "Text", Evaluatable->False] }, Closed]], Cell[CellGroupData[{ Cell[30204, 1151, 128, 3, 35, "Section", Evaluatable->False], Cell[30335, 1156, 1690, 61, 90, "Text", Evaluatable->False] }, Closed]], Cell[CellGroupData[{ Cell[32062, 1222, 92, 2, 35, "Section", Evaluatable->False], Cell[32157, 1226, 509, 18, 50, "Text", Evaluatable->False], Cell[32669, 1246, 269, 7, 23, "Print", Evaluatable->False], Cell[32941, 1255, 280, 7, 23, "Print", Evaluatable->False], Cell[33224, 1264, 588, 21, 31, "Print", Evaluatable->False], Cell[33815, 1287, 155, 4, 32, "Text", Evaluatable->False], Cell[33973, 1293, 66, 1, 27, "Input"], Cell[34042, 1296, 58, 1, 27, "Input"], Cell[34103, 1299, 64, 1, 27, "Input"], Cell[34170, 1302, 58, 1, 27, "Input"], Cell[34231, 1305, 52, 1, 27, "Input"], Cell[34286, 1308, 154, 4, 32, "Text", Evaluatable->False] }, Closed]], Cell[CellGroupData[{ Cell[34477, 1317, 84, 2, 35, "Section", Evaluatable->False], Cell[34564, 1321, 111, 5, 33, "Text"], Cell[34678, 1328, 927, 29, 90, "Text", Evaluatable->False], Cell[35608, 1359, 74, 1, 27, "Input"], Cell[35685, 1362, 813, 26, 90, "Text", Evaluatable->False], Cell[36501, 1390, 405, 15, 33, "Text", Evaluatable->False], Cell[36909, 1407, 57, 1, 27, "Input"], Cell[36969, 1410, 401, 12, 52, "Text", Evaluatable->False], Cell[37373, 1424, 1103, 35, 109, "Text", Evaluatable->False], Cell[38479, 1461, 280, 8, 52, "Text", Evaluatable->False], Cell[38762, 1471, 1242, 50, 53, "Text", Evaluatable->False], Cell[40007, 1523, 64, 1, 27, "Input"], Cell[40074, 1526, 1046, 32, 109, "Text", Evaluatable->False] }, Closed]], Cell[CellGroupData[{ Cell[41157, 1563, 117, 2, 35, "Section", Evaluatable->False], Cell[41277, 1567, 111, 5, 33, "Text"], Cell[41391, 1574, 801, 26, 90, "Text", Evaluatable->False], Cell[42195, 1602, 1704, 59, 128, "Text", Evaluatable->False], Cell[43902, 1663, 72, 1, 27, "Input"], Cell[43977, 1666, 584, 22, 33, "Text", Evaluatable->False], Cell[44564, 1690, 47, 1, 27, "Input"], Cell[44614, 1693, 397, 15, 33, "Text", Evaluatable->False], Cell[45014, 1710, 1006, 35, 71, "Text", Evaluatable->False], Cell[46023, 1747, 897, 34, 52, "Text", Evaluatable->False], Cell[46923, 1783, 2432, 97, 73, "Text", Evaluatable->False], Cell[49358, 1882, 1215, 48, 53, "Text", Evaluatable->False], Cell[50576, 1932, 399, 15, 33, "Text", Evaluatable->False], Cell[50978, 1949, 75, 4, 42, "Input"], Cell[51056, 1955, 887, 31, 71, "Text", Evaluatable->False], Cell[51946, 1988, 67, 1, 27, "Input"], Cell[52016, 1991, 470, 17, 52, "Text", Evaluatable->False], Cell[52489, 2010, 39, 1, 27, "Input"], Cell[52531, 2013, 1023, 38, 52, "Text", Evaluatable->False], Cell[53557, 2053, 44, 1, 27, "Input"], Cell[53604, 2056, 525, 22, 34, "Text", Evaluatable->False], Cell[54132, 2080, 50, 1, 27, "Input"], Cell[54185, 2083, 155, 7, 33, "Text", Evaluatable->False], Cell[54343, 2092, 67, 1, 27, "Input"], Cell[54413, 2095, 386, 15, 33, "Text", Evaluatable->False], Cell[54802, 2112, 45, 1, 27, "Input"], Cell[54850, 2115, 98, 2, 33, "Text", Evaluatable->False], Cell[54951, 2119, 51, 1, 27, "Input"], Cell[55005, 2122, 45, 1, 27, "Input"], Cell[55053, 2125, 66, 2, 33, "Text", Evaluatable->False], Cell[55122, 2129, 52, 1, 27, "Input"], Cell[55177, 2132, 39, 1, 27, "Input"], Cell[55219, 2135, 882, 32, 72, "Text", Evaluatable->False] }, Closed]], Cell[CellGroupData[{ Cell[56138, 2172, 51, 0, 35, "Section"], Cell[56192, 2174, 104, 5, 32, "Text"], Cell[56299, 2181, 1175, 35, 104, "Text", Evaluatable->False], Cell[57477, 2218, 2972, 110, 122, "Text", Evaluatable->False], Cell[60452, 2330, 672, 28, 27, "Print", Evaluatable->False], Cell[61127, 2360, 2006, 77, 69, "Text", Evaluatable->False], Cell[63136, 2439, 51, 1, 27, "Input"], Cell[63190, 2442, 68, 1, 27, "Input"], Cell[63261, 2445, 72, 1, 27, "Input"], Cell[63336, 2448, 1628, 63, 68, "Text", Evaluatable->False], Cell[64967, 2513, 491, 18, 50, "Text", Evaluatable->False], Cell[65461, 2533, 116, 4, 27, "Input"], Cell[65580, 2539, 946, 31, 86, "Text", Evaluatable->False], Cell[66529, 2572, 73, 1, 27, "Input"], Cell[66605, 2575, 517, 15, 68, "Text", Evaluatable->False], Cell[67125, 2592, 43, 1, 27, "Input"], Cell[67171, 2595, 51, 1, 27, "Input"], Cell[67225, 2598, 2151, 73, 140, "Text", Evaluatable->False], Cell[69379, 2673, 59, 0, 32, "Text"], Cell[69441, 2675, 85, 1, 27, "Input"], Cell[69529, 2678, 436, 14, 50, "Text", Evaluatable->False] }, Closed]], Cell[CellGroupData[{ Cell[70002, 2697, 128, 3, 35, "Section", Evaluatable->False], Cell[70133, 2702, 615, 22, 68, "Text", Evaluatable->False] }, Closed]], Cell[CellGroupData[{ Cell[70785, 2729, 107, 2, 35, "Section", Evaluatable->False], Cell[70895, 2733, 112, 5, 33, "Text"], Cell[71010, 2740, 505, 19, 33, "Text", Evaluatable->False], Cell[71518, 2761, 57, 1, 27, "Input"], Cell[71578, 2764, 834, 30, 52, "Text", Evaluatable->False], Cell[72415, 2796, 182, 3, 54, "Text"], Cell[72600, 2801, 889, 28, 72, "Text", Evaluatable->False], Cell[73492, 2831, 185, 3, 51, "Text"], Cell[73680, 2836, 1299, 48, 78, "Text", Evaluatable->False], Cell[74982, 2886, 747, 27, 71, "Text", Evaluatable->False], Cell[75732, 2915, 147, 5, 33, "Text", Evaluatable->False], Cell[75882, 2922, 61, 1, 27, "Input"], Cell[75946, 2925, 55, 1, 27, "Input"], Cell[76004, 2928, 49, 1, 27, "Input"], Cell[76056, 2931, 55, 1, 27, "Input"], Cell[76114, 2934, 50, 1, 27, "Input"], Cell[76167, 2937, 285, 6, 52, "Text", Evaluatable->False], Cell[76455, 2945, 810, 25, 90, "Text", Evaluatable->False], Cell[77268, 2972, 117, 2, 23, "Print", Evaluatable->False], Cell[77388, 2976, 199, 4, 52, "Text", Evaluatable->False], Cell[77590, 2982, 1810, 69, 110, "Text", Evaluatable->False], Cell[79403, 3053, 85, 1, 27, "Input"], Cell[79491, 3056, 54, 1, 27, "Input"], Cell[79548, 3059, 55, 1, 27, "Input"], Cell[79606, 3062, 56, 1, 27, "Input"], Cell[79665, 3065, 57, 1, 27, "Input"], Cell[79725, 3068, 58, 1, 27, "Input"], Cell[79786, 3071, 59, 1, 27, "Input"], Cell[79848, 3074, 1100, 38, 109, "Text", Evaluatable->False] }, Closed]], Cell[CellGroupData[{ Cell[80985, 3117, 128, 3, 35, "Section", Evaluatable->False], Cell[81116, 3122, 242, 5, 52, "Text", Evaluatable->False], Cell[81361, 3129, 392, 15, 25, "Message", Evaluatable->False], Cell[81756, 3146, 312, 8, 71, "Text", Evaluatable->False] }, Closed]], Cell[CellGroupData[{ Cell[82105, 3159, 65, 1, 35, "Section"], Cell[82173, 3162, 160, 4, 50, "Text"], Cell[82336, 3168, 81, 1, 27, "Input"], Cell[82420, 3171, 402, 17, 50, "Text"], Cell[82825, 3190, 231, 8, 42, "Input"], Cell[83059, 3200, 237, 6, 50, "Text"], Cell[83299, 3208, 197, 5, 53, "Print"], Cell[83499, 3215, 286, 10, 50, "Text"], Cell[83788, 3227, 61, 1, 27, "Input"], Cell[83852, 3230, 74, 1, 27, "Input"], Cell[83929, 3233, 242, 5, 50, "Text"], Cell[84174, 3240, 84, 1, 27, "Input"], Cell[84261, 3243, 117, 4, 27, "Input"], Cell[84381, 3249, 693, 24, 86, "Text"], Cell[85077, 3275, 408, 16, 50, "Text"], Cell[85488, 3293, 99, 1, 27, "Input"], Cell[85590, 3296, 861, 28, 105, "Text"], Cell[86454, 3326, 50, 1, 27, "Input"], Cell[86507, 3329, 573, 20, 69, "Text"], Cell[87083, 3351, 41, 1, 27, "Input"], Cell[CellGroupData[{ Cell[87149, 3356, 36, 0, 42, "Subsubsection"], Cell[87188, 3358, 880, 35, 69, "Text"], Cell[88071, 3395, 101, 1, 27, "Input"], Cell[88175, 3398, 102, 1, 27, "Input"], Cell[88280, 3401, 146, 5, 32, "Text"], Cell[88429, 3408, 1063, 41, 87, "Text"], Cell[89495, 3451, 102, 1, 27, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[89634, 3457, 34, 0, 28, "Subsubsection"], Cell[89671, 3459, 340, 15, 32, "Text"], Cell[90014, 3476, 939, 34, 87, "Text"], Cell[90956, 3512, 101, 1, 27, "Input"], Cell[91060, 3515, 102, 1, 27, "Input"], Cell[91165, 3518, 275, 10, 50, "Text"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[91489, 3534, 62, 1, 35, "Section"], Cell[91554, 3537, 286, 9, 52, "Text"], Cell[91843, 3548, 88, 1, 27, "Input"], Cell[91934, 3551, 175, 4, 52, "Text"], Cell[92112, 3557, 480, 11, 109, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[92629, 3573, 59, 0, 35, "Section"], Cell[92691, 3575, 456, 20, 33, "Text"], Cell[93150, 3597, 335, 14, 32, "Text"], Cell[93488, 3613, 434, 19, 32, "Text"], Cell[93925, 3634, 447, 20, 68, "Text"], Cell[94375, 3656, 619, 25, 68, "Text"], Cell[94997, 3683, 1381, 63, 68, "Text"], Cell[96381, 3748, 540, 24, 68, "Text", Evaluatable->False], Cell[96924, 3774, 807, 35, 50, "Text"], Cell[97734, 3811, 957, 44, 50, "Text"], Cell[98694, 3857, 1135, 49, 68, "Text"], Cell[99832, 3908, 488, 22, 32, "Text"] }, Closed]] }, Open ]] } ] *) (******************************************************************* End of Mathematica Notebook file. *******************************************************************)