(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 8.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 157, 7] NotebookDataLength[ 238547, 6729] NotebookOptionsPosition[ 217124, 6047] NotebookOutlinePosition[ 217516, 6064] CellTagsIndexPosition[ 217473, 6061] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["3. Fortgeschrittene Themen", "Subtitle", CellChangeTimes->{{3.529742646748269*^9, 3.52974265373625*^9}, { 3.529742733088111*^9, 3.529742743263365*^9}, {3.5297430130674458`*^9, 3.529743013229287*^9}, 3.529773051416341*^9}], Cell[TextData[{ "Dieses Notebook soll Sie \[UDoubleDot]ber die in \ \[OpenCurlyDoubleQuote]Intro.nb\[CloseCurlyDoubleQuote] aufgezeigten M\ \[ODoubleDot]glichkeiten hinaus vorwiegend mit den Aspekten von ", StyleBox["Mathematica", FontSlant->"Italic"], " vertraut machen, die helfen, Dinge zu automatisieren oder einfach nur \ eleganter und schneller zu erledigen. Ebenso wie das erste Notebook zeigt \ dieses hier bei weitem nicht alle M\[ODoubleDot]glichkeiten auf, bietet aber \ einen gewissen Einblick in bestimmte elementare Themen. Au\[SZ]erdem werden \ einige Kniffe vorgef\[UDoubleDot]hrt, die weniger f\[UDoubleDot]r die \ \[CapitalUDoubleDot]bungsaufgaben als f\[UDoubleDot]r eine eventuelle sp\ \[ADoubleDot]tere Arbeit mit ", StyleBox["Mathematica", FontSlant->"Italic"], " sinnvoll sein k\[ODoubleDot]nnen." }], "Text", PageWidth->PaperWidth, CellChangeTimes->{{3.5307722193636293`*^9, 3.530772234609045*^9}, { 3.530801265259925*^9, 3.530801473969255*^9}, {3.530801505278955*^9, 3.5308016537092333`*^9}, {3.530801704479204*^9, 3.5308017051509943`*^9}, { 3.530871202039857*^9, 3.5308712025562077`*^9}, {3.530871265735161*^9, 3.530871270771305*^9}}], Cell[TextData[{ "Insbesondere m\[ODoubleDot]chte ich hier auf die Erkl\[ADoubleDot]rung von \ Elementen verzichten, die es auch in anderen (Ihnen evtl. schon bekannten) \ Programmiersprachen zu Hauf gibt. Diese werden au\[SZ]erdem auch in der \ Vorlesung behandelt. Wenn sie im Folgenden auftauchen, verweise ich deswegen \ auf die ", StyleBox["Mathematica", FontSlant->"Italic"], "-Referenz." }], "Text", PageWidth->PaperWidth, CellChangeTimes->{{3.530801751379981*^9, 3.530801894869713*^9}, { 3.530801931868331*^9, 3.5308019519561377`*^9}, {3.530801983260076*^9, 3.530802014721878*^9}, 3.530802089592732*^9, {3.530802141126656*^9, 3.530802195629004*^9}, {3.530802228605659*^9, 3.530802257587603*^9}, { 3.5308712370643663`*^9, 3.5308712388336477`*^9}}], Cell[CellGroupData[{ Cell["3.1 \[OpenCurlyDoubleQuote]Pattern Matching\[CloseCurlyDoubleQuote]", \ "Subsubtitle", CellChangeTimes->{{3.529742759690398*^9, 3.529742780136752*^9}, 3.5297730514166594`*^9, {3.530802451954485*^9, 3.5308024530540247`*^9}}], Cell[TextData[{ "Pattern-Objekte sind Platzhalter f\[UDoubleDot]r prinzipiell beliebige \ Konstrukte in ", StyleBox["Mathematica", FontSlant->"Italic"], ". 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StyleBox["Mathematica", FontSlant->"Italic"], " definiert durch Angabe eines Funktionsnamens, dahinter in eckigen Klammern \ Patterns f\[UDoubleDot]r die Parameter, einem SetDelayed \ \[OpenCurlyDoubleQuote]", StyleBox[":=", "Input"], "\[CloseCurlyDoubleQuote] und einem Funktionsk\[ODoubleDot]rper, in dem die \ (benannten) Platzhalter wieder verwendet werden (k\[ODoubleDot]nnen)." }], "Text", PageWidth->PaperWidth, CellChangeTimes->{{3.530778309253889*^9, 3.5307785910268993`*^9}, { 3.530778622962163*^9, 3.5307786320171747`*^9}, {3.530872356191432*^9, 3.530872358012533*^9}}], Cell["Also zum Beispiel:", "Text", CellChangeTimes->{{3.530778637948079*^9, 3.530778650288004*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"f", "[", "x_", "]"}], ":=", RowBox[{ RowBox[{"Sin", "[", "x", "]"}], "*", RowBox[{"Cos", "[", "x", "]"}]}]}]], "Input", CellChangeTimes->{{3.53027971462432*^9, 3.5302797272425327`*^9}}], Cell[TextData[{ "Dabei ist \[CloseCurlyDoubleQuote]", StyleBox["f", "Input"], "\[OpenCurlyDoubleQuote] der Funktionsname und das Pattern \ \[OpenCurlyDoubleQuote]", StyleBox["x_", "Input"], "\[CloseCurlyDoubleQuote] das Argument. Ein Pattern ist Platzhalter (hier f\ \[UDoubleDot]r eine Zahl oder einen beliebigen Ausdruck), der f\[UDoubleDot]r \ all das was man als Argument \[UDoubleDot]bergibt (durch ", StyleBox["SetDelayed", "Input"], "), das \[OpenCurlyDoubleQuote]", StyleBox["x", "Input"], "\[CloseCurlyDoubleQuote] rechts ersetzt. (Das Symbol ", StyleBox["f[x]", "Input"], " ist hier demnach f\[UDoubleDot]r beliebige ", StyleBox["x", "Input"], " definiert.)" }], "Text", PageWidth->PaperWidth, CellChangeTimes->{{3.530279734439062*^9, 3.530280039021722*^9}, { 3.530280099063066*^9, 3.530280153030838*^9}, {3.530293463632206*^9, 3.5302934649741287`*^9}, {3.5304368943184423`*^9, 3.530436896902733*^9}, { 3.530437188915925*^9, 3.530437194077935*^9}, {3.530777513739263*^9, 3.5307775155917788`*^9}, {3.5307787459277678`*^9, 3.5307787897528667`*^9}, { 3.530872419305275*^9, 3.5308724330202303`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"f", "[", RowBox[{"x_", ",", "y_"}], "]"}], ":=", RowBox[{ RowBox[{"Sin", "[", "x", "]"}], "*", RowBox[{"Cos", "[", "y", "]"}]}]}]], "Input", CellChangeTimes->{{3.530280272923573*^9, 3.530280287285726*^9}}], Cell[TextData[{ "Hinter einer Funktionsdefinition braucht kein \[OpenCurlyDoubleQuote]", StyleBox[";", "Input"], "\[CloseCurlyDoubleQuote] zu stehen, da dank \[OpenCurlyDoubleQuote]", StyleBox[":=", "Input"], "\[CloseCurlyDoubleQuote] sowieso keine direkte Auswertung und Ausgabe \ erfolgt." }], "Text", PageWidth->PaperWidth, CellChangeTimes->{{3.530779801789804*^9, 3.5307798632166452`*^9}, 3.5308724664910803`*^9}], Cell[CellGroupData[{ Cell["Ausnahmen", "Subsubsection", CellChangeTimes->{{3.530779142401498*^9, 3.530779144039946*^9}}], Cell["\<\ Bei mehreren Ver\[ADoubleDot]nderlichen funktioniert das eben diskutierte \ analog. Den selben Funktionsnamen zu w\[ADoubleDot]hlen ist \ zul\[ADoubleDot]ssig, genauso wie die Definition von Ausnahmen:\ \>", "Text", PageWidth->PaperWidth, CellChangeTimes->{{3.5302802977829437`*^9, 3.5302804154070997`*^9}, { 3.530280445823415*^9, 3.530280461807472*^9}, {3.530293484651196*^9, 3.5302935000385647`*^9}, {3.5303115401545687`*^9, 3.5303115441021233`*^9}, { 3.5304371316699944`*^9, 3.530437133363084*^9}, {3.53077882688618*^9, 3.530778827059375*^9}, {3.530872495526835*^9, 3.5308725131980543`*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{"f", "[", RowBox[{"x_", ",", "x_"}], "]"}], ":=", "10"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"f", "[", RowBox[{"5", ",", "x_"}], "]"}], ":=", "x"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"f", "[", RowBox[{"1", ",", "1"}], "]"}], ":=", "100"}]}], "Input", CellChangeTimes->{{3.530280475176139*^9, 3.53028048702232*^9}, { 3.530311553420628*^9, 3.530311604165368*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"f", "[", RowBox[{"1", ",", "2"}], "]"}], "\[IndentingNewLine]", RowBox[{"f", "[", RowBox[{"1", ",", "1"}], "]"}], "\[IndentingNewLine]", RowBox[{"f", "[", RowBox[{"5", ",", "67"}], "]"}], "\[IndentingNewLine]", RowBox[{"f", "[", RowBox[{"2", ",", "2"}], "]"}]}], "Input", CellChangeTimes->{{3.530293518129156*^9, 3.5302935511726637`*^9}, { 3.530311633060182*^9, 3.530311644117663*^9}, {3.5614647829157887`*^9, 3.561464785174965*^9}}], Cell[BoxData[ RowBox[{"f", "[", RowBox[{"1", ",", "2"}], "]"}]], "Output", CellChangeTimes->{{3.53029352217336*^9, 3.530293554901773*^9}, 3.530311607996011*^9, 3.5303116445583572`*^9, 3.530778851460507*^9, 3.5308725257621307`*^9, 3.561464796801579*^9}], Cell[BoxData["100"], "Output", CellChangeTimes->{{3.53029352217336*^9, 3.530293554901773*^9}, 3.530311607996011*^9, 3.5303116445583572`*^9, 3.530778851460507*^9, 3.5308725257621307`*^9, 3.561464796803809*^9}], Cell[BoxData["67"], "Output", CellChangeTimes->{{3.53029352217336*^9, 3.530293554901773*^9}, 3.530311607996011*^9, 3.5303116445583572`*^9, 3.530778851460507*^9, 3.5308725257621307`*^9, 3.561464796807033*^9}], Cell[BoxData["10"], "Output", CellChangeTimes->{{3.53029352217336*^9, 3.530293554901773*^9}, 3.530311607996011*^9, 3.5303116445583572`*^9, 3.530778851460507*^9, 3.5308725257621307`*^9, 3.5614647968098497`*^9}] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{"ClearAll", "[", "f", "]"}], ";"}]], "Input", CellChangeTimes->{{3.530314706671769*^9, 3.530314712217634*^9}, { 3.530779666492736*^9, 3.530779666778843*^9}}], Cell["Iterative Funktions-Definition:", "Text", CellChangeTimes->{{3.530314687786394*^9, 3.530314698277306*^9}, { 3.53031479173006*^9, 3.530314797285749*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{"kein", " ", ":=", " ", RowBox[{ "da", " ", "Ergebnis", " ", "nicht", " ", "von", " ", "Argument", " ", "abh\[ADoubleDot]ngt"}]}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"f", "[", "0", "]"}], "=", "0"}], ";"}], "\[IndentingNewLine]", RowBox[{"(*", " ", RowBox[{ "Summe", " ", "von", " ", "0", " ", "bis", " ", "zu", " ", "einer", " ", "ganzen", " ", "Zahl"}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"f", "[", "i_", "]"}], ":=", RowBox[{"i", "+", RowBox[{"f", "[", RowBox[{"i", "-", "1"}], "]"}]}]}]}]}]], "Input", CellChangeTimes->{{3.530314700668112*^9, 3.530314778163855*^9}, { 3.530778895253923*^9, 3.530778949547934*^9}, {3.53077900332187*^9, 3.5307790310332623`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"f", "[", "0", "]"}], ",", RowBox[{"f", "[", "1", "]"}], ",", RowBox[{"f", "[", "2", "]"}], ",", RowBox[{"f", "[", "3", "]"}], ",", RowBox[{"f", "[", "4", "]"}], ",", RowBox[{"f", "[", "5", "]"}], ",", RowBox[{"f", "[", "6", "]"}], ",", RowBox[{"f", "[", "7", "]"}], ",", RowBox[{"f", "[", "8", "]"}], ",", RowBox[{"f", "[", "9", "]"}]}], "}"}]], "Input", CellChangeTimes->{{3.530314769831758*^9, 3.5303147724962187`*^9}, { 3.530778982678376*^9, 3.530778989064899*^9}, {3.5307790399748*^9, 3.5307790758395023`*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ "0", ",", "1", ",", "3", ",", "6", ",", "10", ",", "15", ",", "21", ",", "28", ",", "36", ",", "45"}], "}"}]], "Output", CellChangeTimes->{{3.530314773217313*^9, 3.5303147795840197`*^9}, { 3.530778980030414*^9, 3.5307789893433113`*^9}, {3.5307790366120367`*^9, 3.530779042615107*^9}, 3.530779076315298*^9, 3.530872537525029*^9}] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{"ClearAll", "[", "f", "]"}], ";"}]], "Input", CellChangeTimes->{{3.530314706671769*^9, 3.530314712217634*^9}, { 3.530779672555649*^9, 3.5307796729384336`*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["\[OpenCurlyDoubleQuote]Caching\[CloseCurlyDoubleQuote]", "Subsubsection", CellChangeTimes->{{3.530779107109871*^9, 3.530779109913982*^9}}], Cell["\<\ Ein einfacher Trick, Rechenzeit zu sparen ist es, schon einmal berechnete \ Funktionswerte zu tabellieren, der Funktion also ein \ \[OpenCurlyDoubleQuote]Ged\[ADoubleDot]chtnis\[CloseCurlyDoubleQuote] zu \ geben. Mit dem Konstrukt in der folgenden Funktion wird f\[UDoubleDot]r \ verschiedene auftretende Argumente die Funktionsdefinition jeweils nur ein \ einziges Mal ausgewertet, ein Wert berechnet und gespeichert. 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Bei mehreren optionalen Argumenten werden bei einem Aufruf \ die in der Definition zuerst aufgef\[UDoubleDot]hrten als \ \[UDoubleDot]bergeben angenommen.\ \>", "Text", PageWidth->PaperWidth, CellChangeTimes->{{3.53031271394895*^9, 3.530312832151044*^9}, 3.530313473248502*^9, 3.530437135699318*^9, 3.530437206801581*^9, 3.530722642378059*^9, {3.5307726330396423`*^9, 3.530772664252281*^9}, { 3.5307727145955553`*^9, 3.530772754849798*^9}, {3.530777203506528*^9, 3.5307772265274963`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"f", "[", RowBox[{"x_", ",", RowBox[{"y_:", "100"}]}], "]"}], ":=", RowBox[{ RowBox[{"Sin", "[", "x", "]"}], "*", RowBox[{"Cos", "[", "y", "]"}]}]}]], "Input", CellChangeTimes->{{3.530312838484784*^9, 3.530312877779126*^9}, 3.530313473252129*^9, {3.530772778728817*^9, 3.530772785772936*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"f", "[", RowBox[{"x", ",", "y"}], "]"}], "\[IndentingNewLine]", RowBox[{"f", "[", "x", "]"}]}], "Input", PageWidth->PaperWidth, CellChangeTimes->{{3.530312882359888*^9, 3.530312891721352*^9}, 3.53031347325452*^9, {3.5307729438561087`*^9, 3.530772954426602*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"Cos", "[", "y", "]"}], " ", RowBox[{"Sin", "[", "x", "]"}]}]], "Output", CellChangeTimes->{{3.530772855391487*^9, 3.5307728605305033`*^9}, { 3.530772951459779*^9, 3.530772955849333*^9}, 3.530872836450863*^9}], Cell[BoxData[ RowBox[{ RowBox[{"Cos", "[", "100", "]"}], " ", RowBox[{"Sin", "[", "x", "]"}]}]], "Output", CellChangeTimes->{{3.530772855391487*^9, 3.5307728605305033`*^9}, { 3.530772951459779*^9, 3.530772955849333*^9}, 3.530872836453603*^9}] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{"eigentlich", " ", "unn\[ODoubleDot]tig"}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"ClearAll", "[", "f", "]"}], ";"}]}]], "Input", CellChangeTimes->{{3.5303128982843857`*^9, 3.530312900906023*^9}, 3.530313473248939*^9, {3.530772837796426*^9, 3.530772838096208*^9}, 3.5307729790806303`*^9, {3.5307800277785788`*^9, 3.5307800441472*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"f", "[", RowBox[{"x_", ",", RowBox[{"y_:", "100"}], ",", RowBox[{"z_:", "1000"}]}], "]"}], ":=", RowBox[{ RowBox[{"Sin", "[", "x", "]"}], "*", RowBox[{"Cos", "[", "y", "]"}], "*", RowBox[{"Tan", "[", "z", "]"}]}]}]], "Input", CellChangeTimes->{{3.530772875055086*^9, 3.530772907063858*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"f", "[", RowBox[{"x", ",", "y", ",", "z"}], "]"}], "\[IndentingNewLine]", RowBox[{"f", "[", RowBox[{"x", ",", "y"}], "]"}], "\[IndentingNewLine]", RowBox[{"f", "[", "x", "]"}]}], "Input", CellChangeTimes->{{3.5307729088049994`*^9, 3.530772971855199*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"Cos", "[", "y", "]"}], " ", RowBox[{"Sin", "[", "x", "]"}], " ", RowBox[{"Tan", "[", "z", "]"}]}]], "Output", CellChangeTimes->{{3.530772912460697*^9, 3.530772972296753*^9}, 3.530872836535421*^9}], Cell[BoxData[ RowBox[{ RowBox[{"Cos", "[", "y", "]"}], " ", RowBox[{"Sin", "[", "x", "]"}], " ", RowBox[{"Tan", "[", "1000", "]"}]}]], "Output", CellChangeTimes->{{3.530772912460697*^9, 3.530772972296753*^9}, 3.530872836538212*^9}], Cell[BoxData[ RowBox[{ RowBox[{"Cos", "[", "100", "]"}], " ", RowBox[{"Sin", "[", "x", "]"}], " ", RowBox[{"Tan", "[", "1000", "]"}]}]], "Output", CellChangeTimes->{{3.530772912460697*^9, 3.530772972296753*^9}, 3.530872836541184*^9}] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{"ClearAll", "[", "f", "]"}], ";"}]], "Input", CellChangeTimes->{{3.530772973854211*^9, 3.530772981671926*^9}}], Cell[TextData[{ "Eine einfache M\[ODoubleDot]glichkeit, aufw\[ADoubleDot]ndige \ Fallunterscheidungen bzgl. Anzahl und Art der \[UDoubleDot]bergebenen \ Argumente innerhalb eines Funktionsk\[ODoubleDot]rpers zu umgehen/vermeiden, \ ist das \[CapitalUDoubleDot]berladen von Funktionsdefinitionen. Das heisst: \ man definiert die Funktion (jeweils immer der selbe Name) f\[UDoubleDot]r \ unterschiedliche Patterns (entspricht im Grunde einer st\[UDoubleDot]ckweisen \ Tabellierung) und ", StyleBox["Mathematica", FontSlant->"Italic"], " w\[ADoubleDot]hlt bei Aufruf der Funktion automatisch das Passend(st)e aus." }], "Text", PageWidth->PaperWidth, CellChangeTimes->{{3.530773000381031*^9, 3.530773259659274*^9}, { 3.5307760136797943`*^9, 3.5307760260329447`*^9}, {3.530872848509235*^9, 3.530872883591069*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{ RowBox[{ RowBox[{"zeigt", " ", "jeweils", " ", "die", " ", "ersten"}], " ", "<", "y", ">", " ", "Stellen"}], ",", " ", "Zeichen", ",", " ", RowBox[{ RowBox[{ RowBox[{"bzw", ".", " ", "Elemente"}], " ", "von"}], " ", "<", "x", ">", " ", "an"}]}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"f", "[", RowBox[{ RowBox[{"x_", "?", "NumberQ"}], ",", RowBox[{"y_Integer:", "5"}]}], "]"}], ":=", RowBox[{"Print", "[", 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": \[OpenCurlyDoubleQuote]", StyleBox[";;", "Input"], "\[CloseCurlyDoubleQuote] (der Shortcut f\[UDoubleDot]r ", StyleBox["Span", "Input"], ") wird weiter unten ausf\[UDoubleDot]hrlicher erkl", "\[ADoubleDot]rt." }], "Text", CellChangeTimes->{{3.561702449552063*^9, 3.5617025787371407`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"f", "[", RowBox[{"{", RowBox[{ "e1", ",", "e2", ",", "e3", ",", "e4", ",", "e5", ",", "e6", ",", "e7", ",", "e8", ",", "e9", ",", "e10"}], "}"}], "]"}]], "Input", CellChangeTimes->{{3.530776693049757*^9, 3.5307767579004374`*^9}, { 3.530776817499119*^9, 3.530776822666535*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{"e1", ",", "e2", ",", "e3", ",", "e4", ",", "e5"}], "}"}]], "Print",\ CellChangeTimes->{{3.530776713472622*^9, 3.5307767584408293`*^9}, 3.530776824341722*^9, 3.530776891210101*^9, 3.530872968178134*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"f", "[", "\"\\"", "]"}]], "Input", CellChangeTimes->{{3.530776774472267*^9, 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CellChangeTimes->{{3.530311958000095*^9, 3.530311959927552*^9}}], Cell["\<\ Es sind auch Funktionen/Prozeduren/Subroutinen wie in anderen \ Programmiersprachen m\[ODoubleDot]glich:\ \>", "Text", PageWidth->PaperWidth, CellChangeTimes->{{3.530293579971689*^9, 3.530293620510069*^9}, { 3.5303119663119783`*^9, 3.5303120103521*^9}, {3.530780487471855*^9, 3.530780502644794*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"f", "[", RowBox[{"x_", ",", "y_"}], "]"}], ":=", "\[IndentingNewLine]", RowBox[{"Module", "[", RowBox[{ RowBox[{"{", RowBox[{"z", ",", "i"}], "}"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{"z", "=", RowBox[{"Table", "[", RowBox[{ RowBox[{"RandomReal", "[", RowBox[{"Abs", "[", "x", "]"}], "]"}], ",", RowBox[{"{", RowBox[{"i", ",", "1", ",", RowBox[{"Round", "[", "y", "]"}]}], "}"}]}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"z", "=", RowBox[{"z", "[", RowBox[{"[", RowBox[{"-", "1"}], "]"}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"Return", "[", "z", "]"}]}]}], 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StyleBox["Module", "Input"], " ist eine Liste lokaler Variablen (siehe auch ", StyleBox["Block", "Input"], ", ", StyleBox["With", "Input"], "), das zweite Argument ist der Funktionsk\[ODoubleDot]rper aus Anweisungen. \ R\[UDoubleDot]ckgabewert eines Moduls ist das Ergebnis der letzten Anweisung, \ bzw. das Argument von ", StyleBox["Return", "Input"], ". ", StyleBox["Return", "Input"], " muss nicht am Ende im Funktionsk\[ODoubleDot]rper stehen, denn durch den \ Aufruf von ", StyleBox["Return", "Input"], " wird die weitere Auswertung des Funktionsk\[ODoubleDot]rpers abgebrochen." }], "Text", PageWidth->PaperWidth, CellChangeTimes->{{3.5303122468818417`*^9, 3.530312319674211*^9}, { 3.530312355299363*^9, 3.530312359859459*^9}, {3.53031239181581*^9, 3.530312433804957*^9}, 3.530437205674704*^9, {3.5307806004688597`*^9, 3.530780844241517*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{"simple", " ", "Betragsfunktion", " ", "mit", " ", "Ausgabe"}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"f", "[", "x_", "]"}], ":=", "\[IndentingNewLine]", RowBox[{"Module", "[", RowBox[{ RowBox[{"{", "}"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{"If", "[", RowBox[{ RowBox[{"x", ">", "0"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{"Print", "[", RowBox[{"\"\\"", ",", "x"}], "]"}], ";", "\[IndentingNewLine]", RowBox[{"Return", "[", "x", "]"}]}]}], "\[IndentingNewLine]", "]"}], ";", "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", "x"}], "]"}], ";", "\[IndentingNewLine]", RowBox[{"Return", "[", RowBox[{"-", "x"}], "]"}]}]}], "\[IndentingNewLine]", "]"}]}]}]], "Input", CellChangeTimes->{{3.530780931858695*^9, 3.530781102432575*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"f", "[", RowBox[{"-", "1"}], "]"}]], "Input", CellChangeTimes->{{3.530781081028117*^9, 3.530781082303999*^9}}], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"Zahl kleiner 0: \"\>", "\[InvisibleSpace]", RowBox[{"-", "1"}]}], SequenceForm["Zahl kleiner 0: ", -1], Editable->False]], "Print", CellChangeTimes->{{3.530781082690041*^9, 3.530781118102251*^9}, 3.5308735398854113`*^9}], Cell[BoxData["1"], "Output", CellChangeTimes->{{3.5307810826938753`*^9, 3.530781118107732*^9}, 3.530873539908229*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"f", "[", "1", "]"}]], "Input", CellChangeTimes->{{3.530781084264236*^9, 3.5307810853730183`*^9}}], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"Zahl gr\[ODoubleDot]\[SZ]er 0: \"\>", "\[InvisibleSpace]", "1"}], SequenceForm["Zahl gr\[ODoubleDot]\[SZ]er 0: ", 1], Editable->False]], "Print", CellChangeTimes->{3.530781118655298*^9, 3.5308735406936483`*^9}], Cell[BoxData["1"], "Output", CellChangeTimes->{{3.530781085717053*^9, 3.530781118659087*^9}, 3.53087354071369*^9}] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{"ClearAll", "[", "f", "]"}], ";"}]], "Input", CellChangeTimes->{{3.530780861273466*^9, 3.530780865532915*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Optionen", "Subsection", CellChangeTimes->{{3.530312916802596*^9, 3.530312923342531*^9}}], Cell["\<\ Optionen werden durch die Art der \[CapitalUDoubleDot]bergabe (als Regel) von \ (optionalen) Argumenten unterschieden. 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