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Aus Gr\[UDoubleDot]nden \ der mathematischen Exaktheit l\[ADoubleDot]sst sich das mit den \ \[UDoubleDot]blichen Simplifikationskommandos nicht bewerkstelligen.\n\nDas n\ \[ADoubleDot]chste Kommando extrahiert alle Terme des Produkts bis auf den \ letzten. Dazu wird das Produkt in eine Liste verwandelt, ", StyleBox["DeleteCases", FontWeight->"Bold"], " angewendet und danach das Ergebnis wieder in eine Liste verwandelt. Nat\ \[UDoubleDot]rlich k\[ODoubleDot]nnen Sie alternativ auch die zu \ untersuchenden Teilausdr\[UDoubleDot]cke mit der Maus markieren und in eine \ neue Zelle kopieren." }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"u", "=", RowBox[{"Times", "@@", RowBox[{"DeleteCases", "[", RowBox[{ RowBox[{"List", "@@", "i7"}], ",", "_EllipticPi"}], "]"}]}]}], "\[IndentingNewLine]", RowBox[{ SuperscriptBox["u", "2"], "//", "Together"}]}], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"(", RowBox[{"1", "-", "\[ImaginaryI]"}], ")"}], " ", "x", " ", SqrtBox[ FractionBox[ RowBox[{ RowBox[{"(", RowBox[{"1", "+", "\[ImaginaryI]"}], ")"}], " ", RowBox[{"(", RowBox[{"\[ImaginaryI]", "+", "x"}], ")"}]}], "x"]], " ", SqrtBox[ FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "\[ImaginaryI]"}], "+", "x"}], ")"}], " ", RowBox[{"(", RowBox[{"1", "+", "x"}], ")"}]}], SuperscriptBox["x", "2"]]], " ", SqrtBox[ FractionBox["x", RowBox[{"1", "+", "x", "+", SuperscriptBox["x", "2"], "+", SuperscriptBox["x", "3"]}]]]}]], "Output"], Cell[BoxData[ RowBox[{"2", "-", RowBox[{"2", " ", "\[ImaginaryI]"}]}]], "Output"] }, Open ]], Cell[TextData[{ "Auch h\[ADoubleDot]tte es nicht des Umwegs \[UDoubleDot]ber eine Liste \ bedurft, da ", StyleBox["DeleteCases", FontWeight->"Bold"], " auch auf allgemeinere Ausdr\[UDoubleDot]cke angewendet werden kann. " }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"DeleteCases", "[", RowBox[{"i7", ",", "_EllipticPi"}], "]"}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"(", RowBox[{"1", "-", "\[ImaginaryI]"}], ")"}], " ", "x", " ", SqrtBox[ FractionBox[ RowBox[{ RowBox[{"(", RowBox[{"1", "+", "\[ImaginaryI]"}], ")"}], " ", RowBox[{"(", RowBox[{"\[ImaginaryI]", "+", "x"}], ")"}]}], "x"]], " ", SqrtBox[ FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "\[ImaginaryI]"}], "+", "x"}], ")"}], " ", RowBox[{"(", RowBox[{"1", "+", "x"}], ")"}]}], SuperscriptBox["x", "2"]]], " ", SqrtBox[ FractionBox["x", RowBox[{"1", "+", "x", "+", SuperscriptBox["x", "2"], "+", SuperscriptBox["x", "3"]}]]]}]], "Output"] }, Open ]], Cell["\<\ Das Integral i7 sollte also zum folgenden Ausdruck \[ADoubleDot]quivalent \ sein.\ \>", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"i8", "=", RowBox[{ SqrtBox[ RowBox[{"2", "-", RowBox[{"2", "I"}]}]], RowBox[{"EllipticPi", "[", RowBox[{ RowBox[{"1", "-", "I"}], ",", RowBox[{"ArcSin", "[", SqrtBox[ FractionBox[ RowBox[{ RowBox[{"(", RowBox[{"1", "+", "I"}], ")"}], " ", RowBox[{"(", RowBox[{"1", "+", "x"}], ")"}]}], RowBox[{"2", "x"}]]], "]"}], ",", RowBox[{"-", "I"}]}], "]"}]}]}]], "Input"], Cell[BoxData[ RowBox[{ SqrtBox[ RowBox[{"2", "-", RowBox[{"2", " ", "\[ImaginaryI]"}]}]], " ", RowBox[{"EllipticPi", "[", RowBox[{ RowBox[{"1", "-", "\[ImaginaryI]"}], ",", RowBox[{"ArcSin", "[", FractionBox[ SqrtBox[ FractionBox[ RowBox[{ RowBox[{"(", RowBox[{"1", "+", "\[ImaginaryI]"}], ")"}], " ", RowBox[{"(", RowBox[{"1", "+", "x"}], ")"}]}], "x"]], SqrtBox["2"]], "]"}], ",", RowBox[{"-", "\[ImaginaryI]"}]}], "]"}]}]], "Output"] }, Open ]], Cell[TextData[{ "Alternativ h\[ADoubleDot]tten Sie nat\[UDoubleDot]rlich auch einfach ", Cell[BoxData[ RowBox[{ SuperscriptBox["i7", "2"], " "}]], CellChangeTimes:>{3.35117232134555*^9}], "berechnen k\[ODoubleDot]nnen:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ SuperscriptBox["i7", "2"], "//", "Together"}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"(", RowBox[{"2", "-", RowBox[{"2", " ", "\[ImaginaryI]"}]}], ")"}], " ", SuperscriptBox[ RowBox[{"EllipticPi", "[", RowBox[{ RowBox[{"1", "-", "\[ImaginaryI]"}], ",", RowBox[{"ArcSin", "[", FractionBox[ SqrtBox[ FractionBox[ RowBox[{ RowBox[{"(", RowBox[{"1", "+", "\[ImaginaryI]"}], ")"}], " ", RowBox[{"(", RowBox[{"1", "+", "x"}], ")"}]}], "x"]], SqrtBox["2"]], "]"}], ",", RowBox[{"-", "\[ImaginaryI]"}]}], "]"}], "2"]}]], "Output"] }, Open ]], Cell["\<\ Versuchen wir die Probe, indem wieder die Ableitung der Stammfunktion i6 \ berechnet wird.\ \>", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"a8", "=", RowBox[{"D", "[", RowBox[{"i8", ",", "x"}], "]"}]}]], "Input"], Cell[BoxData[ FractionBox[ RowBox[{ SqrtBox[ RowBox[{"1", "-", "\[ImaginaryI]"}]], " ", RowBox[{"(", RowBox[{ FractionBox[ RowBox[{"1", "+", "\[ImaginaryI]"}], "x"], "-", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{"1", "+", "\[ImaginaryI]"}], ")"}], " ", RowBox[{"(", RowBox[{"1", "+", "x"}], ")"}]}], SuperscriptBox["x", "2"]]}], ")"}]}], RowBox[{"2", " ", SqrtBox[ FractionBox[ RowBox[{ RowBox[{"(", RowBox[{"1", "+", "\[ImaginaryI]"}], ")"}], " ", RowBox[{"(", RowBox[{"1", "+", "x"}], ")"}]}], "x"]], " ", RowBox[{"(", RowBox[{"1", "-", FractionBox[ RowBox[{"1", "+", "x"}], "x"]}], ")"}], " ", SqrtBox[ RowBox[{"1", "-", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ FractionBox["1", "2"], "+", FractionBox["\[ImaginaryI]", "2"]}], ")"}], " ", RowBox[{"(", RowBox[{"1", "+", "x"}], ")"}]}], "x"]}]], " ", SqrtBox[ RowBox[{"1", "-", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ FractionBox["1", "2"], "-", FractionBox["\[ImaginaryI]", "2"]}], ")"}], " ", RowBox[{"(", RowBox[{"1", "+", "x"}], ")"}]}], "x"]}]]}]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"a8", "//", "Simplify"}]], "Input"], Cell[BoxData[ FractionBox["1", RowBox[{ SqrtBox[ RowBox[{ FractionBox["1", "2"], "-", FractionBox["\[ImaginaryI]", "2"]}]], " ", "x", " ", SqrtBox[ FractionBox[ RowBox[{ RowBox[{"(", RowBox[{"1", "+", "\[ImaginaryI]"}], ")"}], " ", RowBox[{"(", RowBox[{"\[ImaginaryI]", "+", "x"}], ")"}]}], "x"]], " ", SqrtBox[ FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "\[ImaginaryI]"}], "+", "x"}], ")"}], " ", RowBox[{"(", RowBox[{"1", "+", "x"}], ")"}]}], SuperscriptBox["x", "2"]]]}]]], "Output"] }, Open ]], Cell["\<\ Leider dasselbe Problem wie oben mit den Wurzeln. Wir quadrieren einfach \ wieder. i6 ist also bis auf das Vorzeichen die richtige Stammfunktion und \ mehr k\[ODoubleDot]nnen wir auch nicht erwarten, da f\[UDoubleDot]r komplexe \ x der Radikand selbst nur bis aufs Vorzeichen eindeutig bestimmt ist. \ \>", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ SuperscriptBox["a8", "2"], "//", "Together"}], "\[IndentingNewLine]", RowBox[{ SuperscriptBox["a8", "2"], "//", "Simplify"}]}], "Input"], Cell[BoxData[ FractionBox["x", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "\[ImaginaryI]"}], "+", "x"}], ")"}], " ", RowBox[{"(", RowBox[{"\[ImaginaryI]", "+", "x"}], ")"}], " ", RowBox[{"(", RowBox[{"1", "+", "x"}], ")"}]}]]], "Output"], Cell[BoxData[ FractionBox["x", RowBox[{"1", "+", "x", "+", SuperscriptBox["x", "2"], "+", SuperscriptBox["x", "3"]}]]], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Mehrfachintegrale", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ UnderoverscriptBox["\[Integral]", "0", "2"], RowBox[{ UnderoverscriptBox["\[Integral]", "0", "3"], RowBox[{"1", RowBox[{"\[DifferentialD]", "y"}], RowBox[{"\[DifferentialD]", "x"}]}]}]}]], "Input"], Cell[BoxData["6"], "Output"] }, Open ]], Cell["\<\ Bei vielen bestimmten Mehrfachintegralen ist die Reihenfolge des \ Abintegrierens relevant, da die Grenzen des inneren Integrals von der \ \[ADoubleDot]u\[SZ]eren Integrationsvariablen abh\[ADoubleDot]ngen k\ \[ODoubleDot]nnen! Die beiden folgenden Integrale sind gleichwertig, einmal \ in der Formelschreibweise, einmal als konventionelles Kommando; beachten Sie \ beim zweiten Beispiel, dass das innere Integral durch den \ \[ADoubleDot]u\[SZ]eren Parameter angegeben wird (w\[ADoubleDot]hrend bei der \ Formelschreibweise das innere Integral nat\[UDoubleDot]rlich innen steht)\ \>", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ UnderoverscriptBox["\[Integral]", "0", "1"], RowBox[{ UnderoverscriptBox["\[Integral]", "0", SqrtBox[ RowBox[{"1", "-", SuperscriptBox["x", "2"]}]]], RowBox[{"1", RowBox[{"\[DifferentialD]", "y"}], RowBox[{"\[DifferentialD]", "x"}]}]}]}]], "Input"], Cell[BoxData[ FractionBox["\[Pi]", "4"]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Integrate", "[", RowBox[{"1", ",", " ", RowBox[{"{", RowBox[{"x", ",", "0", ",", "1"}], "}"}], ",", " ", RowBox[{"{", RowBox[{"y", ",", "0", ",", SqrtBox[ RowBox[{"1", "-", SuperscriptBox["x", "2"]}]]}], "}"}]}], "]"}]], 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