(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 7.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 59464, 1988] NotebookOptionsPosition[ 44756, 1625] NotebookOutlinePosition[ 55417, 1869] CellTagsIndexPosition[ 55339, 1864] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell["Math 421 \[FilledSmallCircle] Fall 2010", "Subsubtitle", CellChangeTimes->{{3.4961540348125*^9, 3.4961540350625*^9}}, TextAlignment->Center], Cell[CellGroupData[{ Cell[TextData[{ "The Cauchy-Riemann equations do ", StyleBox["not", FontSlant->"Italic"], " suffice for differentiability" }], "Subtitle", TextAlignment->Center, TextJustification->0], Cell["18 October 2010", "Subsubtitle", CellChangeTimes->{{3.48893028171875*^9, 3.48893030315625*^9}, { 3.489583451984375*^9, 3.489583455796875*^9}, 3.49028881771875*^9, { 3.4905515529375*^9, 3.490551553125*^9}, 3.490875430171875*^9, { 3.492293113328125*^9, 3.492293120375*^9}, 3.49346163509375*^9, { 3.49410698178125*^9, 3.494106982265625*^9}, 3.494416320359375*^9, { 3.4961541773125*^9, 3.496154183953125*^9}, 3.496418994327352*^9}, TextAlignment->Center, TextJustification->0], Cell["\<\ Copyright \[Copyright] 2004\[Dash]2006 by Murray Eisenberg. All rights \ reserved.\ \>", "SmallText", TextAlignment->Center, TextJustification->0], Cell[TextData[{ "This notebook does ", StyleBox["not", FontSlant->"Italic"], " use ", StyleBox["Presentations", FontSlant->"Italic"], "." }], "Text", CellChangeTimes->{{3.496158111609375*^9, 3.49615812084375*^9}}], Cell[CellGroupData[{ Cell["Introduction", "Section", CellChangeTimes->{{3.4961581305*^9, 3.49615813228125*^9}}], Cell["\<\ This notebook shows, by means of two different examples, that a function can \ satisfy the Cauchy-Riemann equations at a point and yet not be differentiable \ at that point.\ \>", "Text", CellChangeTimes->{{3.496158136484375*^9, 3.496158186*^9}}] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Example 1: ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"f", "(", "z", ")"}], "=", RowBox[{ RowBox[{ FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"z", "\[Conjugate]"}], ")"}], "2"], "z"], " ", "if", " ", "z"}], "\[NotEqual]", "0"}]}], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"f", "(", "0", ")"}], "=", "0"}], TraditionalForm]]] }], "Section", CellChangeTimes->{{3.4958773677786922`*^9, 3.4958773770913115`*^9}, { 3.496154054484375*^9, 3.496154069859375*^9}, 3.49615420975*^9}], Cell["This example is from Mathews & Howell, page 104.", "Text"], Cell[TextData[{ "To form the conjugate notation ", StyleBox["z\[Conjugate]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " in ", StyleBox["Mathematica", FontSlant->"Italic"], ", type ", StyleBox["z\[ThinSpace]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], StyleBox["\[EscapeKey]", FontFamily->"Courier", FontSize->18, FontWeight->"Plain", FontSlant->"Plain"], StyleBox["co", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], StyleBox["\[EscapeKey]", FontFamily->"Courier", FontSize->18, FontWeight->"Plain", FontSlant->"Plain"], StyleBox[". 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Cell[BoxData[ FractionBox[ RowBox[{ RowBox[{"v", "[", RowBox[{"\[CapitalDelta]x", ",", "0"}], "]"}], "-", RowBox[{"v", "[", RowBox[{"0", ",", " ", "0"}], "]"}]}], "\[CapitalDelta]x"]], "Input", CellChangeTimes->{{3.4958776941565695`*^9, 3.4958776978597183`*^9}}], Cell[TextData[{ "Taking the limit of those difference quotients as ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox[ RowBox[{"\[CapitalDelta]", "x"}]], "\[Rule]", "0"}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox[ RowBox[{"\[CapitalDelta]", "y"}]], "\[Rule]", "0"}], TraditionalForm]]], ", respectively, we obtain: \t\n\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ FractionBox[ RowBox[{"\[PartialD]", " ", "u"}], RowBox[{"\[PartialD]", "y"}]], RowBox[{"(", RowBox[{"0", ",", "0"}], ")"}]}], "=", RowBox[{"0", "=", " ", RowBox[{ RowBox[{"-", FractionBox[ RowBox[{"\[PartialD]", " ", "v"}], RowBox[{"\[PartialD]", "x"}]]}], RowBox[{"(", RowBox[{"0", ",", "0"}], ")"}]}]}]}], TraditionalForm]]], "." }], "Text", 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And that is contrary to the fact \ already established\[LongDash]that ", Cell[BoxData[ FormBox["f", TraditionalForm]], FormatType->"TraditionalForm"], " is ", StyleBox["not", FontSlant->"Italic"], " differentiable at ", Cell[BoxData[ FormBox[ RowBox[{"z", "=", "0"}], TraditionalForm]], FormatType->"TraditionalForm"] }], "Text", CellChangeTimes->{{3.496155056078125*^9, 3.496155085453125*^9}, { 3.496155496765625*^9, 3.496155599296875*^9}}, ScriptSizeMultipliers->{1.}, CellTags->"spurious limit"], Cell[TextData[{ "This is only an apparent contradiction, which is due to the way ", StyleBox["Mathematica", FontSlant->"Italic"], " calculates limits. This is one place that ", StyleBox["Mathematica", FontSlant->"Italic"], " treats variables as if they were real rather than complex. 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We have to help ", StyleBox["Mathematica", FontSlant->"Italic"], " here!" }], "Text", CellChangeTimes->{{3.49615765028125*^9, 3.4961576809375*^9}, { 3.4961579821875*^9, 3.496157992515625*^9}}], Cell[TextData[{ "To remedy the situation, first rewrite ", StyleBox["u[x,\[ThinSpace]y]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " and ", StyleBox["v[x,\[ThinSpace]y]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " so they no longer involve ", StyleBox["Abs", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ":" }], "Text", CellChangeTimes->{{3.496157685859375*^9, 3.496157747046875*^9}, 3.4961582111875*^9}], Cell[BoxData[{ RowBox[{ RowBox[{"uu", "[", RowBox[{"x_", ",", "y_"}], "]"}], ":=", RowBox[{ RowBox[{ RowBox[{"u", "[", RowBox[{"x", ",", "y"}], "]"}], "/.", RowBox[{ RowBox[{"Abs", "[", RowBox[{"x", "+", RowBox[{"\[ImaginaryI]", " ", "y"}]}], "]"}], "\[Rule]", SqrtBox[ RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox["y", 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