Re: Re: Boundary cond. at Infinity
- To: mathgroup@smc.vnet.net
- Subject: [mg12202] Re: [mg12160] Re: [mg12097] Boundary cond. at Infinity
- From: Bob Hanlon <BobHanlon@aol.com>
- Date: Fri, 1 May 1998 03:08:51 -0400
I believethat this notebook is a clearer and more accurate response than my first response. Bob Hanlon __________________________ Notebook[{ Cell[BoxData[ \(TraditionalForm\`Needs["\<Algebra`InequalitySolve`\>"]\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{\(f(x_, a_)\), ":=", TagBox[ RowBox[{\(\(\[ThinSpace]\_2\) F\_1\), "(", RowBox[{ RowBox[{ TagBox[\(\(a + 1\)\/2\), (Editable -> True)], ",", TagBox["1", (Editable -> True)]}], ";", TagBox[\(\(a + 1\)\/2 + 1\), (Editable -> True)], ";", TagBox[\(-\[ExponentialE]\^\(2\ x\)\), (Editable -> True)]}], ")"}], InterpretTemplate[ Hypergeometric2F1[ #, #2, #3, #4]&]]}], TraditionalForm]], "Input"], Cell["For n=0,1,2,3,... there is a discontinuity at", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(TraditionalForm\`Solve[\(a + 1\)\/2 + 1 == \(-n\), a]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`{{a \[Rule] \(-2\)\ n - 3}}\)], "Output"] }, Open ]], Cell["That is, there is a discontinuity for a = -3,-5,-7,...", "Text"], Cell[BoxData[ \(TraditionalForm \`\(Plot3D[f[x, a], \ {a, \(-8.1\), 4.1}, {x, \(-1.5\), 4}, PlotPoints \[Rule] 35, AxesLabel \[Rule] {"\<a\>", "\<x\>", None}, PlotRange -> {\(-50\), 50}, \ ImageSize -> {450, 365}]; \)\)], "Input"], Cell[TextData[{ "Since the argument (-", Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^\(2 x\)\)]], ") is negative, the terms of the hypergeometric series alternate signs. \ This is not a convenient form for determining the limit. Using a linear \ transformation (Abramowitz and Stegun, 15.3.5) to obtain a positive \ argument:" }], "Text"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"trans", "=", RowBox[{ TagBox[ RowBox[{\(\(\[ThinSpace]\_2\) F\_1\), "(", RowBox[{ RowBox[{ TagBox["a_", (Editable -> True)], ",", TagBox["b_", (Editable -> True)]}], ";", TagBox["c_", (Editable -> True)], ";", TagBox["z_", (Editable -> True)]}], ")"}], InterpretTemplate[ Hypergeometric2F1[ #, #2, #3, #4]&]], "\[Rule]", FractionBox[ TagBox[ RowBox[{\(\(\[ThinSpace]\_2\) F\_1\), "(", RowBox[{ RowBox[{ TagBox["b", (Editable -> True)], ",", TagBox[\(c - a\), (Editable -> True)]}], ";", TagBox["c", (Editable -> True)], ";", TagBox[\(z\/\(z - 1\)\), (Editable -> True)]}], ")"}], InterpretTemplate[ Hypergeometric2F1[ #, #2, #3, #4]&]], \(\((1 - z)\)\^b\)]}]}], ";"}], TraditionalForm]], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(TraditionalForm\`f2\ = \ Simplify[f(x, a) /. trans]\)], "Input"], Cell[BoxData[ FormBox[ FractionBox[ TagBox[ RowBox[{\(\(\[ThinSpace]\_2\)F\_1\), "(", RowBox[{ RowBox[{ TagBox["1", (Editable -> True)], ",", TagBox["1", (Editable -> True)]}], ";", TagBox[\(\(a + 3\)\/2\), (Editable -> True)], ";", TagBox[\(\[ExponentialE]\^\(2\ x\)\/\(1 + \[ExponentialE]\^\(2\ x\)\)\), (Editable -> True)]}], ")"}], InterpretTemplate[ Hypergeometric2F1[ #, #2, #3, #4]&]], \(1 + \[ExponentialE]\^\(2\ x\)\)], TraditionalForm]], "Output"] }, Open ]], Cell["As x approaches +Infinity, and for", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(TraditionalForm\`InequalitySolve(\(a + 3\)\/2 - 1 - 1 > 0, a)\)], "Input"], Cell[BoxData[ \(TraditionalForm\`a > 1\)], "Output"] }, Open ]], Cell["the numerator will tend towards", "Text"], Cell[CellGroupData[{ Cell[BoxData[ FormBox[ TagBox[ RowBox[{\(\(\[ThinSpace]\_2\) F\_1\), "(", RowBox[{ RowBox[{ TagBox["1", (Editable -> True)], ",", TagBox["1", (Editable -> True)]}], ";", TagBox[\(\(a + 3\)\/2\), (Editable -> True)], ";", TagBox["1", (Editable -> True)]}], ")"}], InterpretTemplate[ Hypergeometric2F1[ #, #2, #3, #4]&]], TraditionalForm]], "Input"], Cell[BoxData[ \(TraditionalForm\`\(a + 1\)\/\(a - 1\)\)], "Output"] }, Open ]], Cell["\<\ And the denominator tends to Infinity. For a>1 the limit as x goes \ to +Infinity is then zero.\ \>", "Text"], Cell[BoxData[ \(TraditionalForm \`\(Plot3D[f[x, a], {a, \(-1\), 4.1}, {x, \(-1.5\), 4}, PlotPoints \[Rule] 35, AxesLabel \[Rule] {"\<a\>", "\<x\>", None}, ImageSize -> {450, 365}]; \)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(TraditionalForm\`f2\ /. \ x -> \(-Infinity\)\)], "Input"], Cell[BoxData[ \(TraditionalForm\`1\)], "Output"] }, Open ]], Cell[BoxData[ \(TraditionalForm \`\(Plot3D[f[x, a], {a, \(-8.1\), 4.1}, {x, \(-15\), \(-3\)}, PlotPoints \[Rule] 35, AxesLabel \[Rule] {"\<a\>", "\<x\>", None}, PlotRange -> {.975, 1.025}, \ ImageSize -> {450, 365}]; \)\)], "Input"] }, FrontEndVersion->"Macintosh 3.0", ScreenRectangle->{{0, 1024}, {0, 748}}, WindowSize->{729, 701}, WindowMargins->{{20, Automatic}, {Automatic, 6}}, ShowCellLabel->False, MacintoshSystemPageSetup->"\<\ 00<0001804P000000]P2:?oQon82n@960dL5:0?l0080001804P000000]P2:001 0000I00000400`<300000BL?00400@0000000000000006P801T1T00000000000 00000000000000000000000000000000\>" ]