Re: Re: Re: Boundary cond. at Infinity
- To: mathgroup@smc.vnet.net
- Subject: [mg12253] Re: [mg12202] Re: [mg12160] Re: [mg12097] Boundary cond. at Infinity
- From: "Jrgen_Tischer" <jtischer@pitagoras.univalle.edu.co>
- Date: Tue, 5 May 1998 03:29:49 -0400
Hopefully nobody will get mad about me, leaving all that stuff in my
message. The thing is, up to now I was not once able to read those
added notebooks. Am I alone with that problem? I tried to copy the
notebook in a file to read it afterwards, I tried to fill it directly
in a new notebook, the result is always the same. Mathematica is quite
willing to read and encounters a syntax error in line (in this case)
97. And while complaining, a lot of the messages (mine included) come
out as sancocho (which is a local dish comparable with a stew). And
lately the timestamps obviously are mixed up, while reading messages I
get deja vus en masse.
I would be glad to find some fellow sufferers, or even better some
advice to do better. I am subscribed to Mathgroup, there is no
(reasonable) way for me to connect via newsgroup.
Jürgen
Universidad del Valle
Cali
Colombia
-----Original Message-----
From: Bob Hanlon <BobHanlon@aol.com> To: mathgroup@smc.vnet.net
Subject: [mg12253] [mg12202] Re: [mg12160] Re: [mg12097] Boundary cond. at
Infinity
>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\>"
>]
>