(* 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[     22979,        631]
NotebookOptionsPosition[     21720,        585]
NotebookOutlinePosition[     22065,        600]
CellTagsIndexPosition[     22022,        597]
WindowFrame->Normal*)

(* Beginning of Notebook Content *)
Notebook[{
Cell[BoxData[
 RowBox[{"Quit", "[", "]"}]], "Input",
 CellChangeTimes->{{3.7776368928813496`*^9, 3.7776368936851254`*^9}}],

Cell[BoxData[
 RowBox[{"(*", 
  RowBox[{"Paquete", " ", "para", " ", "correr"}], "*)"}]], "Input",
 CellChangeTimes->{{3.7776502042363377`*^9, 3.7776502190208445`*^9}}],

Cell[BoxData[
 RowBox[{"Needs", "[", "\"\<PlotLegends`\>\"", "]"}]], "Input"],

Cell[BoxData[
 StyleBox[
  RowBox[{"(*", " ", 
   RowBox[{
    RowBox[{
     RowBox[{
     "Consideremos", " ", "las", " ", "soluciones", " ", "que", " ", 
      "encontramos", " ", "para", " ", "el", " ", "problema", " ", "que", " ",
       "hicimos", " ", "en", " ", 
      RowBox[{"clase", ".", " ", "El"}], " ", "problema", " ", 
      "consist\[IAcute]a", " ", "en", " ", "un", " ", "plano", " ", "ubicado",
       " ", "en", " ", "z"}], "=", 
     RowBox[{
      RowBox[{
       RowBox[{
       "0", " ", "que", " ", "estaba", " ", "completamente", " ", "a", " ", 
        "tierra", " ", "excepto", " ", "por", " ", "un", " ", "circulo", " ", 
        "de", " ", "radio", " ", "\"\<a\>\"", " ", "centrado", " ", "que", 
        " ", "estaba", " ", "a", " ", "potencial", " ", 
        RowBox[{"V0", ".", " ", "Con"}], " ", "el", " ", "metodo", " ", "de", 
        " ", "Green", " ", "pudimos", " ", "obtener", " ", "el", " ", 
        "potencial", " ", "en", " ", "el", " ", "semi"}], "-", 
       RowBox[{"eje", " ", "z"}]}], ">", 
      RowBox[{
       RowBox[{"0", " ", "y", " ", "en", " ", "el", " ", "semi"}], "-", 
       RowBox[{"eje", " ", "z"}]}], "<", 
      RowBox[{"0.", " ", "Luego"}]}]}], ",", " ", 
    RowBox[{
    "usando", " ", "la", " ", "extensi\[OAcute]n", " ", "anal\[IAcute]tica", 
     " ", "pudimos", " ", "obtener", " ", "el", " ", "potencial", " ", "en", 
     " ", "todo", " ", "el", " ", "espacio", " ", "como", " ", "una", " ", 
     "serie", " ", "en", " ", "\"\<r\>\""}], ",", " ", 
    RowBox[{
    "con", " ", "polinomios", " ", "de", " ", "Legendre", " ", "del", " ", 
     RowBox[{
      RowBox[{"Cos", "[", "\[Theta]", "]"}], ".", " ", "Por"}], " ", "otra", 
     " ", "parte", " ", "vimos", " ", "tambi\[EAcute]n", " ", "que", " ", 
     "el", " ", "mismo", " ", "problema", " ", "pod\[IAcute]a", " ", "ser", 
     " ", "atacado", " ", "usando", " ", "separaci\[OAcute]n", " ", "de", " ",
      "variables", " ", "en", " ", "coordenadas", " ", "cil\[IAcute]ndricas", 
     " ", "obteniendo", " ", "una", " ", "expresi\[OAcute]n", " ", "del", " ",
      "mismo", " ", "como", " ", "una", " ", "transformada", " ", "de", " ", 
     "Hankel"}]}], " ", "*)"}],
  FontSize->16]], "Input",
 CellChangeTimes->{{3.7776346172557917`*^9, 3.7776346580118036`*^9}, {
   3.777634729616378*^9, 3.777634746281397*^9}, 3.7776348031868305`*^9, {
   3.777636455347522*^9, 3.7776366298460016`*^9}, {3.777636821698852*^9, 
   3.777636824597107*^9}, {3.777649360268688*^9, 3.7776493945540175`*^9}, {
   3.777650303477969*^9, 3.7776503076119184`*^9}}],

Cell[CellGroupData[{

Cell[TextData[{
 "Definimos el potencial que obtuvimos por el primer m\[EAcute]todo. Este est\
\[AAcute] dado por una serie infinita que decidimos cortar en el valor \
\[OpenCurlyDoubleQuote]n\[CloseCurlyDoubleQuote]. A su vez, como tenemos \
simetr\[IAcute]a de revoluci\[OAcute]n, nos interesar\[AAcute] graficar el \
potencial en una superficie fija (e imaginar que su forma tiene simetr\
\[IAcute]a de revoluci\[OAcute]n) y elegimos el plano x=0. As\[IAcute], ",
 Cell[BoxData[
  FormBox[
   RowBox[{"r", "=", 
    SqrtBox[
     RowBox[{
      SuperscriptBox["y", "2"], "+", 
      SuperscriptBox["z", "2"]}]]}], TraditionalForm]]],
 "y ",
 Cell[BoxData[
  FormBox[
   RowBox[{
    RowBox[{"Cos", "[", "\[Theta]", "]"}], "=", 
    FractionBox["z", 
     SqrtBox[
      RowBox[{
       SuperscriptBox["y", "2"], "+", 
       SuperscriptBox["z", "2"]}]]]}], TraditionalForm]]],
 ". Por \[UAcute]ltimo, elegimos fijar V0=1 y a=1 de manera que medimos el \
potencial en unidades de V0 y medimos las distancias en unidades de \
\[OpenCurlyDoubleQuote]a\[CloseCurlyDoubleQuote]. Como vimos en la \
pr\[AAcute]ctica debemos definir el potencial por tramos separando para r<a y \
r>a :"
}], "Subsection",
 CellChangeTimes->{{3.7776366379878283`*^9, 3.7776368878214064`*^9}, {
  3.777638815388977*^9, 3.77763881613056*^9}, {3.777649455888994*^9, 
  3.777649458884981*^9}}],

Cell[BoxData[{
 RowBox[{
  RowBox[{
   RowBox[{"\[Phi]1", "[", 
    RowBox[{"y_", ",", "z_", ",", "n_"}], "]"}], ":=", 
   RowBox[{
    RowBox[{"1", "-", 
     RowBox[{"1", 
      RowBox[{"Sign", "[", "z", "]"}], 
      RowBox[{"Sum", "[", 
       RowBox[{
        RowBox[{
         FractionBox[
          RowBox[{
           RowBox[{"Pochhammer", "[", 
            RowBox[{
             RowBox[{"1", "/", "2"}], ",", "l"}], "]"}], 
           SuperscriptBox[
            RowBox[{"(", 
             RowBox[{"-", "1"}], ")"}], "l"]}], 
          RowBox[{"l", "!"}]], 
         SuperscriptBox[
          RowBox[{"(", 
           SqrtBox[
            RowBox[{
             SuperscriptBox["y", "2"], "+", 
             SuperscriptBox["z", "2"]}]], ")"}], 
          RowBox[{
           RowBox[{"2", "l"}], "+", "1"}]], 
         RowBox[{"LegendreP", "[", 
          RowBox[{
           RowBox[{
            RowBox[{"2", "l"}], "+", "1"}], ",", 
           FractionBox["z", 
            SqrtBox[
             RowBox[{
              SuperscriptBox["y", "2"], "+", 
              SuperscriptBox["z", "2"]}]]]}], "]"}]}], ",", 
        RowBox[{"{", 
         RowBox[{"l", ",", "0", ",", "n"}], "}"}]}], "]"}]}]}], "/;", 
    RowBox[{
     RowBox[{
      SuperscriptBox["y", "2"], "+", 
      SuperscriptBox["z", "2"]}], "<", "1"}]}]}], 
  ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"\[Phi]1", "[", 
   RowBox[{"y_", ",", "z_", ",", "n_"}], "]"}], ":=", 
  RowBox[{
   RowBox[{
    RowBox[{"-", "1"}], 
    RowBox[{"Sign", "[", "z", "]"}], 
    RowBox[{"Sum", "[", 
     RowBox[{
      RowBox[{
       FractionBox[
        RowBox[{
         RowBox[{"Pochhammer", "[", 
          RowBox[{
           RowBox[{"1", "/", "2"}], ",", "l"}], "]"}], 
         SuperscriptBox[
          RowBox[{"(", 
           RowBox[{"-", "1"}], ")"}], "l"]}], 
        RowBox[{"l", "!"}]], 
       SuperscriptBox[
        RowBox[{"(", 
         FractionBox["1", 
          SqrtBox[
           RowBox[{
            SuperscriptBox["y", "2"], "+", 
            SuperscriptBox["z", "2"]}]]], ")"}], 
        RowBox[{"2", "l"}]], 
       RowBox[{"LegendreP", "[", 
        RowBox[{
         RowBox[{
          RowBox[{"2", "l"}], "-", "1"}], ",", 
         FractionBox["z", 
          SqrtBox[
           RowBox[{
            SuperscriptBox["y", "2"], "+", 
            SuperscriptBox["z", "2"]}]]]}], "]"}]}], ",", 
      RowBox[{"{", 
       RowBox[{"l", ",", "1", ",", "n"}], "}"}]}], "]"}]}], "/;", 
   RowBox[{
    RowBox[{
     SuperscriptBox["y", "2"], "+", 
     SuperscriptBox["z", "2"]}], "\[GreaterEqual]", "1"}]}]}]}], "Input",
 CellChangeTimes->{{3.777633890432028*^9, 3.7776338905913963`*^9}, {
  3.7776339478856306`*^9, 3.7776340033674984`*^9}, {3.7776340345913305`*^9, 
  3.777634134821755*^9}, {3.777635999627184*^9, 3.7776359997968426`*^9}, {
  3.777646313030835*^9, 3.77764631920019*^9}}]
}, Open  ]],

Cell[CellGroupData[{

Cell["\<\
Podemos entonces graficar la intensidad del potencial en el plano x=0. Vemos \
que en z=0 el potencial es nulo para r>a y muy intenso para r<a. Visto desde \
lejos r>>a el potencial es parecido al \[OpenCurlyDoubleQuote]doble hongo\
\[CloseCurlyDoubleQuote] de un doble dipolo (como veremos m\[AAcute]s abajo).\
\>", "Subsection",
 CellChangeTimes->{{3.7776372651590395`*^9, 3.7776373395878763`*^9}, {
  3.7776375693524942`*^9, 3.7776375719465246`*^9}, {3.777638122036089*^9, 
  3.777638133002202*^9}, {3.7776490788029184`*^9, 3.777649081261346*^9}, {
  3.7776492897476764`*^9, 3.7776492898175287`*^9}}],

Cell[BoxData[
 RowBox[{"ShowLegend", "[", 
  RowBox[{
   RowBox[{"DensityPlot", "[", 
    RowBox[{
     RowBox[{"\[Phi]1", "[", 
      RowBox[{"y", ",", "z", ",", "10"}], "]"}], ",", 
     RowBox[{"{", 
      RowBox[{"y", ",", 
       RowBox[{"-", "3"}], ",", "3"}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{"z", ",", 
       RowBox[{"-", "3"}], ",", "3"}], "}"}], ",", 
     RowBox[{"ColorFunction", "\[Rule]", "\"\<SunsetColors\>\""}]}], "]"}], 
   ",", 
   RowBox[{"{", 
    RowBox[{
     RowBox[{
      RowBox[{
       RowBox[{"ColorData", "[", "\"\<SunsetColors\>\"", "]"}], "[", 
       RowBox[{"1", "-", "#1"}], "]"}], "&"}], ",", "10", ",", "\"\< Max\>\"",
      ",", "\"\<Min\>\"", ",", 
     RowBox[{"LegendPosition", "\[Rule]", 
      RowBox[{"{", 
       RowBox[{"1.1", ",", 
        RowBox[{"-", ".4"}]}], "}"}]}]}], "}"}]}], "]"}]], "Input",
 CellChangeTimes->{{3.7776343373916826`*^9, 3.777634429431342*^9}, {
   3.7776344604219575`*^9, 3.777634502812387*^9}, {3.7776345549052286`*^9, 
   3.7776345853136587`*^9}, {3.777634807230543*^9, 3.777634818980336*^9}, {
   3.777635246119358*^9, 3.777635258787628*^9}, {3.777635499490532*^9, 
   3.777635500102043*^9}, {3.7776373943993397`*^9, 3.777637394483428*^9}, {
   3.777649730684574*^9, 3.7776497494304523`*^9}, {3.777649788243662*^9, 
   3.777649802541459*^9}, 3.7776498788480663`*^9, {3.777649979918604*^9, 
   3.777650024134508*^9}, {3.7776504913996167`*^9, 3.7776504916948285`*^9}}]
}, Open  ]],

Cell[CellGroupData[{

Cell["\<\
Si graficamos la intensidad del potencial cerca del plano z=0 y en bien \
dentro de la regi\[OAcute]n interior (r<a) obtenemos el potencial de un plano \
infinito:\
\>", "Subsection",
 CellChangeTimes->{{3.7776490857583165`*^9, 3.777649138458272*^9}}],

Cell[BoxData[
 RowBox[{"ShowLegend", "[", 
  RowBox[{
   RowBox[{"DensityPlot", "[", 
    RowBox[{
     RowBox[{"\[Phi]1", "[", 
      RowBox[{"y", ",", "z", ",", "10"}], "]"}], ",", 
     RowBox[{"{", 
      RowBox[{"y", ",", 
       RowBox[{
        RowBox[{"-", "1"}], "/", "4"}], ",", 
       RowBox[{"1", "/", "4"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{"z", ",", 
       RowBox[{
        RowBox[{"-", "1"}], "/", "4"}], ",", 
       RowBox[{"1", "/", "4"}]}], "}"}], ",", 
     RowBox[{"ColorFunction", "\[Rule]", "\"\<SunsetColors\>\""}]}], "]"}], 
   ",", 
   RowBox[{"{", 
    RowBox[{
     RowBox[{
      RowBox[{
       RowBox[{"ColorData", "[", "\"\<SunsetColors\>\"", "]"}], "[", 
       RowBox[{"1", "-", "#1"}], "]"}], "&"}], ",", "10", ",", "\"\< Max\>\"",
      ",", "\"\<Min\>\"", ",", 
     RowBox[{"LegendPosition", "\[Rule]", 
      RowBox[{"{", 
       RowBox[{"1.1", ",", 
        RowBox[{"-", ".4"}]}], "}"}]}]}], "}"}]}], "]"}]], "Input",
 CellChangeTimes->{{3.777648929845415*^9, 3.7776489398545074`*^9}, {
  3.777649062264143*^9, 3.7776490671999474`*^9}, {3.7776500539108877`*^9, 
  3.777650089492737*^9}}]
}, Open  ]],

Cell[CellGroupData[{

Cell["\<\
Aqu\[IAcute] definimos el modo k-esimo de la soluci\[OAcute]n exterior r>a (y \
seteamos a cero la soluci\[OAcute]n interior r<a porque solo nos interesa el \
desarrollo multipolar exterior). Como vemos de arriba, las contribuciones son \
todas impares en k. Luego graficamos con \[OpenCurlyDoubleQuote]Manipulate\
\[CloseCurlyDoubleQuote] para ver las contribuciones de cada modo. Es \
interesante ver el patron de nodos y m\[AAcute]ximos que surge al aumentar k. \
Esta claro de este an\[AAcute]lisis, al comparar con el gr\[AAcute]fico de \
arriba, que es el modo k=1 el que domina la forma de la soluci\[OAcute]n \
exterior.\
\>", "Subsection",
 CellChangeTimes->{{3.7776381882783527`*^9, 3.777638266977609*^9}, {
  3.77763887154248*^9, 3.7776388842102656`*^9}, {3.77764841763236*^9, 
  3.7776484666502404`*^9}, {3.7776485256793976`*^9, 3.777648569598188*^9}, {
  3.777648809908737*^9, 3.7776488348461475`*^9}, {3.7776495146668673`*^9, 
  3.7776495518683558`*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{
   RowBox[{"\[Phi]mode", "[", 
    RowBox[{"y_", ",", "z_", ",", "k_"}], "]"}], ":=", 
   RowBox[{"0", "/;", 
    RowBox[{
     RowBox[{
      SuperscriptBox["y", "2"], "+", 
      SuperscriptBox["z", "2"]}], "<", "1"}]}]}], ";", 
  RowBox[{
   RowBox[{"\[Phi]mode", "[", 
    RowBox[{"y_", ",", "z_", ",", "k_"}], "]"}], ":=", 
   RowBox[{
    RowBox[{
     RowBox[{"-", "1"}], 
     RowBox[{"Sign", "[", "z", "]"}], 
     FractionBox[
      RowBox[{
       RowBox[{"Pochhammer", "[", 
        RowBox[{
         RowBox[{"1", "/", "2"}], ",", 
         FractionBox[
          RowBox[{"k", "+", "1"}], "2"]}], "]"}], 
       SuperscriptBox[
        RowBox[{"(", 
         RowBox[{"-", "1"}], ")"}], 
        FractionBox[
         RowBox[{"k", "+", "1"}], "2"]]}], 
      RowBox[{
       RowBox[{"(", 
        FractionBox[
         RowBox[{"k", "+", "1"}], "2"], ")"}], "!"}]], 
     SuperscriptBox[
      RowBox[{"(", 
       FractionBox["1", 
        SqrtBox[
         RowBox[{
          SuperscriptBox["y", "2"], "+", 
          SuperscriptBox["z", "2"]}]]], ")"}], 
      RowBox[{"k", "+", "1"}]], 
     RowBox[{"LegendreP", "[", 
      RowBox[{"k", ",", 
       FractionBox["z", 
        SqrtBox[
         RowBox[{
          SuperscriptBox["y", "2"], "+", 
          SuperscriptBox["z", "2"]}]]]}], "]"}]}], "/;", 
    RowBox[{
     RowBox[{
      SuperscriptBox["y", "2"], "+", 
      SuperscriptBox["z", "2"]}], "\[GreaterEqual]", "1"}]}]}]}]], "Input",
 CellChangeTimes->{{3.7776483500996294`*^9, 3.7776483607172565`*^9}, {
   3.7776485969490128`*^9, 3.777648619207492*^9}, {3.7776486973919907`*^9, 
   3.77764871296889*^9}, 3.777648762527687*^9, {3.777648805171419*^9, 
   3.7776488065018625`*^9}}],

Cell[BoxData[
 RowBox[{"Manipulate", "[", 
  RowBox[{
   RowBox[{"ShowLegend", "[", 
    RowBox[{
     RowBox[{"DensityPlot", "[", 
      RowBox[{
       RowBox[{"\[Phi]mode", "[", 
        RowBox[{"y", ",", "z", ",", "k"}], "]"}], ",", 
       RowBox[{"{", 
        RowBox[{"y", ",", 
         RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", 
       RowBox[{"{", 
        RowBox[{"z", ",", 
         RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", 
       RowBox[{"PlotPoints", "\[Rule]", "100"}], ",", 
       RowBox[{"ColorFunction", "\[Rule]", "\"\<SunsetColors\>\""}]}], "]"}], 
     ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{
         RowBox[{"ColorData", "[", "\"\<SunsetColors\>\"", "]"}], "[", 
         RowBox[{"1", "-", "#1"}], "]"}], "&"}], ",", "10", ",", 
       "\"\< Max\>\"", ",", "\"\<Min\>\"", ",", 
       RowBox[{"LegendPosition", "\[Rule]", 
        RowBox[{"{", 
         RowBox[{"1.1", ",", 
          RowBox[{"-", ".4"}]}], "}"}]}]}], "}"}]}], "]"}], ",", 
   RowBox[{"{", 
    RowBox[{"k", ",", "1", ",", "11", ",", "2"}], "}"}]}], "]"}]], "Input",
 CellChangeTimes->{{3.7776350578344727`*^9, 3.777635088213995*^9}, 
   3.777635657718528*^9, {3.7776357502350454`*^9, 3.7776357629825954`*^9}, {
   3.7776358489644036`*^9, 3.7776358675224886`*^9}, {3.7776361437778616`*^9, 
   3.7776361946789274`*^9}, {3.777636269192482*^9, 3.777636300425773*^9}, {
   3.777637626205959*^9, 3.7776376265984735`*^9}, {3.777637698689543*^9, 
   3.777637740392474*^9}, {3.777645670408416*^9, 3.7776456986868415`*^9}, {
   3.7776496872686763`*^9, 3.7776496915681763`*^9}, {3.777650112758523*^9, 
   3.7776501148967676`*^9}}]
}, Open  ]],

Cell[CellGroupData[{

Cell["\<\
Aqu\[IAcute] en cambio definimos nuestro potencial como una integral n\
\[UAcute]merica que aproxima al potencial que obtuvimos como una transformada \
de Hankel. Vemos que este gr\[AAcute]fico es computacionalmente mucho m\
\[AAcute]s costoso que la serie de polinomios de Legendre.\
\>", "Subsection",
 CellChangeTimes->{{3.77763827894414*^9, 3.7776383511701717`*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"\[Phi]2", "[", 
   RowBox[{"y_", ",", "z_"}], "]"}], ":=", 
  RowBox[{"NIntegrate", "[", 
   RowBox[{
    RowBox[{
     RowBox[{"BesselJ", "[", 
      RowBox[{"1", ",", "k"}], "]"}], 
     RowBox[{"BesselJ", "[", 
      RowBox[{"0", ",", " ", 
       RowBox[{"k", " ", 
        RowBox[{"Abs", "[", "y", "]"}]}]}], "]"}], 
     SuperscriptBox["\[ExponentialE]", 
      RowBox[{
       RowBox[{"-", "k"}], " ", 
       RowBox[{"Abs", "[", "z", "]"}]}]]}], ",", 
    RowBox[{"{", 
     RowBox[{"k", ",", "0", ",", "\[Infinity]"}], "}"}]}], "]"}]}]], "Input",
 CellChangeTimes->{{3.777635334545019*^9, 3.7776354233557687`*^9}}],

Cell[BoxData[
 RowBox[{"p2", "=", 
  RowBox[{"ShowLegend", "[", 
   RowBox[{
    RowBox[{"DensityPlot", "[", 
     RowBox[{
      RowBox[{"\[Phi]2", "[", 
       RowBox[{"y", ",", "z"}], "]"}], ",", 
      RowBox[{"{", 
       RowBox[{"y", ",", 
        RowBox[{"-", "3"}], ",", "3"}], "}"}], ",", 
      RowBox[{"{", 
       RowBox[{"z", ",", 
        RowBox[{"-", "3"}], ",", "3"}], "}"}], ",", 
      RowBox[{"ColorFunction", "\[Rule]", "\"\<SunsetColors\>\""}]}], "]"}], 
    ",", 
    RowBox[{"{", 
     RowBox[{
      RowBox[{
       RowBox[{
        RowBox[{"ColorData", "[", "\"\<SunsetColors\>\"", "]"}], "[", 
        RowBox[{"1", "-", "#1"}], "]"}], "&"}], ",", "10", ",", 
      "\"\< Max\>\"", ",", "\"\<Min\>\"", ",", 
      RowBox[{"LegendPosition", "\[Rule]", 
       RowBox[{"{", 
        RowBox[{"1.1", ",", 
         RowBox[{"-", ".4"}]}], "}"}]}]}], "}"}]}], "]"}]}]], "Input",
 CellChangeTimes->{{3.7776354265076733`*^9, 3.777635428975958*^9}, {
  3.7776355055606318`*^9, 3.7776355066348433`*^9}, {3.777650139243713*^9, 
  3.7776501451778316`*^9}}]
}, Open  ]],

Cell[CellGroupData[{

Cell["\<\
Aqu\[IAcute] graficamos ambos resultados a la par. Esto es, el resultado como \
una serie en \[OpenCurlyDoubleQuote]r\[CloseCurlyDoubleQuote] con polinomios \
de Legendre y el resultado como una transformada de Henkel. Vemos la \
similitud entre ambos gr\[AAcute]ficos.\
\>", "Subsection",
 CellChangeTimes->{{3.7776383586438427`*^9, 3.777638380307536*^9}, {
  3.777649236422279*^9, 3.7776492646917176`*^9}, {3.7776503778253202`*^9, 
  3.777650387704899*^9}}],

Cell[BoxData[
 RowBox[{"GraphicsRow", "[", 
  RowBox[{"{", 
   RowBox[{
    RowBox[{"DensityPlot", "[", 
     RowBox[{
      RowBox[{"\[Phi]1", "[", 
       RowBox[{"y", ",", "z", ",", "10"}], "]"}], ",", 
      RowBox[{"{", 
       RowBox[{"y", ",", 
        RowBox[{"-", "3"}], ",", "3"}], "}"}], ",", 
      RowBox[{"{", 
       RowBox[{"z", ",", 
        RowBox[{"-", "3"}], ",", "3"}], "}"}], ",", 
      RowBox[{"ColorFunction", "\[Rule]", "\"\<SunsetColors\>\""}]}], "]"}], 
    ",", 
    RowBox[{"DensityPlot", "[", 
     RowBox[{
      RowBox[{"\[Phi]2", "[", 
       RowBox[{"y", ",", "z"}], "]"}], ",", 
      RowBox[{"{", 
       RowBox[{"y", ",", 
        RowBox[{"-", "3"}], ",", "3"}], "}"}], ",", 
      RowBox[{"{", 
       RowBox[{"z", ",", 
        RowBox[{"-", "3"}], ",", "3"}], "}"}], ",", 
      RowBox[{"ColorFunction", "\[Rule]", "\"\<SunsetColors\>\""}]}], "]"}]}],
    "}"}], "]"}]], "Input",
 CellChangeTimes->{{3.7776355150196943`*^9, 3.777635549506809*^9}, {
  3.7776504447144833`*^9, 3.777650455965405*^9}}]
}, Open  ]],

Cell[CellGroupData[{

Cell["\<\
El gr\[AAcute]fico izquierdo a continuaci\[OAcute]n es el doble hongo que me \
refer\[IAcute]a m\[AAcute]s arriba. Corresponde al campo de un dipolo \
apuntando hacia arriba (si lo miro desde arriba) y un dipolo apuntando hacia \
abajo (si lo miro desde abajo). Grafico a la derecha la funcion \[Phi]1[y,z] \
(nuestra primer solucion) con una escala que va -10 a 10 en ambas variables, \
como si estuviera haciendo un \[OpenCurlyDoubleQuote]zoom back\
\[CloseCurlyDoubleQuote] de nuestra soluci\[OAcute]n. Vemos que ambos gr\
\[AAcute]ficos son muy similares.\
\>", "Subsection",
 CellChangeTimes->{{3.777638447946154*^9, 3.7776384986384077`*^9}, {
  3.77764606817378*^9, 3.7776461012273216`*^9}, {3.77764615669676*^9, 
  3.7776462288729744`*^9}, {3.777646420049168*^9, 3.7776464251880765`*^9}}],

Cell[BoxData[
 RowBox[{"GraphicsRow", "[", 
  RowBox[{"{", 
   RowBox[{
    RowBox[{"DensityPlot", "[", 
     RowBox[{
      FractionBox[
       RowBox[{"Abs", "[", "z", "]"}], 
       SuperscriptBox[
        RowBox[{"(", 
         SqrtBox[
          RowBox[{
           SuperscriptBox["y", "2"], "+", 
           SuperscriptBox["z", "2"]}]], ")"}], "3"]], ",", 
      RowBox[{"{", 
       RowBox[{"y", ",", 
        RowBox[{"-", "10"}], ",", "10"}], "}"}], ",", 
      RowBox[{"{", 
       RowBox[{"z", ",", 
        RowBox[{"-", "10"}], ",", "10"}], "}"}], ",", 
      RowBox[{"PlotPoints", "\[Rule]", "100"}], ",", 
      RowBox[{"ColorFunction", "\[Rule]", "\"\<SunsetColors\>\""}]}], "]"}], 
    ",", 
    RowBox[{"DensityPlot", "[", 
     RowBox[{
      RowBox[{"\[Phi]1", "[", 
       RowBox[{"y", ",", "z", ",", "10"}], "]"}], ",", 
      RowBox[{"{", 
       RowBox[{"y", ",", 
        RowBox[{"-", "10"}], ",", "10"}], "}"}], ",", 
      RowBox[{"{", 
       RowBox[{"z", ",", 
        RowBox[{"-", "10"}], ",", "10"}], "}"}], ",", 
      RowBox[{"ColorFunction", "\[Rule]", "\"\<SunsetColors\>\""}], ",", 
      RowBox[{"PlotPoints", "\[Rule]", "100"}]}], "]"}]}], "}"}], 
  "]"}]], "Input",
 CellChangeTimes->{{3.7776380397373767`*^9, 3.7776380741181154`*^9}, {
  3.777638517000923*^9, 3.777638522290079*^9}, {3.777646037757348*^9, 
  3.777646052181508*^9}, {3.7776461128724365`*^9, 3.777646147298676*^9}}]
}, Open  ]]
},
WindowSize->{950, 504},
WindowMargins->{{24, Automatic}, {Automatic, 31}},
FrontEndVersion->"8.0 for Microsoft Windows (64-bit) (February 23, 2011)",
StyleDefinitions->"Default.nb"
]
(* End of Notebook Content *)

(* Internal cache information *)
(*CellTagsOutline
CellTagsIndex->{}
*)
(*CellTagsIndex
CellTagsIndex->{}
*)
(*NotebookFileOutline
Notebook[{
Cell[557, 20, 122, 2, 31, "Input"],
Cell[682, 24, 168, 3, 31, "Input"],
Cell[853, 29, 77, 1, 31, "Input"],
Cell[933, 32, 2573, 48, 264, "Input"],
Cell[CellGroupData[{
Cell[3531, 84, 1367, 32, 128, "Subsection"],
Cell[4901, 118, 2880, 90, 140, "Input"]
}, Open  ]],
Cell[CellGroupData[{
Cell[7818, 213, 613, 9, 70, "Subsection"],
Cell[8434, 224, 1454, 33, 52, "Input"]
}, Open  ]],
Cell[CellGroupData[{
Cell[9925, 262, 261, 5, 53, "Subsection"],
Cell[10189, 269, 1144, 32, 52, "Input"]
}, Open  ]],
Cell[CellGroupData[{
Cell[11370, 306, 978, 15, 104, "Subsection"],
Cell[12351, 323, 1740, 55, 96, "Input"],
Cell[14094, 380, 1650, 38, 72, "Input"]
}, Open  ]],
Cell[CellGroupData[{
Cell[15781, 423, 380, 6, 70, "Subsection"],
Cell[16164, 431, 658, 19, 33, "Input"],
Cell[16825, 452, 1069, 29, 52, "Input"]
}, Open  ]],
Cell[CellGroupData[{
Cell[17931, 486, 469, 8, 53, "Subsection"],
Cell[18403, 496, 1038, 29, 52, "Input"]
}, Open  ]],
Cell[CellGroupData[{
Cell[19478, 530, 805, 12, 87, "Subsection"],
Cell[20286, 544, 1418, 38, 95, "Input"]
}, Open  ]]
}
]
*)

(* End of internal cache information *)