Permalink
Cannot retrieve contributors at this time
This commit does not belong to any branch on this repository, and may belong to a fork outside of the repository.
425 lines (413 sloc)
15.3 KB
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
(* Content-type: application/vnd.wolfram.mathematica *) | |
(*** Wolfram Notebook File ***) | |
(* http://www.wolfram.com/nb *) | |
(* CreatedBy='Mathematica 12.0' *) | |
(*CacheID: 234*) | |
(* Internal cache information: | |
NotebookFileLineBreakTest | |
NotebookFileLineBreakTest | |
NotebookDataPosition[ 158, 7] | |
NotebookDataLength[ 15435, 415] | |
NotebookOptionsPosition[ 14733, 396] | |
NotebookOutlinePosition[ 15070, 411] | |
CellTagsIndexPosition[ 15027, 408] | |
WindowFrame->Normal*) | |
(* Beginning of Notebook Content *) | |
Notebook[{ | |
Cell[BoxData[ | |
RowBox[{ | |
RowBox[{"d", "[", "z_", "]"}], ":=", | |
RowBox[{"Sqrt", "[", | |
RowBox[{ | |
RowBox[{ | |
RowBox[{"(", | |
RowBox[{"kappa", "-", | |
RowBox[{"I", "*", "rho", "*", "sigma", "*", "z"}]}], ")"}], "^", "2"}], | |
"+", | |
RowBox[{ | |
RowBox[{"(", | |
RowBox[{ | |
RowBox[{"z", "*", "z"}], "+", | |
RowBox[{"I", "*", "z"}]}], ")"}], "*", | |
RowBox[{"sigma", "^", "2"}]}]}], "]"}]}]], "Input", | |
CellLabel->"In[10]:=",ExpressionUUID->"b4481ef6-407a-491d-941b-4a889e6f1688"], | |
Cell[BoxData[ | |
RowBox[{ | |
RowBox[{"g", "[", "z_", "]"}], ":=", | |
RowBox[{ | |
RowBox[{"(", | |
RowBox[{"kappa", "-", | |
RowBox[{"I", "*", "rho", "*", "sigma", "*", "z"}], "-", | |
RowBox[{"d", "[", "z", "]"}]}], ")"}], "/", | |
RowBox[{"(", | |
RowBox[{"kappa", "-", | |
RowBox[{"I", "*", "rho", "*", "sigma", "*", "z"}], "+", | |
RowBox[{"d", "[", "z", "]"}]}], ")"}]}]}]], "Input", | |
CellLabel->"In[11]:=",ExpressionUUID->"e4490652-4d06-41c8-996b-9c580e66fa02"], | |
Cell[BoxData[ | |
RowBox[{ | |
RowBox[{"phi", "[", "z_", "]"}], ":=", | |
RowBox[{"Exp", "[", | |
RowBox[{ | |
RowBox[{ | |
RowBox[{"v0", "/", | |
RowBox[{"(", | |
RowBox[{"sigma", "^", "2"}], ")"}]}], "*", | |
RowBox[{ | |
RowBox[{"(", | |
RowBox[{"1", "-", | |
RowBox[{"Exp", "[", | |
RowBox[{ | |
RowBox[{"-", | |
RowBox[{"d", "[", "z", "]"}]}], "*", "t"}], "]"}]}], ")"}], "/", | |
RowBox[{"(", | |
RowBox[{"1", "-", | |
RowBox[{ | |
RowBox[{"g", "[", "z", "]"}], "*", | |
RowBox[{"Exp", "[", | |
RowBox[{ | |
RowBox[{"-", | |
RowBox[{"d", "[", "z", "]"}]}], "*", "t"}], "]"}]}]}], ")"}]}], | |
"*", | |
RowBox[{"(", | |
RowBox[{"kappa", "-", | |
RowBox[{"I", "*", "rho", "*", "sigma", "*", "z"}], "-", | |
RowBox[{"d", "[", "z", "]"}]}], ")"}]}], "+", | |
RowBox[{"kappa", "*", | |
RowBox[{"theta", "/", | |
RowBox[{"sigma", "^", "2"}]}], "*", | |
RowBox[{"(", | |
RowBox[{ | |
RowBox[{ | |
RowBox[{"(", | |
RowBox[{"kappa", "-", | |
RowBox[{"I", "*", "rho", "*", "sigma", "*", "z"}], "-", | |
RowBox[{"d", "[", "z", "]"}]}], ")"}], "*", "t"}], "-", | |
RowBox[{"2", "*", | |
RowBox[{"Log", "[", | |
RowBox[{ | |
RowBox[{"(", | |
RowBox[{"1", "-", | |
RowBox[{ | |
RowBox[{"g", "[", "z", "]"}], "*", | |
RowBox[{"Exp", "[", | |
RowBox[{ | |
RowBox[{"-", | |
RowBox[{"d", "[", "z", "]"}]}], "*", "t"}], "]"}]}]}], ")"}], | |
"/", | |
RowBox[{"(", | |
RowBox[{"1", "-", | |
RowBox[{"g", "[", "z", "]"}]}], ")"}]}], "]"}]}]}], ")"}]}]}], | |
"]"}]}]], "Input", | |
CellLabel->"In[12]:=",ExpressionUUID->"6e208680-ed06-4137-b530-3d8b98048402"], | |
Cell[CellGroupData[{ | |
Cell[BoxData[ | |
RowBox[{"FullSimplify", "[", | |
RowBox[{ | |
RowBox[{"Series", "[", | |
RowBox[{ | |
RowBox[{"phi", "[", "z", "]"}], ",", " ", | |
RowBox[{"{", | |
RowBox[{"sigma", ",", " ", "0", ",", " ", "2"}], "}"}]}], "]"}], ",", | |
RowBox[{ | |
RowBox[{"kappa", ">", "0"}], "&&", | |
RowBox[{"theta", ">", "0"}], "&&", | |
RowBox[{"v0", ">", "0"}], "&&", | |
RowBox[{"sigma", ">", "0"}], "&&", | |
RowBox[{ | |
RowBox[{"rho", "[", "Element", "]"}], " ", | |
RowBox[{"{", | |
RowBox[{ | |
RowBox[{"-", "1"}], ",", "1"}], "}"}]}]}]}], "]"}]], "Input", | |
CellLabel->"In[13]:=",ExpressionUUID->"09368f72-2407-487f-82b4-11da81445837"], | |
Cell[BoxData[ | |
InterpretationBox[ | |
RowBox[{ | |
SuperscriptBox["\[ExponentialE]", | |
RowBox[{"-", | |
FractionBox[ | |
RowBox[{ | |
SuperscriptBox["\[ExponentialE]", | |
RowBox[{ | |
RowBox[{"-", "kappa"}], " ", "t"}]], " ", | |
RowBox[{"(", | |
RowBox[{"theta", "-", "v0", "+", | |
RowBox[{ | |
SuperscriptBox["\[ExponentialE]", | |
RowBox[{"kappa", " ", "t"}]], " ", | |
RowBox[{"(", | |
RowBox[{ | |
RowBox[{ | |
RowBox[{"(", | |
RowBox[{ | |
RowBox[{"-", "1"}], "+", | |
RowBox[{"kappa", " ", "t"}]}], ")"}], " ", "theta"}], "+", | |
"v0"}], ")"}]}]}], ")"}], " ", "z", " ", | |
RowBox[{"(", | |
RowBox[{"\[ImaginaryI]", "+", "z"}], ")"}]}], | |
RowBox[{"2", " ", "kappa"}]]}]], "+", | |
RowBox[{ | |
FractionBox["1", | |
RowBox[{"2", " ", | |
SuperscriptBox["kappa", "2"]}]], | |
RowBox[{ | |
SuperscriptBox["\[ExponentialE]", | |
RowBox[{ | |
RowBox[{ | |
RowBox[{"-", "kappa"}], " ", "t"}], "-", | |
FractionBox[ | |
RowBox[{ | |
SuperscriptBox["\[ExponentialE]", | |
RowBox[{ | |
RowBox[{"-", "kappa"}], " ", "t"}]], " ", | |
RowBox[{"(", | |
RowBox[{"theta", "-", "v0", "+", | |
RowBox[{ | |
SuperscriptBox["\[ExponentialE]", | |
RowBox[{"kappa", " ", "t"}]], " ", | |
RowBox[{"(", | |
RowBox[{ | |
RowBox[{ | |
RowBox[{"(", | |
RowBox[{ | |
RowBox[{"-", "1"}], "+", | |
RowBox[{"kappa", " ", "t"}]}], ")"}], " ", "theta"}], "+", | |
"v0"}], ")"}]}]}], ")"}], " ", "z", " ", | |
RowBox[{"(", | |
RowBox[{"\[ImaginaryI]", "+", "z"}], ")"}]}], | |
RowBox[{"2", " ", "kappa"}]]}]], " ", "rho", " ", | |
RowBox[{"(", | |
RowBox[{ | |
RowBox[{"2", " ", "theta"}], "+", | |
RowBox[{"kappa", " ", "t", " ", "theta"}], "-", "v0", "-", | |
RowBox[{"kappa", " ", "t", " ", "v0"}], "+", | |
RowBox[{ | |
SuperscriptBox["\[ExponentialE]", | |
RowBox[{"kappa", " ", "t"}]], " ", | |
RowBox[{"(", | |
RowBox[{ | |
RowBox[{ | |
RowBox[{"(", | |
RowBox[{ | |
RowBox[{"-", "2"}], "+", | |
RowBox[{"kappa", " ", "t"}]}], ")"}], " ", "theta"}], "+", | |
"v0"}], ")"}]}]}], ")"}], " ", | |
RowBox[{"(", | |
RowBox[{"1", "-", | |
RowBox[{"\[ImaginaryI]", " ", "z"}]}], ")"}], " ", | |
SuperscriptBox["z", "2"], " ", "sigma"}]}], "+", | |
RowBox[{ | |
FractionBox["1", | |
RowBox[{"16", " ", | |
SuperscriptBox["kappa", "4"]}]], | |
RowBox[{ | |
SuperscriptBox["\[ExponentialE]", | |
RowBox[{ | |
RowBox[{ | |
RowBox[{"-", "2"}], " ", "kappa", " ", "t"}], "-", | |
FractionBox[ | |
RowBox[{ | |
SuperscriptBox["\[ExponentialE]", | |
RowBox[{ | |
RowBox[{"-", "kappa"}], " ", "t"}]], " ", | |
RowBox[{"(", | |
RowBox[{"theta", "-", "v0", "+", | |
RowBox[{ | |
SuperscriptBox["\[ExponentialE]", | |
RowBox[{"kappa", " ", "t"}]], " ", | |
RowBox[{"(", | |
RowBox[{ | |
RowBox[{ | |
RowBox[{"(", | |
RowBox[{ | |
RowBox[{"-", "1"}], "+", | |
RowBox[{"kappa", " ", "t"}]}], ")"}], " ", "theta"}], "+", | |
"v0"}], ")"}]}]}], ")"}], " ", "z", " ", | |
RowBox[{"(", | |
RowBox[{"\[ImaginaryI]", "+", "z"}], ")"}]}], | |
RowBox[{"2", " ", "kappa"}]]}]], " ", | |
SuperscriptBox["z", "2"], " ", | |
RowBox[{"(", | |
RowBox[{"\[ImaginaryI]", "+", "z"}], ")"}], " ", | |
RowBox[{"(", | |
RowBox[{ | |
RowBox[{ | |
RowBox[{"-", "2"}], " ", | |
SuperscriptBox["rho", "2"], " ", | |
SuperscriptBox[ | |
RowBox[{"(", | |
RowBox[{ | |
RowBox[{"2", " ", "theta"}], "+", | |
RowBox[{"kappa", " ", "t", " ", "theta"}], "-", "v0", "-", | |
RowBox[{"kappa", " ", "t", " ", "v0"}], "+", | |
RowBox[{ | |
SuperscriptBox["\[ExponentialE]", | |
RowBox[{"kappa", " ", "t"}]], " ", | |
RowBox[{"(", | |
RowBox[{ | |
RowBox[{ | |
RowBox[{"(", | |
RowBox[{ | |
RowBox[{"-", "2"}], "+", | |
RowBox[{"kappa", " ", "t"}]}], ")"}], " ", "theta"}], "+", | |
"v0"}], ")"}]}]}], ")"}], "2"], " ", | |
SuperscriptBox["z", "2"], " ", | |
RowBox[{"(", | |
RowBox[{"\[ImaginaryI]", "+", "z"}], ")"}]}], "+", | |
RowBox[{"2", " ", "kappa", " ", "v0", " ", | |
RowBox[{"(", | |
RowBox[{ | |
RowBox[{"-", "\[ImaginaryI]"}], "-", "z", "+", | |
RowBox[{ | |
SuperscriptBox["\[ExponentialE]", | |
RowBox[{"2", " ", "kappa", " ", "t"}]], " ", | |
RowBox[{"(", | |
RowBox[{"\[ImaginaryI]", "+", "z", "+", | |
RowBox[{"4", " ", | |
SuperscriptBox["rho", "2"], " ", "z"}]}], ")"}]}], "-", | |
RowBox[{"2", " ", | |
SuperscriptBox["\[ExponentialE]", | |
RowBox[{"kappa", " ", "t"}]], " ", | |
RowBox[{"(", | |
RowBox[{ | |
RowBox[{"2", " ", | |
SuperscriptBox["rho", "2"], " ", "z"}], "+", | |
RowBox[{"kappa", " ", "t", " ", | |
RowBox[{"(", | |
RowBox[{"\[ImaginaryI]", "+", "z", "+", | |
RowBox[{ | |
SuperscriptBox["rho", "2"], " ", | |
RowBox[{"(", | |
RowBox[{"2", "+", | |
RowBox[{"kappa", " ", "t"}]}], ")"}], " ", "z"}]}], | |
")"}]}]}], ")"}]}]}], ")"}]}], "+", | |
RowBox[{"kappa", " ", "theta", " ", | |
RowBox[{"(", | |
RowBox[{"\[ImaginaryI]", "+", "z", "+", | |
RowBox[{ | |
SuperscriptBox["\[ExponentialE]", | |
RowBox[{"2", " ", "kappa", " ", "t"}]], " ", | |
RowBox[{"(", | |
RowBox[{ | |
RowBox[{ | |
RowBox[{"-", "5"}], " ", "\[ImaginaryI]"}], "-", | |
RowBox[{"5", " ", "z"}], "-", | |
RowBox[{"24", " ", | |
SuperscriptBox["rho", "2"], " ", "z"}], "+", | |
RowBox[{"2", " ", "kappa", " ", "t", " ", | |
RowBox[{"(", | |
RowBox[{"\[ImaginaryI]", "+", "z", "+", | |
RowBox[{"4", " ", | |
SuperscriptBox["rho", "2"], " ", "z"}]}], ")"}]}]}], ")"}]}], | |
"+", | |
RowBox[{"4", " ", | |
SuperscriptBox["\[ExponentialE]", | |
RowBox[{"kappa", " ", "t"}]], " ", | |
RowBox[{"(", | |
RowBox[{"\[ImaginaryI]", "+", "z", "+", | |
RowBox[{"6", " ", | |
SuperscriptBox["rho", "2"], " ", "z"}], "+", | |
RowBox[{"kappa", " ", "t", " ", | |
RowBox[{"(", | |
RowBox[{"\[ImaginaryI]", "+", "z", "+", | |
RowBox[{ | |
SuperscriptBox["rho", "2"], " ", | |
RowBox[{"(", | |
RowBox[{"4", "+", | |
RowBox[{"kappa", " ", "t"}]}], ")"}], " ", "z"}]}], | |
")"}]}]}], ")"}]}]}], ")"}]}]}], ")"}], " ", | |
SuperscriptBox["sigma", "2"]}]}], "+", | |
InterpretationBox[ | |
SuperscriptBox[ | |
RowBox[{"O", "[", "sigma", "]"}], "3"], | |
SeriesData[$CellContext`sigma, 0, {}, 0, 3, 1], | |
Editable->False]}], | |
SeriesData[$CellContext`sigma, 0, { | |
E^(Rational[-1, 2] | |
E^(-$CellContext`kappa $CellContext`t) $CellContext`kappa^(-1) \ | |
($CellContext`theta - $CellContext`v0 + | |
E^($CellContext`kappa $CellContext`t) ((-1 + $CellContext`kappa \ | |
$CellContext`t) $CellContext`theta + $CellContext`v0)) $CellContext`z ( | |
Complex[0, 1] + $CellContext`z)), Rational[1, 2] | |
E^(-$CellContext`kappa $CellContext`t + | |
Rational[-1, 2] | |
E^(-$CellContext`kappa $CellContext`t) $CellContext`kappa^(-1) \ | |
($CellContext`theta - $CellContext`v0 + | |
E^($CellContext`kappa $CellContext`t) ((-1 + $CellContext`kappa \ | |
$CellContext`t) $CellContext`theta + $CellContext`v0)) $CellContext`z ( | |
Complex[0, | |
1] + $CellContext`z)) $CellContext`kappa^(-2) $CellContext`rho ( | |
2 $CellContext`theta + $CellContext`kappa $CellContext`t \ | |
$CellContext`theta - $CellContext`v0 - $CellContext`kappa $CellContext`t \ | |
$CellContext`v0 + | |
E^($CellContext`kappa $CellContext`t) ((-2 + $CellContext`kappa \ | |
$CellContext`t) $CellContext`theta + $CellContext`v0)) (1 + | |
Complex[0, -1] $CellContext`z) $CellContext`z^2, Rational[1, 16] | |
E^((-2) $CellContext`kappa $CellContext`t + | |
Rational[-1, 2] | |
E^(-$CellContext`kappa $CellContext`t) $CellContext`kappa^(-1) \ | |
($CellContext`theta - $CellContext`v0 + | |
E^($CellContext`kappa $CellContext`t) ((-1 + $CellContext`kappa \ | |
$CellContext`t) $CellContext`theta + $CellContext`v0)) $CellContext`z ( | |
Complex[0, | |
1] + $CellContext`z)) $CellContext`kappa^(-4) $CellContext`z^2 ( | |
Complex[0, | |
1] + $CellContext`z) ((-2) $CellContext`rho^2 ( | |
2 $CellContext`theta + $CellContext`kappa $CellContext`t \ | |
$CellContext`theta - $CellContext`v0 - $CellContext`kappa $CellContext`t \ | |
$CellContext`v0 + | |
E^($CellContext`kappa $CellContext`t) ((-2 + $CellContext`kappa \ | |
$CellContext`t) $CellContext`theta + $CellContext`v0))^2 $CellContext`z^2 ( | |
Complex[0, 1] + $CellContext`z) + | |
2 $CellContext`kappa $CellContext`v0 (Complex[0, -1] - $CellContext`z + | |
E^(2 $CellContext`kappa $CellContext`t) ( | |
Complex[0, 1] + $CellContext`z + | |
4 $CellContext`rho^2 $CellContext`z) - 2 | |
E^($CellContext`kappa $CellContext`t) ( | |
2 $CellContext`rho^2 $CellContext`z + $CellContext`kappa \ | |
$CellContext`t ( | |
Complex[0, 1] + $CellContext`z + $CellContext`rho^2 ( | |
2 + $CellContext`kappa $CellContext`t) $CellContext`z))) + \ | |
$CellContext`kappa $CellContext`theta (Complex[0, 1] + $CellContext`z + | |
E^(2 $CellContext`kappa $CellContext`t) (Complex[0, -5] - | |
5 $CellContext`z - 24 $CellContext`rho^2 $CellContext`z + | |
2 $CellContext`kappa $CellContext`t (Complex[0, 1] + $CellContext`z + | |
4 $CellContext`rho^2 $CellContext`z)) + | |
4 E^($CellContext`kappa $CellContext`t) ( | |
Complex[0, 1] + $CellContext`z + | |
6 $CellContext`rho^2 $CellContext`z + $CellContext`kappa \ | |
$CellContext`t ( | |
Complex[0, 1] + $CellContext`z + $CellContext`rho^2 ( | |
4 + $CellContext`kappa $CellContext`t) $CellContext`z))))}, 0, 3, | |
1], | |
Editable->False]], "Output", | |
CellChangeTimes->{3.79741834385044*^9}, | |
CellLabel->"Out[13]=",ExpressionUUID->"77d6afba-14b1-4c80-bf7b-a471094d8260"] | |
}, Open ]] | |
}, | |
WindowSize->{808, 897}, | |
WindowMargins->{{Automatic, -2320}, {73, Automatic}}, | |
FrontEndVersion->"12.0 for Linux ARM (32-bit) (June 23, 2019)", | |
StyleDefinitions->"Default.nb" | |
] | |
(* End of Notebook Content *) | |
(* Internal cache information *) | |
(*CellTagsOutline | |
CellTagsIndex->{} | |
*) | |
(*CellTagsIndex | |
CellTagsIndex->{} | |
*) | |
(*NotebookFileOutline | |
Notebook[{ | |
Cell[558, 20, 518, 16, 31, "Input",ExpressionUUID->"b4481ef6-407a-491d-941b-4a889e6f1688"], | |
Cell[1079, 38, 473, 12, 31, "Input",ExpressionUUID->"e4490652-4d06-41c8-996b-9c580e66fa02"], | |
Cell[1555, 52, 1798, 55, 124, "Input",ExpressionUUID->"6e208680-ed06-4137-b530-3d8b98048402"], | |
Cell[CellGroupData[{ | |
Cell[3378, 111, 652, 18, 55, "Input",ExpressionUUID->"09368f72-2407-487f-82b4-11da81445837"], | |
Cell[4033, 131, 10684, 262, 367, "Output",ExpressionUUID->"77d6afba-14b1-4c80-bf7b-a471094d8260"] | |
}, Open ]] | |
} | |
] | |
*) | |
(* End of internal cache information *) | |