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Instead, the superposition itself has to be \ transformed for the proper results.\nThe Imaginary part of the superposition \ can then be compared." }], "Text", CellChangeTimes->{{3.6709081343505793`*^9, 3.670908266765153*^9}, 3.670908384707899*^9, {3.6709086535072737`*^9, 3.670908672890382*^9}, { 3.6709087048342094`*^9, 3.6709087461145706`*^9}, {3.670920033931196*^9, 3.670920121498205*^9}, {3.6709201707700233`*^9, 3.6709201757063055`*^9}, { 3.670920242937151*^9, 3.6709205279424524`*^9}, {3.702284992766015*^9, 3.7022849974382825`*^9}}], Cell[CellGroupData[{ Cell["imaginary error function", "Subsection", CellChangeTimes->{{3.673934309605707*^9, 3.673934319540275*^9}}], Cell[TextData[{ "The error function for complex arguments, erf(z), can be approximated (see \ Stegun or old notebooks for details) and holds well for arguments around -10 \ to 10. This approximation can be simplified erf(z)=erf(x+\[ImaginaryI]y) with \ x=0.\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"erf", "(", RowBox[{"0", "+", RowBox[{"\[ImaginaryI]", " ", "y"}]}], ")"}], "=", RowBox[{ FractionBox[ RowBox[{"\[ImaginaryI]", " ", "y"}], "\[Pi]"], "+", RowBox[{ FractionBox[ RowBox[{"2", "\[ImaginaryI]"}], "\[Pi]"], "\[CapitalSigma]", FractionBox[ RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{ RowBox[{"-", SuperscriptBox["n", "2"]}], "/", "4"}]], "*", " ", RowBox[{"sinh", "[", RowBox[{"n", "*", "y"}], "]"}]}], "n"]}]}]}], TraditionalForm]]], "\nThe imaginary error function, as used in the gaussian definition, is \ given by erfi(z)= -\[ImaginaryI] erf(\[ImaginaryI] z). Given that the \ argument passed to the imaginary error function in a complete antisymmetrized \ gaussian will always be real (cause we don\[CloseCurlyQuote]t do imaginary \ frequencies), the argument passed onto the error function will always have a \ real part of 0 and the aforementioned approximation can be used.\nAs such:\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"erfi", "(", RowBox[{ SubscriptBox["x", "i"], "+", RowBox[{"\[ImaginaryI]", "*", "0"}]}], ")"}], "=", " ", RowBox[{ RowBox[{ RowBox[{"-", "\[ImaginaryI]"}], " ", RowBox[{"erf", "(", RowBox[{"\[ImaginaryI]", "*", RowBox[{"(", RowBox[{ SubscriptBox["x", "i"], "+", RowBox[{"\[ImaginaryI]", "*", "0"}]}], ")"}]}], ")"}]}], "=", RowBox[{ RowBox[{ RowBox[{"-", "\[ImaginaryI]"}], " ", RowBox[{"erf", "(", RowBox[{ RowBox[{"\[ImaginaryI]", " ", SubscriptBox["x", "i"]}], "-", "0"}], ")"}]}], "=", RowBox[{ RowBox[{ RowBox[{"-", "\[ImaginaryI]"}], " ", RowBox[{"erf", "(", RowBox[{"\[ImaginaryI]", " ", SubscriptBox["x", "i"]}], ")"}]}], "=", " ", RowBox[{ RowBox[{ RowBox[{"-", "\[ImaginaryI]"}], " ", "*", RowBox[{"(", RowBox[{ FractionBox[ RowBox[{"\[ImaginaryI]", " ", SubscriptBox["x", "i"]}], "\[Pi]"], "+", RowBox[{ FractionBox[ RowBox[{"2", "\[ImaginaryI]"}], "\[Pi]"], "\[CapitalSigma]", FractionBox[ RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{ RowBox[{"-", SuperscriptBox["n", "2"]}], "/", "4"}]], "*", " ", RowBox[{"sinh", "[", RowBox[{"n", "*", SubscriptBox["x", "i"]}], "]"}]}], "n"]}]}], ")"}]}], "=", RowBox[{ FractionBox[ SubscriptBox["x", "i"], "\[Pi]"], "+", RowBox[{ FractionBox["2", "\[Pi]"], "\[CapitalSigma]", FractionBox[ RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{ RowBox[{"-", SuperscriptBox["n", "2"]}], "/", "4"}]], "*", " ", RowBox[{"sinh", "[", RowBox[{"n", "*", SubscriptBox["x", "i"]}], "]"}]}], "n"]}]}]}]}]}]}]}], TraditionalForm]]] }], "Text", CellChangeTimes->{{3.6738782985530577`*^9, 3.6738788572530136`*^9}, 3.702285004182668*^9}], Cell[BoxData[ RowBox[{ RowBox[{"erfiI0CD", "=", RowBox[{"Compile", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"xI0CD", ",", "_Real"}], "}"}], ",", RowBox[{"{", RowBox[{"nI0CD", ",", "_Real"}], "}"}], ",", RowBox[{"{", RowBox[{"inversepi", ",", "_Real"}], "}"}]}], "}"}], ",", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"xI0CD", "*", "inversepi"}], ")"}], "+", RowBox[{"(", RowBox[{"2.", "*", "inversepi", "*", RowBox[{"Sum", "[", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"Exp", "[", RowBox[{ RowBox[{"-", "0.25"}], "*", RowBox[{"(", RowBox[{"n", "^", "2"}], ")"}]}], "]"}], "*", RowBox[{ RowBox[{"Sinh", "[", RowBox[{"n", "*", "xI0CD"}], "]"}], "/", "n"}]}], ")"}], ",", RowBox[{"{", RowBox[{"n", ",", "1.", ",", "nI0CD", ",", "1."}], "}"}]}], "]"}]}], ")"}]}], ")"}]}], "]"}]}], ";"}]], "Input", CellChangeTimes->{{3.673877974042497*^9, 3.673878011369632*^9}, { 3.6738788869477124`*^9, 3.6738790363862596`*^9}, 3.673938383351712*^9}] }, Open ]], Cell[CellGroupData[{ Cell["complete Brownian oscillator", "Subsection", CellChangeTimes->{{3.6684127614541545`*^9, 3.6684127705116725`*^9}, { 3.668778615474676*^9, 3.6687786170187645`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"brownian", "[", RowBox[{"afBO_", ",", "o0fBO_", ",", "gammafBO_", ",", "omegafBO_"}], "]"}], ":=", RowBox[{ RowBox[{"(", RowBox[{"afBO", "*", RowBox[{"(", RowBox[{"o0fBO", "^", "2."}], ")"}]}], ")"}], "/", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"o0fBO", "^", "2."}], ")"}], "-", RowBox[{"omegafBO", RowBox[{"(", RowBox[{"omegafBO", "+", RowBox[{"(", RowBox[{"\[ImaginaryI]", " ", "2.", "gammafBO"}], ")"}]}], ")"}]}]}], ")"}]}]}], ";"}]], "Input", CellChangeTimes->{{3.668768672939996*^9, 3.668768677881279*^9}, { 3.66876875955195*^9, 3.6687691585337706`*^9}, {3.6687703254455137`*^9, 3.668770348860853*^9}, {3.6687703894291735`*^9, 3.66877040247692*^9}, { 3.668771491773224*^9, 3.6687715039729223`*^9}, 3.668776585801586*^9, 3.668851579456978*^9, {3.668862118461774*^9, 3.6688621188537965`*^9}, { 3.670756327524726*^9, 3.670756328508782*^9}, {3.6709237722720175`*^9, 3.67092377354309*^9}, {3.6739351247913327`*^9, 3.6739351273344784`*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["complete antisymmetric Gaussian", "Subsection", CellChangeTimes->{{3.668412774046875*^9, 3.668412790606822*^9}, { 3.6687786192028894`*^9, 3.668778621163002*^9}}], Cell["\<\ Cutoffs are inserted to avoid errors from numbers becoming too big/small. \ They are justified once the values involved become negligible, especially \ since an approximation is involved anyway and the data fitted has some \ assumptions and a not too high signal to noise.\[LineSeparator]The cutoff \ depends on the value becoming less than a thousandth and as such negligible, \ due to only minimal changes in the tail vs computational efficiency For the exponential, if \\!\\(TraditionalForm\\`x >= \\ 2.63\\), then the \ term goes to zero. Use the absolute because it is squared anyway For the product of the exponential and the imaginary error function the \ cutoff is \\!\\(TraditionalForm\\`x >= \\ 11.71\\), after which it can go to \ zero. Use the absolute, since the imaginary error function is uneven. The \ values will only switch sign, which doesn' t matter\ \>", "Text", CellChangeTimes->{{3.6743787747653847`*^9, 3.6743788255722904`*^9}, { 3.674379438775364*^9, 3.6743795031670465`*^9}}], Cell[TextData[Cell[BoxData[ FormBox[ FractionBox[ RowBox[{"\[Omega]", " ", RowBox[{"Im", "[", "f", "]"}]}], RowBox[{"c", " ", RowBox[{"Re", "[", SqrtBox["f"], "]"}]}]], TraditionalForm]]]], "Text", CellChangeTimes->{{3.674306533936445*^9, 3.674306585314384*^9}}], Cell["\<\ We make the transformation after adding a real constant, so while the \ imaginary part can go to zero the real part shouldn' t just because of the \ cutoff. 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