(* ::Package:: *)

(************************************************************************)\012(* This file was generated automatically by the Mathematica front end.  *)\012(* It contains Initialization cells from a Notebook file, which         *)\012(* typically will have the same name as this file except ending in      *)\012(* ".nb" instead of ".m".                                               *)\012(*                                                                      *)\012(* This file is intended to be loaded into the Mathematica kernel using *)\012(* the package loading commands Get or Needs.  Doing so is equivalent   *)\012(* to using the Evaluate Initialization Cells menu command in the front *)\012(* end.                                                                 *)\012(*                                                                      *)\012(* DO NOT EDIT THIS FILE.  This entire file is regenerated              *)\012(* automatically each time the parent Notebook file is saved in the     *)\012(* Mathematica front end.  Any changes you make to this file will be    *)\012(* overwritten.                                                         *)\012(************************************************************************)\012\012

(* ::Input::Initialization:: *)
CF[\[Omega]_. \[ScriptCapitalE]_\[DoubleStruckCapitalE]]:=CF[\[Omega]]CF/@\[ScriptCapitalE];
CF[\[ScriptCapitalE]_List]:=CF/@\[ScriptCapitalE];
CF[q_]:=Expand@Collect[q,\[Epsilon]|Subscript[(p|x|\[Pi]|\[Xi]), _],F]/.F->Factor;


(* ::Input::Initialization:: *)
\[DoubleStruckCapitalE]/:\[DoubleStruckCapitalE][A_]\[DoubleStruckCapitalE][B_]:=\[DoubleStruckCapitalE][A+B];


(* ::Input::Initialization:: *)
$\[Pi]=Identity; 


(* ::Input::Initialization:: *)
Unprotect[Integrate]; 
\[Integral]\[Omega]_. \[DoubleStruckCapitalE][L_]\[DifferentialD](vs_List):=Module[{n,L0,Q,\[CapitalDelta],G,Z0,Z,\[Lambda],DZ,DDZ,FZ,a,b},
n=Length@vs; L0=L/.\[Epsilon]->0;
Q=Table[(-\!\(
\*SubscriptBox[\(\[PartialD]\), \(vs[[\(a\)\(]\)], vs[[\(b\)\(]\)]\)]L0\))/.Thread[vs->0]/.Subscript[(p|x), __]->0,{a,n},{b,n}];
If[(\[CapitalDelta]=Det[Q])==0,Return@"\!\(\*
StyleBox[\"Degenerate\",\nFontColor->RGBColor[1, 0, 0]]\)\!\(\*
StyleBox[\" \",\nFontColor->RGBColor[1, 0, 0]]\)\!\(\*
StyleBox[\"Q\",\nFontColor->RGBColor[1, 0, 0]]\)\!\(\*
StyleBox[\"!\",\nFontColor->RGBColor[1, 0, 0]]\)"];
Z=Z0=CF@$\[Pi][L+vs . Q . vs/2]; G=Inverse[Q];
FixedPoint[(DZ=Table[\!\(
\*SubscriptBox[\(\[PartialD]\), \(v\)]Z\),{v,vs}];
DDZ=Table[\!\(
\*SubscriptBox[\(\[PartialD]\), \(u\)]DZ\),{u,vs}];
FZ=Sum[G[[a,b]](DDZ[[a,b]]+DZ[[a]]DZ[[b]]),{a,n},{b,n}]/2;
Z=CF[Z0+\!\(
\*SubsuperscriptBox[\(\[Integral]\), \(0\), \(\[Lambda]\)]\($\[Pi][FZ]\ \[DifferentialD]\[Lambda]\)\)])&,Z];
PowerExpand@Factor[\[Omega] \[CapitalDelta]^(-1/2)]\[DoubleStruckCapitalE][CF[Z/.\[Lambda]->1/.Thread[vs->0]]]];
Protect[Integrate];



