(* ::Package:: *)

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



(* ::Input::Initialization:: *)
Subscript[DP, x_->Subscript[D, \[Alpha]_],y_->Subscript[D, \[Beta]_]][P_][f_]:=Total[CoefficientRules[P,{x,y}]/.({m_,n_}->c_):>c D[f,{\[Alpha],m},{\[Beta],n}]]


(* ::Input::Initialization:: *)
Subscript[O\[CapitalLambda], k_][e_,l_,f_,\[Alpha]_,\[Beta]_,\[Delta]_] := Subscript[O\[CapitalLambda], k][e,l,f,\[Alpha],\[Beta],\[Delta]]=Module[
{\[Rho]h,\[Rho]e,\[Rho]l,\[Rho]f,eqn,a,b,c,sol,q,\[Lambda],\[Nu]},
\[Rho]h=({
 {-1, 0},
 {0, -1}
});  \[Rho]e=({
 {0, 0},
 {-\[Epsilon], 0}
});  \[Rho]l=({
 {-(1+1/\[Epsilon])/2, 0},
 {0, (1-1/\[Epsilon])/2}
}); \[Rho]f=({
 {0, 1},
 {0, 0}
});
eqn=MatrixExp[\[Alpha] \[Rho]f].MatrixExp[\[Beta] \[Rho]e]==MatrixExp[a \[Rho]e].MatrixExp[c (\[Rho]h-2\[Epsilon] \[Rho]l)].MatrixExp[b \[Rho]f];
Echo[sol=Solve[Thread[Flatten/@eqn], {a,b,c}][[1]] /. C[1]->0];
\[Lambda]=Simplify[E^(-f \[Alpha]-e \[Beta]+h \[Alpha] \[Beta]) Normal@Series[E^(c h + a e - 2\[Epsilon] c l + b f)/.sol, {\[Epsilon],0,k}]];
q=E^(\[Nu](f \[Alpha]+e \[Beta]-h \[Alpha] \[Beta]+e f \[Delta]));
Collect[q^-1 Subscript[DP, \[Alpha]->Subscript[D, f],\[Beta]->Subscript[D, e]][\[Lambda]][q] /. \[Nu]->(1+h \[Delta])^-1,\[Epsilon],Simplify]];


(* ::Input::Initialization:: *)
Subscript[\[CapitalLambda], k_][e_,l_,f_,\[Alpha]_,\[Beta]_,\[Delta]_] :=Module[{\[Lambda]},
\[Lambda]=Echo[Expand@Normal@Series[E^((f \[Alpha]+e \[Beta])/(1-\[Alpha] \[Beta] \[Epsilon])) (1-\[Alpha] \[Beta] \[Epsilon])^(-2 l+h/\[Epsilon]), {\[Epsilon],0,k}]/.E^_->1];
Collect[Subscript[DP, \[Alpha]->Subscript[D, f],\[Beta]->Subscript[D, e]][\[Lambda]][E^((f \[Alpha]+e \[Beta]+e f \[Delta])/(1+h \[Delta]))]/.E^_->1,\[Epsilon],Simplify]];



