Pensieve header: Delta function games.

Integration

As in Talks/Beijing-2407

CF[ω_.ℰ_𝔼]:=CF[ω]CF/@ℰ;
CF[ℰ_List]:=CF/@ℰ;
CF[q_,vp_,h_]:= Module[{H},Expand@Collect[q,vp,H]/.H->h];
CF[q_,vp_]:=CF[q,vp,Factor];
CF[q_]:=CF[q,ϵ|(Subscript[(p|x|π|ξ), _])|(SubPlus[(p|x|π|ξ)])|(SubMinus[(p|x|π|ξ)])];

𝔼/:𝔼[A_]𝔼[B_]:=𝔼[A+B];

$π=Identity;

Unprotect[Integrate]; 
∫ω_.𝔼[L_]\[DifferentialD](vs_List):=Module[{n,L0,Q,Δ,G,Z0,Z,λ,DZ,DDZ,FZ,a,b},
n=Length@vs; L0=L/.ϵ->0;
Q=Table[(-(∂_(vs[[a]],vs[[b]]))L0)/.Thread[vs->0]/.(Subscript[(p|x), __])->0,{a,n},{b,n}];
If[(Δ=Det[Q])==0,Return@"Degenerate Q!"];
Z=Z0=CF@$π[L+vs.Q.vs/2]; G=Inverse[Q];
FixedPoint[(DZ=Table[(∂_v)Z,{v,vs}];
DDZ=Table[(∂_u)DZ,{u,vs}];
FZ=Sum[G[[a,b]](DDZ[[a,b]]+DZ[[a]]DZ[[b]]),{a,n},{b,n}]/2;
Z=CF[Z0+(∫_0^λ)$π[FZ] \[DifferentialD]λ])&,Z];
PowerExpand@Factor[ω (Δ^(-1/2))]𝔼[CF[Z/.λ->1/.Thread[vs->0]]]];
Protect[Integrate];

∫𝔼[- μ (x^2)/2+I ξ x]\[DifferentialD]{x}

FofG=∫𝔼[- μ (((x-a))^2)/2+I ξ x]\[DifferentialD]{x}

∫FofG 𝔼[-I ξ x] \[DifferentialD]{ξ}

L=-(1/2){(Subscript[x, 1]),(Subscript[x, 2])}.(({{a, b}, {b, c}})).{(Subscript[x, 1]),(Subscript[x, 2])}+{(Subscript[ξ, 1]),(Subscript[ξ, 2])}.{(Subscript[x, 1]),(Subscript[x, 2])};
Z12=∫𝔼[L]\[DifferentialD]{(Subscript[x, 1]),(Subscript[x, 2])}

{Z1=∫𝔼[L]\[DifferentialD]{(Subscript[x, 1])}, Z12==∫Z1 \[DifferentialD]{(Subscript[x, 2])}}

$π=Normal[#+((O[ϵ])^13)]&; ∫𝔼[-(ϕ^2)/2+ϵ (ϕ^3)/6]\[DifferentialD]{ϕ}

(∫𝔼[I x y+λ((x^2)+(y^2))+I α y]\[DifferentialD]{x,y})/.λ->0

$π=Normal[#+((O[ϵ])^3)]&;
(∫𝔼[I x y+λ((x^2)+(y^2))+ϵ (x^3)+I α y]\[DifferentialD]{x,y})/.λ->0

$π=Normal[#+((O[ϵ])^6)]&;
(∫𝔼[I x y+λ((x^2)+(y^2))+ϵ (x^3)+I α y]\[DifferentialD]{x,y})/.λ->0

$π=Normal[#+((O[ϵ])^2)]&;
(∫𝔼[I x y+λ((x^2)+(y^2))+ϵ (x^4)+I α y]\[DifferentialD]{x,y})/.λ->0

$π=Normal[#+((O[ϵ])^3)]&;
(∫𝔼[I x y+λ((x^2)+(y^2))+ϵ (x^4)+I α y]\[DifferentialD]{x,y})/.λ->0