Pensieve Header: Semi-Symmetrized calculus in the 2D quotient.

Continues “A 2D B-Picture, IV.nb”, uses computations from “BCH in Blobs.nb”.

SetDirectory["C:\\drorbn\\AcademicPensieve\\Projects\\w-Computations"]

ar[i_,j_] := t[i]h[j]

HMultiply[x_,y_,z_][mix_] := Module[
{ξ, η,j},
ξ = D[mix, h[x]] /. t[s_] :> c[s];
η = D[mix, h[y]] /. t[s_] :> c[s];
j[ζ_] := If[ζ===0, 1, ((((E^ζ)-1))/ζ)];
(mix /. {h[x]->0, h[y]->0})+
 ((j[ξ] h[z]D[mix, h[x]])/(j[ξ+η])) + (E^ξ)((j[η] h[z]D[mix, h[y]])/(j[ξ+η])) 
]

BDisplay[expr_] := Assuming[
And @@ ((#>0)& /@ Union[Cases[expr, _c, Infinity]]),
Collect[expr, _h, Collect[#, _t, FullSimplify]&]
]

HMultiply[3,4,5][t[1]h[3]+t[2]h[4]] // BDisplay

a=t[1]h[4]+t[2]h[5]+t[3]h[6]

HMultiply[4,5,7][a] // BDisplay

HMultiply[7,6,8][HMultiply[4,5,7][a]] // BDisplay

HMultiply[4,7,8][HMultiply[5,6,7][a]] // BDisplay

(HMultiply[7,6,8][HMultiply[4,5,7][a]] - HMultiply[4,7,8][HMultiply[5,6,7][a]]) // BDisplay

HFactorize[z_, xtails_List -> x_, y_][mix_] := Module[
{ytails},
ytails = Complement[
Union[Cases[mix, t[s_] :> s, Infinity]],
xtails
];
HFactorize[z,x, ytails -> y][mix]
];
HFactorize[z_, x_, ytails_List -> y_][mix_] := Module[
{ζ, ζ0, α, α0, β, β0},
ζ0 = D[mix, h[z]];
α0 = ζ0 /. ((t[#]->0)& /@ ytails);
β0 = ζ0-α0;
{ζ, α, β} = {ζ0, α0, β0} /. t[s_] :> c[s];
Expand[mix-ζ0 h[z]+((Log[((α (E^ζ)+β)/ζ)])/α)α0 h[x]-((Log[((α +β (E^(-ζ)))/ζ)])/β)β0 h[y]]
]

HMultiply[3,4,5][ar[1,3]+ar[2,4]] // BDisplay

HMultiply[3,4,5][ar[1,3]+ar[2,4]] // HFactorize[5,{1}->3, 4] // FunctionExpand // BDisplay