Scatter and Glow - BCH Formulas

Pensieve Header: Several BCH mod CC formulas.

Failed Tests

Dynamic[$FailLog]

Testing Code

SetAttributes[{Test, Expect}, {HoldAll}];
Test[expr_] := If[TrueQ[Check[expr, False]], True,
If[Head[$FailLog] =!= List, $FailLog={}];
AppendTo[$FailLog, "On "<>ToString[Date[]]<>" failed in "<>ToString[HoldForm[expr]]];
Print[Last[$FailLog]]
];
Expect[val_, expr_] := If[TrueQ[Test[val==expr]], val];

SetDirectory["C:\\drorbn\\AcademicPensieve\\Projects\\ScatterAndGlow"]
<< ScatterAndGlow.m

Test Test

Test[0==1]

exp(x)exp(y)=exp(x+y+bch1)

(lhs1=S[sigma[1,3],sigma[2,3]]) // Short

(rhs1 = S[Exp[Ar[1,3]+Ar[2,3]+Y[1,2,3,bc]]]) // Short

bch1 = bc /. First[HSolve[
Coefficient[
Ar[0,1] // lhs1,
Y[0,1,3]
] == Coefficient[
Ar[0,1] // rhs1,
Y[0,1,3]
],
bc
]]

Test[CanonicalForm[
S[Exp[Ar[1,3]+Ar[2,3]+Y[1,2,3,AH[bch1]]]] == S[sigma[1,3],sigma[2,3]]
]]

Compare with Kurlin

Test[Simplify[(bch1 /. {x[1]->x, x[2]->y}) ==(1/y)(1-(((E^x)-1)/x) ((x+y)/((E^(x+y))-1)))]]

exp(x)exp(y)=exp(x+y)exp(bch2)

(lhs2 = S[sigma[1,3],sigma[2,3]]) // Short

rhs2 = S[Exp[Ar[1,3]+Ar[2,3]]]**S[Exp[Y[1,2,3,bc2]]]

bch2 = bc2 /. First[HSolve[
Coefficient[
Ar[0,1] // lhs2,
Y[0,1,3]
] == Coefficient[
Ar[0,1] // rhs2,
Y[0,1,3]
],
bc2
]]

Simplify[bch2 /. {x[1]->x, x[2]->y}]

exp(x)exp(y)exp(-x)exp(-y)=exp(bch3)

(lhs3 = S[sigma[1,3],sigma[2,3], sigbar[1,3], sigbar[2,3]]) // Short

rhs3 = S[Exp[Y[1,2,3,bc3]]]

bch3 = bc3 /. First[HSolve[
Coefficient[
Ar[0,1] // lhs3,
Y[0,1,3]
] == Coefficient[
Ar[0,1] // rhs3,
Y[0,1,3]
],
bc3
]]

Simplify[bch3 /. {x[1]->x, x[2]->y}]

Test[CanonicalForm[
S[Exp[Y[1,2,3,AH[bch3]]]] == S[sigma[1,3],sigma[2,3], sigbar[1,3], sigbar[2,3]]
]]