CTdec
CTdec[{{qi, a}, {qj, b}, ...}, {p1, p2, ...}]
or CTdec[exp, {{qi, a}, {qj, b}, ...}, {p1, p2, ...}]
calculates the tensorial decomposition formulas for Cartesian integrals. The more common ones are saved in TIDL.
See also
Overview, Tdec, TIDL, TID.
Examples
Check that ∫dD−1qf(p,q)qi=p2pi∫dD−1qf(p,q)p⋅q
{{X1→p⋅q,X2→p2},X2X1pi}
p2pi(p⋅q)
CTdec[{{q, i}}, {p}, List -> False]
p2pi(p⋅q)
This calculates integral transformation for any ∫dD−1q1dD−1q2dD−1q3f(p,q1,q2,q3)q1iq2jq3k.
CTdec[{{Subscript[q, 1], i}, {Subscript[q, 2], j}, {Subscript[q, 3], k}}, {p}, List -> False]
(2−D)p4pkδij(p⋅q3)((p⋅q1)(p⋅q2)−p2(q1⋅q2))+(2−D)p4pjδik(p⋅q2)((p⋅q1)(p⋅q3)−p2(q1⋅q3))+(2−D)p4piδjk(p⋅q1)((p⋅q2)(p⋅q3)−p2(q2⋅q3))−(2−D)p6pipjpk((D−1)(p⋅q1)(p⋅q2)(p⋅q3)+2(p⋅q1)(p⋅q2)(p⋅q3)−p2(q1⋅q2)(p⋅q3)−p2(q1⋅q3)(p⋅q2)−p2(q2⋅q3)(p⋅q1))
Contract[% CVD[p, i] CVD[p, j] CVD[p, k]] // Factor
(p⋅q1)(p⋅q2)(p⋅q3)