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x
=
α
0
+
α
1
q
+
β
1
p
+
α
2
(
q
2
p
2
)+2
β
2
itqp
+
−
1)
s
r
q
r−
2
s
p
2
s
+
[
2
]
N
+
α
r
(
−
(D.28)
2
s
r
=3
s
=0
(
−
1)
s
+1
r
q
r−
2
s
+1
p
2
s−
1
,
[
r
+1
2
]
N
β
r
+
2
s −
1
r
=3
s
=1
y
=
β
0
+
β
1
q
+
α
1
p
+
β
2
(
q
2
p
2
)+2
α
2
qp
−
−
1)
s
r
q
r−
2
s
p
2
s
+
[
2
]
N
−
β
r
(
−
(D.29)
2
s
r
=3
s
=0
1)
s
+1
r
q
r−
2
s
+1
p
2
s−
1
.
[
r
+1
2
]
N
+
α
r
(
−
2
s
−
1
r
=3
s
=1
It should be mentioned that the fundamental solution is
not
in the class of separation of variables,
namely of type
f
(
q
)
g
(
p
). Accordingly, we present an alternative fundamental solution of the
d'Alembert-Euler equations (Cauchy-Riemann equations) subject to the integrability conditions
of the harmonicity type now in the class of
separation of variables
.
End of Proof.
The fundamental solution (
D.15
), (
D.16
), (
D.28
), and (
D.29
) of the equations which govern
conformal mapping of type isometric cha-cha-cha can be interpreted as following. In matrix
notation, namely based upon the Kronecker-Zehfuss product, we write
x
y
=
α
0
+(
α
1
I+
β
1
A)
q
+
β
0
p
+
vec
α
2
,
vec
−β
2
q
p
q
p
+O
3
,
β
2
α
2
⊗
(D.30)
β
2
−α
2
α
2
−β
2
where we identify the transformation group of motion (translation (
α
0
,β
0
), rotation
β
1
,in
total three parameters), the transformation group of dilatation (one parameter
α
1
)andthe
special-conformal transformation (two parameters (
α
2
,β
2
)), actually the six-parameter O(2
,
2)
sub-algebra of the infinite dimensional conformal algebra C(
2
.
We here note in passing that the “small rotation parameter”
β
1
operates on the antisymmetric
matrix (
D.31
), while the matrices
{
H
1
,
H
2
∞
) in two dimensions
{
q,p
}∈
R
}
, which generate the special-conformal transforma-
tion are traceless and symmetric, a property being enjoyed by all coecient matrices of confor-
mal transformations of higher order. A more detailed information is
Boulware et al.
(
1970
)and
Ferrara et al.
(
1972
).
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