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M
x
(
q,p
)=
[A
m
exp(
mq
)cos
mp
+
B
m
exp(
mq
)sin
mp
+
(D.35)
m
=1
+
C
m
exp(
−
mq
)cos
mp
+
D
m
exp(
−
mq
)sin
mp
]+
x
0
,
M
[
A
m
exp(
m
q
)cos
m
p
+
B
m
exp(
m
q
)sin
m
p
+
y
(
q,p
)=
(D.36)
m
=1
m
q
)sin
m
p
+
y
0
,
+
C
m
exp(
m
q
)cos
m
p
+
D
m
exp(
−
−
M
x
q
=
[
mA
m
exp(
mq
)cos
mp
+
mB
m
exp(
mq
)sin
mp−
(D.37)
m
=1
−
mC
m
exp(
−
mq
)cos
mp
−
mD
m
exp(
−
mq
)sin
mp
]
,
M
m
A
m
exp(
m
q
)sin
m
p
+
m
B
m
exp(
m
q
)cos
m
p
y
p
=
[
−
−
(D.38)
m
=1
m
C
m
exp(
m
q
)sin
m
p
+
m
D
m
exp(
m
q
)cos
m
p
]
−
−
−
⇔
A
m
=
B
m
,B
m
=
A
m
,C
m
=
D
m
,D
m
=
C
m
.
−
−
End of Proof.
Lemma D.4 (Fundamental solution of the d'Alembert-Euler equations subject to integrability
conditions of harmonicity, separation of variables).
A fundamental solution of the d'Alembert-Euler equations (Cauchy-Riemann equations) subject
to the integrability conditions of harmonicity type in the class of
separation of variables
is
x
(
q,p
) =
(D.39)
M
M
=
x
0
+
[
I
m
cosh
mq
+
K
m
sinh
mq
]cos
mp
+
[
J
m
cosh
mq
+
L
m
sinh
mq
]sin
mp ,
m
=1
m
=1
y
(
q,p
) =
(D.40)
M
M
=
y
0
+
[
L
m
cosh
mq
+
J
m
sinh
mq
]cos
mp
+
[
−
K
m
cosh
mq
−
I
m
sinh
mq
]sin
mp .
m
=1
m
=1
End of Lemma.
Proof.
By
separation-of-variables
,namelybyusing
x
(
q,p
)=
f
(
q
)
g
(
p
)and
y
(
q,p
)=
F
(
q
)
G
(
p
), the vec-
torial Laplace-Beltrami equation leads to (
D.34
) and to similar equations for
F
(
q
)and
G
(
p
).
f
f
+
g
g
f
f
g
g
=:
m
2
f
=
m
2
f
=0
⇒
=
−
⇒
⇒
(D.41)
g
=
m
2
g
f
=
k
m
cosh
mq
+
l
m
sinh
mq
⇒
−
⇒
g
=
i
m
cos
mp
+
j
m
sin
mp .
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