D-2 D’Alembert–Euler (Cauchy–Riemann) equations 533
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) =
= x 0 +
M
m=1
[I m cosh mq + K m sinh mq] cos mp +
M
m=1
[J m cosh mq + L m sinh mq] sin mp ,
(D.39)
y(q, p) =
= y 0 +
M
m=1
[L m cosh mq + J m sinh mq] cos mp +
M
m=1
[−K m cosh mq − I m sinh mq] sin mp .
(D.40)
End of Lemma.
Proof.
By separation-of-variables, namely by using x(q, p) = f (q)g(p) and y(q, p) = F (q)G(p), the vectorial
Laplace–Beltrami equation leads to (D.34) and to similar equations for F (q) and G(p).
f
f
+
g
g
= 0 ⇒
f
f
= −
g
g
=: m
2
⇒ f
= m
2 f
⇒
f = ¯
k m cosh mq + ¯ l m sinh mq ⇒ g
= −m
2 g ⇒ g = ¯ i m cos mp + ¯ j m sin mp .
(D.41)
Superposition of base functions gives the setups (D.42) and (D.43). The d’Alembert–Euler equations
(Cauchy–Riemann equations) x p = y q and x q = −y p then are specified by (D.44) and (D.45).
x(q, p) =
M
m=1
I m cosh mq cos mp + J m cosh mq sin mp+
+K m sinh mq cos mp + L m sinh mq sin mp
+ x 0 ,
(D.42)
y(q, p) =
M
m =1
I
m cosh m
q cos m
p + J
m cosh m
q sin m
p+
+K
m sinh m
q cos m
p + L
m sinh m
q sin m
p
+ y 0 ,
(D.43)
x q =
M
m=1
mI m sinh mq cos mp + mJ m sinh mq sin mp+
+mK m cosh mq cos mp + mL m cosh mq sin mp
,
(D.44)
y p =
M
m =1
− m
I
m cosh m
q sin m
p + m
J
m cosh m
q cos m
p−
−m
K
m sinh m
q sin m
p + m
L
m sinh m
q cos m
p
⇔
I
m = L m , J
m = −K m , K
m = J m , L
m = −I m .
(D.45)
End of Proof.