Korn–Lichtenstein and d’Alembert–Euler Equations - Map Projections: Cartographic Information Systems - page 707

A:= 01

,

− 10

(D.31)

H 1 := α 2

, H 2 := − .

β 2 β 2 α 2

β 2 − α 2 α 2 β 2

Lemma D.3 (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

M

x ( q,p )= x 0 +

[ A m exp( mq )+ C m exp( −mq )] cos mp +

m =1

(D.32)

M

+ [ B m exp( mq )+ D m exp( − mq )] sin mp ,

m =1

M

y ( q,p )= y 0 + [ B m exp( mq )+ D m exp( − mq )] cos mp +

m =1

(D.33)

M

+ [ − A m exp( mq )+ C m exp( − mq )] sin mp .

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.34)

g =

m 2 g

f = c m exp( mq )+ d m exp( − mq ) ⇒ − ⇒ g = a m cos mp + b m sin mp .

Superposition of base functions gives the setups ( D.35 )and( D.36 ). The d'Alembert-Euler equa-
tions (Cauchy-Riemann equations) x p = y q and x q = − y p then are specified by ( D.37 )and( D.38 ).