(
)
(1)
,,
(
)
1
0,
:
inN
mn
mn
n
n
d
Nn
x
e
CS
N
n
l
+
=−
+
−
=
.
(55)
When
0
= the functional equation (54) is transformed in (6). Introduction of
coefficients
,
mn
y
according to formula
(
)
(1)
,
,
,
(
)
1
mn
mn
mn
n
y
Nn
x
n
=
+
−
(56)
leads (54), (55) to the form, respectively,
(
)
0
(1)
,
,
,
1
2
1
,
(
)
sin
im
in
in
mn
mn
mn
n
n
n
y
e
y
e
e
Nn
n
+
+
=−
=−
−
−
=
+
(57)
:,
d
S
l
and
,
0,
:
.
in
mn
n
d
y
e
CS
l
+
=−
=
(58)
After introduction of the function
,
()
,
,
in
mn
n
ye
+
=−
=
−
,
(59)
and application of the algorithm of reducing functional equations to SIE,
introduced in the case of the first boundary condition, we obtain SIE in the case of the
mixed boundary condition (50) or required function
()
(59):
0
(1)
1
1
(
)()
(
)
()
2
2
2
()
,
,
sin
m
S
S
im
F
d
K
d
eS
−
+
−
−
−
−
=
(60),
where
()
F
−
and
()
K
−
are defined in (15), (16).
When
0
= SIE (60) is transformed to the form
0
1
1
ˆ
ln2sin
()
(
)()
2
2
2
2
()
,
.
sin
m
S
S
im
d
K
d
eS
−
+
−
+
−
=−
(61)
Assuming in (60)
0
=, we obtain SIE (14) in the particular case of the first
boundary condition on the cone strips (4). The procedure of numerical implementation
of the SIE solution (60), (61) can be performed for example with the method of discrete
singularities [26,27], as it was done in the case of applying of the second boundary
condition (26) on the strips of the conical gratings in (43)-(49).
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