The singular integral equations (14) are defined on the strips of the cone. We
will obtain SIE defined on slots CS. For this purpose we will transform system (6), (7)
to the form
(1)
,
0,
in
mn
n
y
e
S
=−
=,
(17)
(
)
(
)
(1)
(1)
(1)
0
,
,
0
,
0
(
)
1
(
)
1
,
,
in
in
mn
mn
mn
n
m
n
Nn
y
e
m
e
CS
n
m
+
=−
+
−
=−
+
−
where
0
(1)
(1)
,,
m
mn
mn
n
yx
=−
. It is necessary to differentiate both parts of the equation
(17) with respect to and to add the additional condition when
=
. Then we
obtain such system of the functional equations
(1)
,
(
)
0
in
mn
n
n
y
e
=−
+=,
S
, (18)
(
)
(
)
0
0
(1)
(1)
(1)
0
,
,
0
,
0
(
)
1
(
)
1
,
,
im
in
mn
mn
mm
n
m
n
Nn
y
e
m
e
CS
n
m
+
=−
+
−
=−
+
−
(19)
with the additional condition
(1)
,
(1)
0
n
mn
n
y
=−
−=
()
=
,
(20)
Let us introduce into consideration the function
(1)
(
)
1
,
()
(
)
in
mn
n
F
i
n
y
e
+
=−
=+
, (21)
when
,
−
.
Because of (18) we have
(1)
(
)
,
1
1
()
2
in
mn
CS
i
y
F
e
d
n
−+
=−
+
, (22)
where
[
,]\
CS
S
=−
,
0
n. From the additional condition (20) we have
(1)
(1)
(
)
,
1
,0
0
0
(1)
(1)
()
.
2
n
n
in
mn
m
n
n
CS
i
y
y
F
e
d
n
−+
−
=−
−
=
+
(23)
Using (19), (22), (23), and also formulas
0
(1)
1
sin
n
in
i
n
ee
n
−
−
=−
+
,
(
)
0
2
in
n
n
e
ictg
n
−
−
=−
, we come to SIE for finding
1
F (21), when
CS
:
(
)
0
0
1
11
(1)
0
0
,
0
1
()
1
(
)
()
(
)
1
,
i
i
CS
CS
im
mm
eF
d
K
e
F
d
m
me
m
−
−
+
−
=
−
=
+
−
(24)
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