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problem for the Laplace equation was solved for an infinite layer in
n
-
dimensional space. The solutions of this problem for the bandwidth and
the infinite layer in three-dimensional space obtained by the method of
reflection are known [5]. In this work, the problem has been solved by
the Fourier transform method. The recurrence relation between the integral
kernels for n- dimensional and (
n
+ 2
)-dimensional layers was obtained
and for
n
= 2
,
3
,
4
these kernels are expressed in terms of elementary
functions (for
n
= 3
Bessel functions are also used).
Notations. Statement of the problem.
Let us introduce the following
notation
x
= (
x
1
, . . . , x
n
)
∈
R
n
,
(
x, y
) = (
x
1
, . . . , x
n
, y
)
∈
R
n
+1
, y
∈
R
;
|
x
|
=
q
x
2
1
+
. . .
+
x
2
n
,
h
x, t
i
=
x
1
t
1
+
∙ ∙ ∙
+
x
n
t
n
, dx
=
dx
1
∙ ∙ ∙
dx
n
;
Δ
u
(
x, y
) = Δ
u
=
u
x
1
x
1
+
. . .
+
u
x
n
x
n
+
u
yy
— Laplacian;
F
(
t
) =
F
[
f
] (
t
) =
Z
R
n
f
(
x
)
e
i
h
x,t
i
dx
— Fourier transform method of a summable function
f
(
x
)
.
If the function
f
(
x, y
)
summable on
x
depends on the variables
x
and
y
, then its Fourier
transform on
x
will be denoted as
F
x
[
f
] (
t, y
) =
Z
R
n
f
(
x, y
)
e
i
h
x,t
i
dx.
The inverse Fourier transform of a summable function
F
(
t
)
is defined
similarly
f
(
x
) =
F
−
1
[
F
] (
x
) =
1
(2
π
)
n
Z
R
n
F
(
t
)
e
−
i
h
x,t
i
dt,
as well as the Fourier transform of the function
F
(
t, y
)
summable on
t
F
−
1
t
[
F
] (
x, y
) =
1
(2
π
)
n
Z
R
n
F
(
t, y
)
e
−
i
h
x,t
i
dt.
The definition of the Fourier transform of tempered distribution is given
in [7].
Let us consider the mixed boundary value problem: to find a function
u
(
x, y
)
of
n
+ 1
variable harmonic in
D
=
{
(
x, y
) : 0
< y < a
} ⊂
R
n
+1
;
4
ISSN 1812-3368. Herald of the BMSTU. Series “Natural Sciences”. 2015. No. 1