Асимптотические оценки. Мицик М.Ф. - 11 стр.

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ИСОиП ДГТУ | Шахты

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( )
( )
( )
.
2
2
1
1
0
4
0
−
+
′′
= λ
λ
π
π
O
xS
exf
i
(16)
 (15) (16) .
.
0)(
0
<
′
′
xS
, , , :
( ) ( )
( )
( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
.
2
2
1
0
4
0
1
0
4
0
0
0
−
−
−
−
−
+
′′
=
=+
′′
=
===
∫∫
λ
λ
π
λ
λ
π
λ
π
λ
π
λ
λλ
O
xS
eexf
O
xS
eexf
dxexfdxexfF
i
xSi
i
xSi
xSi
b
a
xSi
b
a
 7. .
( )
( )
∞→=
∫
−
xdexJ
nxi
n

2
1
2
0
sin
ϕ
π
π
ϕϕ
.

( )
ϕ
ϕ
n
ef
−
=
, 
ϕ
ϕ
sin)(
=
S


2
3
,
2
21
π
ϕ
π
ϕ ==
. , 
.

( ) ( ) ( ) ( )
,1,1,1,1
2211
=
′
′
−=−=
′
′
= ϕϕϕϕ SSSS

( )
( )
∞→+
−−=
−
xxO
n
x
x
xJ
n

42
cos
2
1
ππ
π
.
.

( ) ( )
( )
dzezfF
zSλ
γ
λ
∫
=
, (17)

−
γ

C
, 
zf

)(zS

D
,
                             iπ
                                        2π
         = f ( x0 )e 4
          1
          2                           λS ′′( x0 )
                                                  + O λ−1 .          ( )                                                     (16)

                                              (15) (16)                                                                      .
                         .                 S ′′( x0 ) < 0 , ,                                                   ,             :
                                  b                                          b
                F (λ ) = ∫ f ( x ) e                 iλ S ( x )
                                                                   dx = ∫ f ( x )eiλ ( −S ( x ))dx =
                                  a                                          a
                                                         iπ
                                                                     2π
                = f ( x0 )e           −iλ S ( x0 )
                                                     e    4
                                                                   λS ′′( x0 )
                                                                               + O λ−1 =     ( )
                                                         iπ
                                                                     2π
                = f ( x0 )e           iλ S ( x0 )
                                                e
                                                     −
                                                          4
                                                                   λS ′′( x0 )
                                                                               + O λ−1 .     ( )
                    7.                                                                                    .
                                                              2π
                             J n (x ) =
                                         1                         i ( x sin ϕ − nϕ )

                                        2π                    ∫e
                                                              0
                                                                                        dϕ               x → ∞.

                                         f (ϕ ) = e − nϕ ,                                               S (ϕ ) = sin ϕ
                                               π        3π
                                      ϕ1 =       ,ϕ 2 =    .                                         ,
                                               2         2
                                                                                                                         .
                             S (ϕ1 ) = 1, S ′′(ϕ1 ) = −1, S (ϕ 2 ) = −1, S ′′(ϕ 2 ) = 1,


                                                    πn π 
                J n (x ) =
                                         2
                                        πx
                                              
                                           cos x −   −  + O x −1                             ( )                  x→∞.
                                                    2 4

                                                                                               .


                             F (λ ) = ∫ f ( z ) e λ S ( z )dz ,                                                              (17)
                                                γ



    γ−                                                                                                        C,
f (z )     S (z )                                                                                                   D,

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