Асимптотические оценки. Мицик М.Ф. - 10 стр.

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( ) ( ) ( )
( )
.0
1
1
0
−
=
′
++≤
∫
λ
αλ
Odxxggdg
d
 (14) 
.

.
.4.
( )
xf

)(xS


];[ ba
, 
)(xS


];[
0
bax ∈

0)(
0
>
′
′
xS
. 
+∞
→
λ
( ) ( )
( )
( )
( )
( )
( )
1
0
4
0
2
0
−
+
′′
==
∫
λ
λ
π
λ
π
λ
λ
O
xS
eexfdxexfF
i
xSi
xSi
b
a
.

)(xS
′

0
x

0)(
0
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′
′
xS
, 

0
xxa <≤

0)(
0
<
′
xS

)(xS
,

bxx ≤<
0

0)(
0
>
′
xS

)(xS
.
 - 
[ ] [ ]
ba;;: →βαϕ
,
, 
[ ]
( )
0
0,;0 x=∈ ϕβα

( )
2
0
yxSyS +=ϕ
. 
( )
( )
0
2
0
xS
′′
=
′
ϕ
. , 
( )
( )
( )( ) ( )
dyeyyfeF
yi
xSi
2
0
λ
β
α
λ
ϕϕλ
∫
′
=
.

( )( ) ( )
( )( ) ( )
( )
( )
( )
( )
.
2
2
1
00
2
1
1
0
4
0
1
4
0
2
−
−
+
′′
=
=+
′
=
′
∫
λ
λ
π
λ
λ
π
ϕϕϕϕ
π
π
λ
β
O
xS
exf
Oefdyeyyf
i
i
yi
(15)

u
y
−
=
, 
[ ]
0;α

( )( )() ( )( )( )
=−
′
−=
′
∫∫
−
dueuufdyeyyf
uiyi
22
0
0
λ
α
λ
α
ϕϕϕϕ
                      1                                  
                                                                                                            ( )
                                            d
                   ≤      g (d ) + g (0) + ∫ g ′( x ) dx  = O λ−1 .
                     αλ                   0
                                                          
                                                          
                                                                                        (14)
          .

                                                                                                      .
                       .4.                                           f (x )         S (x )
                                                              [ a; b ] ,                        S (x )
                                         x 0 ∈ [ a; b ]              S ′′( x0 ) > 0 .                                  λ → +∞
                                                                                                iπ
                                                                                                        2π
                                                                                                                               ( )
                       b
          F (λ ) = ∫ f ( x ) e               iλ S ( x )
                                                          dx = f ( x0 )e      iλ S ( x0 )
                                                                                            e    4
                                                                                                                  + O λ−1
                       a
                                                                                                      λS ′′( x0 )
                                                                                                 .
                               S ′(x )                                                                                 x0    S ′′( x0 ) > 0 ,
     a ≤ x < x0                                            S ′( x0 ) < 0                                  S (x )                                   ,
      x0 < x ≤ b                                            S ′( x0 ) > 0           S (x )                                              .
                                                  -                                                                         ϕ : [α ; β ] → [a; b ],
      ,          0 ∈ [α ; β ], ϕ (0 ) = x0                       S (ϕ ( y )) = S ( x0 ) + y 2 .

ϕ ′(0 ) =
                   2
               S ′′( x0 )
                          .                                                                                        ,

                                         β
          F (λ ) = e       iλ S ( x0 )
                                         ∫    f (ϕ ( y ))ϕ ′( y ) e iλ y dy .
                                                                                2



                                         α


          β                                                                                          iπ
                                                                                                            π
          ∫   f (ϕ ( y ))ϕ ′( y ) e            iλ y 2
                                                        dy = f (ϕ (0))ϕ ′(0)e 4
                                                            1
                                                            2                                               λ
                                                                                                              + O λ−1 =     ( )
          0
                                iπ
                                               2π
                                                                      ( )
                                                                                                                                            (15)
          = f ( x0 )e 4
           1
                                                         + O λ−1 .
           2                                 λS ′′( x0 )
                                                                  y = −u ,                                                          [α ;0]
          0                                                     −α

          ∫   f (ϕ ( y ))ϕ ′( y ) e            iλ y 2
                                                        dy =     ∫    f (ϕ (− u ))ϕ ′(− u ) e iλ u du =
                                                                                                                   2



          α                                                      0

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