Асимптотические оценки. Мицик М.Ф. - 8 стр.

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 ,  
( )
xf

)(xS
 ,  
    ,   (1) 
.
 3. 
( )
xf
– 

];[ ba
,
)(xS
–    
];[ ba
0)(
≠
′
xS

];[ bax
∈
. 
+∞
→
λ
( ) ( )
( )
( )
( )
( )
( )
( )
1
1
|
−
−
+
′
==
∫
λλλ
λλ
o
xS
xf
eidxexfF
b
a
xSixSi
b
a
.
.
, :
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
dxe
xS
xf
dx
d
i
xS
xf
eied
xSi
xf
F
xSi
b
a
b
a
xSixSi
b
a
λ
λλ
λ
λ
λ
λ
∫
∫
′
+
+
′
=
′
=
−
−
1
1
|

0)(
≠
′
xS
, 
)(xS

];[ ba
.

)(xSy
=
, 

( )
dyeyh
yiλ
β
α
∫
,
,  – 
+∞
→
λ
,
.
2. .
 1 :
0)(
≠
′
xS

];[ bax
∈
, 
. 

λF
.
, 
( )
xf

)(xS
.
 3.(). 
( )
xf


];0[ d

0
>
α
. 
+∞
→
λ
( ) ( ) ( )
( )
1
42
0
0
2
2
1
2
−
+==
∫
λ
αλ
π
λ
π
αλ
Oefdxexf
i
x
id
.
                         ,                                     f (x )         S (x )                                  ,
                                                                              ,                                           (1)
                                                                                                 .
                             3.                 f (x ) –
            [ a; b ] ,       S (x ) –                                                                                           [ a; b ]
S ′( x ) ≠ 0             x ∈ [ a; b ] .                             λ → +∞
                                                                           iλ S ( x ) f ( x )  b
                                                                                                        ( )
                       b
         F (λ ) = ∫ f ( x ) e            iλ S ( x )
                                                      dx = (iλ )
                                                                     −1
                                                                          e                     | + o λ−1 .
                       a                                                              S ′( x )  a
                                                                                            .
                                                      ,                 :
                       f (x )                          −1  iλS ( x ) f ( x ) 
                                                 (             )
                       b
         F (λ ) = ∫                             = (iλ ) e
                                                                                 b
                                     iλ S ( x )

                      λ   ′(   )
                                 d e
                                                                        ′(   )  a| +
                  a
                    i   S    x                                       S    x   
                          d  f ( x )  iλ S ( x )
                       b
         + (iλ )       ∫a dx  S ′(x ) e dx
                  −1



                                  S ′( x ) ≠ 0 ,                            S (x )                                        [ a; b ] .
                                                                                                y = S (x) ,

                                            β

                                            ∫ h( y ) e
                                                            iλ y
                                                                   dy ,
                                            α

           ,                                                   –                                                     λ → +∞ ,
                                                                                            .
                                 2.                                                                                   .
                                                  1                                                            : S ′( x ) ≠ 0
x ∈ [ a; b ] ,
                             .
                                            F (λ )                            .
                                                                                                                ,
f (x )     S (x )                                                                                          .
                    3.(                                   ).                                f (x )
                                                          [0; d ]       α > 0.                         λ → +∞
                                                                                       iπ
                                                           1 2π
                                                                                                     ( )
                       d                 i
                                           αλ x 2
           (λ ) = ∫ f (x ) e             2
                                                      dx =      f (0 )e 4 + O λ−1 .
                       0
                                                           2 αλ

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