Асимптотические оценки. Мицик М.Ф. - 4 стр.

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
.
5
.  
( )
)( xSxf
  
Rba
⊂
];[
, 
( )
0≡xf
,
constxS
≡
)(
.    
0
>
λ
,
(
)
+∞
→
λ
.

( ) ( )
( )
dxexfF
xS
b
a
λ
λ
∫
=
(6)
.
 1.
)(xS

Rba
⊂
];[
, .
[ ]
baxMxS ;,)( ∈≤
.    (6)   

0
λ
. 
0
λ

( )
λ
λ
M
eF =
, (7)

−
.
.

)(xS
, 
( )
[ ]
baxeCe
M
xS
;,
0
)(
0
∈≤
−
λ
λλ
, 
( )
.
00
0
)( MxS
eCe
λλλ −−
=≤
,
( ) ( )
( ) ( ) ( )
( )
( )
dxexfedxeexfF
xS
b
a
M
xSxS
b
a
000
0
λ
λ
λλλ
λ
∫∫
≤=
−
 (7) .
 1. 
( )
)( xSxf
  
 [a;b],     
0
=
x
,  
)(xS
  :
( ) ( )
,aSxS
<

a
x
≠

( )
.0
≠
′
aS

( )
( )
( )
( )
( )
.,
1
∞→
+
′
−
=
∫
λ
λλ
λ
λ
Oaf
aS
e
dxexf
aS
xS
b
a
(8)
.

( )
0
≠
′
aS
, 
0
>
δ
, 
( )
δ
+
≤
≤
≠
′
axa0aS
.  (6) :
( ) ( )
( )
( )
( )
( ) ( )
λλλ
λ
δ
λ
δ
21
FFdxexfdxexfF
xS
b
a
xS
a
a
+=+=
∫∫
+
+
. (9)
                                                                                                            .
    5.                                        f (x )       S ( x)                                                             [ a; b ] ⊂ R ,
f (x ) ≡ 0 ,            S ( x ) ≡ const .                                                                                                       λ > 0,
                                                                                                        b

(    .        λ → +∞ )                                                                  F (λ ) = ∫ f ( x ) e λ S ( x )dx                        (6)
                                                                                                        a

                                                                    .
                      1.                                         S (x)                                                         [ a; b ] ⊂ R ,     .
S ( x ) ≤ M , x ∈ [a; b ] .                                                                         (6)
                 λ0 .                                λ ≥ λ0
                                                               F (λ ) = e Mλ ,                                                                  (7)
         −                         .
                                                                                                    .
             S (x)                                               ,
         e (λ −λ0 )S ( x ) ≤ C0 e Mλ , x ∈ [a; b] ,                                e (λ −λ0 )S ( x ) ≤ C0 = e −λ0M .
                        ,
             b                                                                         b
F (λ ) =     ∫ f (x ) e
                            λ0 S ( x )
                                         e   (λ −λ0 )S ( x )
                                                               dx ≤       0   e   Mλ
                                                                                       ∫ f (x ) e
                                                                                                        λ0 S ( x )
                                                                                                                     dx
             a                                                                         a

                                                                                                             (7)                    .
                            1.                                           f (x )            S ( x)
             [a;b],
x = 0,                                               S (x)                                                                          : S ( x ) < S (a ),
         x≠a         S ′(a ) ≠ 0.
                                        e λS ( a )              1 
         b

         ∫ f (x ) e                                 f (a ) + O  ,
                       λ S (x )
                                  dx =                                                                                        λ → ∞.
         a
                                       − λS ′(a )               λ 
                                                                                                                                                (8)

                                                                                                    .
             S ′(a ) ≠ 0 ,                                                    δ >0                      ,
S ′(a ) ≠ 0           a ≤ x ≤ a +δ .                                                                (6)                   :
                      a +δ                                           b
         F (λ ) =       ∫ f (x ) e
                                              λ S (x)
                                                        dx +      ∫ f (x ) e
                                                                                       λ S (x )
                                                                                                  dx = F1 (λ ) + F2 (λ ) .                      (9)
                        a                                        a +δ

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