Плоские разложения и их приложения

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ИПМ им.М.В.Келдыша РАН • Электронная библиотека
Препринты ИПМ • Препринт № 23 за 2014 г.
Варин В.П.
Плоские разложения и их
приложения
Рекомендуемая форма библиографической ссылки: Варин В.П. Плоские разложения и их
приложения // Препринты ИПМ им. М.В.Келдыша. 2014. № 23. 25 с. URL:
http://library.keldysh.ru/preprint.asp?id=2014-23
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C H(x) 6 &%
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# 2 )8,. x = x0 x0 < x# B A )8,. #
B P S(B) # 2 P &% ## % ρ < 1 ||P (y1 ) − P (y2 )|| < ρ||y1 − y2||,
y1 , y2 ∈ S(B).
( ||.|| C[x0, ∞]#
y1 = y2 + h ! |h(x)| ≤ ||h|| ≤ B # 7! A = 0 ' P |P (y1 ) − P (y2 )| ≤
|P2 (y2, h)| + |P2 (h, y2)| + |P2 (h, h)| ≤ P2 (B, |h|) + P2 (|h|, B) + P2 (B, |h|) ≤
(2 P2(B, 1) + P2 (1, B)) ||h|| = 3 B P (1) ||h|| < ρ||h||.
2 3/4 B H(x) < ρ,
)80.
! ! B &! 0 < ρ < 1 &!
x0 ∈ R#
)80. 0 < rhs(42) ## B H(x) < 4#
7 &% ## % w = P (w) S(B)
'
% A B # ? "
wn+1 = P (wn) &! !
w0 ∈ S(B)#
2 A = 4 C w0 = 0# 7!
w1 = 4 C,
w2 = 4 C + C 2 v2 exp(x2), . . .
2 & wn exp(−x2) )0@. C n
& # 2 wn → w n → ∞ )0@. '
#
)0@.
$ vn(x) < B exp(−x2) '
∞
! ## x (.)dt #
C )0A. '
C > 0 yn (x) &
) ' ! .# /' C < 0
# Z
'
% )0A. &
'"
! # ; )#
*++ +,-. y(x) 0 ' "
! ! ! ! ' ! #
? ' U& !
V# B ' #
$ ! '"
)0A. ! y(x) #
7 ! ' E# ! ' "
! y(x) & ' )0A.#
( ' )8,. )80. )0A. &
"
## ' # B&
B ' C<
2 1
,
9 H(x0)
)88.
' )0A. x0 → −∞
O(1/x20)#
2 ' ) 0,
.#
; s ≈ 0.664114672430392 & ≈ 30 "
' *+8- ) ,.# $ y(0) =
y (0) = 0 y (0) = s ! ),:. x1 = 16# 2 y (x1) ≈ 1.33 × 10−100 ## ! # 2
b = x1 − y(x1 ) ≈ 0.860393828760251.
2! ),:. x1 = 4# (
b )0A.5 y(x) ≈ x + C y1(x) + C 2 y2(x) C = C∗ ≈ 0.11686381064255.
' & ) ! . "
*+, # 8#+- ! B = −2 b κ = s/2 Q = 2 C ' S = 2 R# > Q "!#
7 x0 = 0 C∗ < 2/9/H(x0) ≈ 0.125375463 ## E ! '
! )
b.5
x∗ = R − b ≈ 1.9846# 2 2/9/H(x∗) ≈ 151.4#
7 ! )# !.
## ' )0A. [−b = x0, ∞]#
C ! )0A. C 3# B "
C = C∗ ! E ) b .
)# # +. "
#
! !
' &
x = x0 = −b#
b C s # C ' U"
& EV ## [x0, ∞] C
)0A. )&% & . '
y(x0) = y(x0) = 0# ? ' ' x0
C # / % & [J\PQ C ≈ 0.11742437764239,
b = −x0 ≈ 0.88673978127115.
2 ! s = y(x0) ≈ 0.57468729191270#
7 )0A. '
[−b, ∞] '
' "
#
7 )0A. )=.#
7 + ) .
% ! ! "
! )=. %' # 2
& R(x) #
7 a = C e−x 2
3
15
105
945
10395
135135
+
−
+
−
+
+
.
.
.
4
6
8
10
12
14
2x
4x
8x
16 x
32 x
64 x
1
21
423
9375
234405
6618465
209814255
2
+a 4 x5 − 16 x7 + 64 x9 − 256 x11 + 1024 x13 − 4096 x15 + 16384 x17 + . . .
14059
369973
30379175
2669238655
+a3 725x8 − 288173x10 + 3456
−
+
−
+
.
.
.
12
14
16
18
x
13824 x
165888 x
1990656 x
17
1633
110609
6756695
3656281775
4
+a 864 x11 − 6912 x13 + 55296 x15 − 442368 x17 + 31850496 x19 + . . . + . . .
y(x) = x + a
1
x2
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