Bohye.Woo
/
bo-graduation
1
0
Fork 0
Code Issues Pull Requests Releases Wiki Activity
You cannot select more than 25 topics Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
master
Branches Tags
${ item.name }
Create tag ${ searchTerm }
Create branch ${ searchTerm }
from '01d040344a'
${ noResults }
bo-graduation/nltk-book/pattern-master/examples/03-en/texts/1701.00100.txt

1723 lines
17 KiB
Plaintext
Raw Blame History Unescape Escape

This file contains invisible Unicode characters!

This file contains invisible Unicode characters that may be processed differently from what appears below. If your use case is intentional and legitimate, you can safely ignore this warning. Use the Escape button to reveal hidden characters.

This file contains ambiguous Unicode characters that may be confused with others in your current locale. If your use case is intentional and legitimate, you can safely ignore this warning. Use the Escape button to highlight these characters.

1
. .
arXiv:1701.00100v1 [math.CA] 31 Dec 2016
, x ( ) ln-1 x (
) x i , i = -1, R, = 0 ( ). . , .
On properties of the coefficients of the complicated and exotic expansions of the solutions of the sixth Painlev´e
equation
I. V. Goryuchkina
It is known, that among the formal solutions of the sixth Painlev´e equation there met series with integer power exponents of the independent variable x with coefficients in form of formal Laurent series (with finite main parts) in log-1 x (complicated expansions), or in x i , where
i = -1, R, = 0 (exotic expansions). These coefficients can be computed consecutively. Here we research analytic properties of the series, that are the coefficients of the complicated and exotic expansions of the solutions of the sixth Painlev´e equation.
1. .
(y )2 1 1
1
11
1
y=
+
+
-y +
+
+
2 y y-1 y-x
x x-1 y-x
y(y - 1)(y - x)
x x - 1 x(x - 1)
+ x2(x - 1)2
a
+
by2
+
c (y
-
1)2
+
d
(y
-
x)2
,
(1)
a, b, c, d ­ , x y ­ ,
y = dy/dx. x = 0, x =
1 x = , ,
1) x = z, y = z/w, 2) x = 1/z, y = 1/w, 3) x = 1 - z, y = 1 - w,
(. [1]). . x = 0 (1),
2
, .
x , ln-1 x(- ) x i ( R, i = -1) ( ) , . [2]. , x , ln-1 x x i ( = 0). , , (1) , .
y = k(x) xk,
(2)
k=0
k(x) ­
k(x) =
ckj j,
j=0
ckj C, Z,
= ln-1 x = x i ( = 0).
(3)
-
, (2), ­ , [2].
, [3] () , . , [3] , ( ) , . -,
3
. [3] . , [3] . , . ( [3], ) , [3], , . , y = 0(x) + u , 0(x) ­ . ( ) , , u = 1(x)x. , (2), . .
, k(x) (2) (1), , , . , , . . (. . ), . .
k(x) .
() 0(x) x i , = 0 (x = 0, 1, ) . [4] , (. .
4
). , .. k(x) .
(. [5], [6]) , ( , ), . .
: k(x) (2) (1).
, , , , [3] . : [3] , , .
2. . -
, .
(x) x = 0 r R {},
ln |(x)|
lim
= r,
(4)
x0 ln |x|
xD
D ­ ,
. (x) x =
r R {},
ln |(x)|
lim
= r,
(5)
x xD
ln |x|
5
D ­ , .
(x), r R {} , , r.
(2) .
g(x, u, u , . . . , u(n)) = 0,
(6)
(x) xq1uq20(xu )q21 . . . (xnu(n))q2n,
(7)
(x) ­ -
(.. ), q1 C, q21, . . . , q2n Z+, (n)(x) 0, (n)(x) -n.
(6) u =
= k(x)xk,
(8)
k=0
k(x) ­ , k R, (kn)(x) 0, k(n)(x) -n, , k+1 > k.
(7) (6) -
( ) (q1, q2),
q2 = q20 + . . . + q2n. [3]
(x) xq1q20(x )q21 . . . (xn(n))q2n
q1 + q20. {Qi = (q1i , q2i ), i = 0, . . . , m} ­ -
(6), . . ,
, R ­ (1, 0). Qi, R = ci R. c = min ci.
i=0,...,m
(7) (6) -
, Qi, R = c R, (. [7]), g^(x, u0, . . . , un),
g^(x, u0, . . . , un) = 0
(9)
6
­ .
1. (6) u = , (8), (9) ()
u = ^, ^ = 0(x).
(10)
. 0 = 0, (6)
u = x0v.
(11)
G(x, v, v , . . . , v(n)) = 0,
(12)
v = , = x0 , .
, (11)
(6)
(x) xq1uq20(xu )q21 . . . (xnu(n))q2n
(x) xq1+q20vq~20(xv )q~21 . . . (xnv(n))q~2n,
(13)
q~20, . . . , q~2n Z+, q~20 + . . . + q~2n = q2. , (11) g^(x, u, u . . . , u(n)) xc P0(x, v, v . . . , v(n)), c ­ q1 +q20, P0(x, v, v . . . , v(n)) ­ v, v . . . , v(n) , (6) (13) q1 + q20 > c. (12)
xc [P0(x, v0, . . . , vn) + x1P1(x, v0, . . . , vn) + . . . + xtPt(x, v0, . . . , vn)] = 0,
vj = xjv(j), P0(x, v0, . . . , vn), . . . , Pt(x, v0, . . . , vn) ­ v0, . . . , vn , 1, . . . , n R+, 1, . . . , n = 0. xc,
P0(x, v0, . . . , vn) + x1P1(x, v0, . . . , vn) + . . . + xtPt(x, v0, . . . , vn) = 0, (14)
G(x, v0, . . . , vn) = 0. (14) 1 ­ 3 [3] , p G(x, 0, . . . , n) 0 (
7
, Pj x, 0, . . . , n = 0 p Pj(x, 0, . . . , n) = 0, Pj x, 0, . . . , n ).
u = ,
= ~ k0(x1, . . . , xN ) k0(x)
k0=0
(14)
u = ^ + w, ^ = ~ 0(x1, . . . , xN ) 0(x).
(15)
(15) (14), ( , (14) v0, . . . , vn) ,
P0(x, ^ 0, . . . , ^ n) +
P0(x,
^ 0, . v0
.
.
,
^ n)
w0
+
.
.
.
+
P0(x,
^ 0, . vn
.
.
,
^ n)
wn
+
+ . . . + x1P1(x, ^ 0, . . . , ^ n) + · · · = 0, ^ j = xj ^ (j), wj = xj w(j). (16)
(16), -
, P0(x, ^ 0, . . . , ^ n), p(P0(x, ^ 0, . . . , ^ n)) , - ( P0(x, ^ 0, . . . , ^ n) = 0).
(16) . , p(wj) > p(^ j) = 0, P0(x, ^ 0, . . . , ^ n), . . . , Pt(x, ^ 0, . . . , ^ n) v0, . . . , vn ,
, 2 3 [3], n > · · · > 1 > 0. -
(14), (16) , . . P0(x, ^ 0, . . . , ^ n) = 0. -
P0(x, v0, . . . , vn) = 0 (11) x c, ,
g^(x, ^0, . . . , ^n) = 0, ^j = xj ^(j), ^ = x0 ^ . 2
3. . (1) . x2(x - 1)2y(y - 1)(y - x), .
2x2(x - 1)2y(y - 1)(y - x)y - x2(x - 1)2(3y2 - 2xy - 2y + x)y 2+, (17)
8
+2xy(x - 1)(y - 1)(2xy - x2 - y)y - 2y6a + 4a(x + 1)y5-
-2 (a + d)x2 + (4a + b + c - d)x + (a - c) y4+
+4x ((a + b + c + d)x + (a + b - c - d)) y3-
-2 (b + c)x3 + (a + 4b - c + d)x2 + (b - d)x y2 + 4bx2(x + 1)y - 2bx3 = 0,
, (1), . [2]. [2] , 0(x)
(2) ­ , = ln-1 x = x i . x , . , , (2) .
(17)
y = 0(x) + xu.
(18)
L 0, 0, ¨ 0, U + xM x, 0, 0, ¨ 0, U + H x, 0, 0, ¨ 0 = 0, (19)
U = (u0, u1, u2), uj = xju(j),
0
=
0(x),
0
=
x d0(x) , dx
¨ 0
=
x2
d20(x) dx2
,
L 0, 0, ¨ 0, U = 220 (0 - 1) u2 +
20 320 - 30 0 - 30 + 2 0 u1 -
(20)
2 6a50 - 10a40 + 4a30 - 4c30 - 30 - 320¨ 0 + 30 20 + 20 + 20¨ 0 - 20 u0,
M x, 0, 0, ¨ 0, U H x, 0, 0, ¨ 0 . (19)
u = k+1(x) xk.
(21)
k=0
9
4. . (19) c (21), . (19) u0, u1, u2 , L 0, 0, ¨ 0, U ) xM (x, 0, 0, ¨ 0, U ). x = 0 , L 0, 0, ¨ 0, U . , L
, u0, u1 u2 , M ­ .
[2] a = c, a, c = 0 (2) k(x), ln-1 x
2(c - a)
0(x)
=
(c
-
a)2(ln x
+
C )2
-
. 2a
(22)
(21) (22) (19)
k=1
Lk(0, 0, ¨ 0, k, k, ¨ k) - Nk(0, 0, . . . , ¨ 0, k-1, k-1, ¨ k-1)
j
=
j (x),
j
=
x dj (x) , dx
¨ j
=
x2
d2j (x) dx2
,
Lk 0, 0, ¨ 0, k, k, ¨ k =
xk,
(23) (24)
= L 0, 0, ¨ 0, k, k + (k - 1)k, ¨ k + 2(k - 1) k + (k - 1)(k - 2)k ,
Nk 0, 0, ¨ 0, . . . , k-1, k-1, ¨ k-1 . , , k(x) -
k
Lk 0, 0, ¨ 0, k, k, ¨ k = Nk 0, 0, ¨ 0, . . . , k-1, k-1, ¨ k-1 . (25)
(25) x ,
= ( + C)-1 = (ln x + C)-1, C C.
(26)
, (26) ,
x dy = -2 dy ,
dx
d
x2
d2y dx2
=
4
d2y d2
+ (23
+
2) dy . d
10
(-2a2 + a2 - 2ac + c2)4
(-c + a)64
.
Lk
,
^k(),
d^k() d
,
d2^k() d2
= Nk(),
(27)
Lk
,
^k(),
d^k() d
,
d2^k() d2
=
4P2()
d2^k() d2
+
2P1()
d^k() d
+
P0()^k(),
^k() = k(x), P2 P0 ­ , P1 ­ , P2(0), P1(0), P0(0) = 0, Nk() ­ .
,
= 0 . ,
a2()2
d2 d2
+
d a1() d
+
a0(),
a2(), a1(), a0() ­ ,
a2() a1(), a0().
, k(x), ln-1(x) c , . ,
-
(21) -
(25) ( 0(x) (22)). (27) , ..
Lk
,
^k(),
d^k() d
,
d2^k() d2
= -8k2^k() + . . . ,
­ ( ), . k(x) ­ ln-1 x.
a = c = 0 (2) c
1
0(x)
=
2a
(ln x
+
, C)
C C.
(28)
,
11
(2) c (22). .
.
1. k(x) (k 1) (2) c (22) (28) ln-1 x,
.
[8] [9].
, . , , . , ( ) k(x) ln-1 x , , .
5. .
(21) (19) k(x) xi
-4C0(2a - 2c + C1)
, (29)
x 2c-2a-C1(C12 + 8C1a + 16a2 - 16ac) - 2C1C0 + C02x- 2c-2a-C1
C0, C1 ­ , C0 = 0, 2c-2a-C1 R, 2c - 2a - C1 < 0, = sgn(Im 2c - 2a - C1).
(., , [2]) B0 , B1 , B2 , B6 B7 . , ( C0 C1) (2) -
(29) (1). ,
· B0
0(x)
=
2c
- C3 2a
cos2(ln(C2x) C3-2c/2)
1 +
sin2(ln(C2x)C3-2c/2) ,
(30)
a = 0, C0 =
C32
+
4C3a
+
4a2
-
16ac ,
C2 2c-C3
C1 = C3 - 2a,
C32 +
4C3a + 4a2 - 16ac = 0, C3 = 2c, C2 = 0, 2c - C3 R, 2c - C3 < 0,
12
­ 2at2 + (C3 - 2a)t + 2c - C3 = 0, = sgn(Im 2c - C3);
· B1
1 - c/a
0(x)
= 1
-
,
C2x
2c-
2a
(31)
a = c = 0, C0 = 8 a( c - a) C2, C1 = 4 a( c - a), C2 = 0,
Re( 2c - 2a) = 0, = sgn(Im( 2c - 2a));
· B2
1 + c/a
0(x)
=
1
-C2x 2c+ 2a
,
(32)
a = c = 0, C0 = -8 a( c + a) C2, C1 = -4 a( c + a),
C2 = 0, Re( 2c + 2a) = 0, = sgn(Im( 2c + 2a));
· B6
1
0(x)
=
1
+
, C2x 2a
(33)
a = 0, c = 0, C0 = 8aC2, C1 = -4a, C2 = 0, = sgn(Im 2a);
· B7
0(x)
=
2c - C1 C1
1
sin2(ln(C2x) C1-2c/2)
a = 0,C0 = -C1/C2 2c-C1, C2 = 0, 2c - C1 R, = sgn(Im 2c - C1).
(34) 2c - C1 < 0,
( C0 C1) (1), (2) (29). , (29) Cxi.
1(x). , . (21) (19), , x.
L1 0, 0, ¨ 0, 1, 1, ¨ 1 + N1 0, 0, ¨ 0 = 0,
(35)
0, 0, ¨ 0, 1, 1, ¨ 1 (23),
L1 0, 0, ¨ 0, 1, 1, ¨ 1 = 220(0 - 1)¨ 1 + 20(320 - 30 0 - 30 + 2 0) 1
13
-2(6a50 - 10a40 + 4a30 - 4c30 + 30 - 320¨ 0 + 30 20 + 20 + 20¨ 0 - 20)1, N1 0, 0, ¨ 0 = 4a50 - 2(4a + b + c - d)40 + 4(a + b - c - d)30 - 630 0
-430¨ 0 + 620 20 - 2(b - d)20 + 620 0 + 220¨ 0 - 20 20 + 20¨ 0 - 20.
(29) (35) x = C = Cxi, R, = 0. , ,
dy
dy
x = i ,
dx
d
x2
d2y dx2
=
-22
d2y d2
-
(
+
dy i) ,
d
. . -
. -
,
. (29), 0, 0, ¨ 0 -
Cxi = Cxi
42
0
=
A2
+
B
+
, 1
A = 4 + 4(a + c)2 + 4(a - c)2, B = 22 - 4(a - c),
0
=
4 i 3(A2 - 1) -(A2 + B + 1)2
,
(36)
¨ 0
=
43(A2(i
-
)4
+
AB(i + )3 + 6A2 (A2 + B + 1)3
-
B(i
-
)
-
i
-
) .
(35) y = 1(x)
(A2 + B + 1)6
- 16 4 2
8
p2j
j+2
d2y d2
+
p1j
j+1
dy d
+
p0j
j y
+ tj j = 0,
(37)
j=0
p2j, p1j, p0j, tj C, p20 p28 = 0. (37) ,
y = C1 y1() + C2 y2() + y3(),
C1, C2 ­ , y1(), y2(), y3() ­ .
, (35) = Cxi ­ (A2 + B + 1)6,
14
, -16 4 2 (37), d2y
d2
-
(A2
+ B 84
+
1)6
20(0
-
1).
(38)
(38) ­ , 5 = 0, a1, a2, a3, a4 . , = 0 ­ 2, = a1 = a2 ­ 1, a1 a2 ­ (A-42)2+B+1 = 0, a3 a4 ­ 3, a3 a4 ­ A2+B+1 = 0. (37) = 0, , a1, a2, a3, a4. p20 p28 = 0, (37) , . (37) = + aj, , (37) . , , . [10]. , , . , (37)
y = CiFi( - aj)( - aj)i lnµi ( - aj)+
i=1,2
+ F3( - aj)( - aj)3 lnµ3 ( - aj),
(39)
F1(), F2(), F3() C{}, 1, 2, 3 C, µ1, µ2, µ3 Z. ,
. , [11].
(37) . , y3() ­ , y1() y2() ­ , = 0 = .
2. 1(x) (21) (19) (29) Cxi.
15
, 1(x) = y3(Cxi), y3() ­ (37).
2. -
(37) , ,
(39), , -
. -
1,
(37), -
. -
-.
, ( -
), ,
( ), -
(- -
)
,
(37).
(37) , -
. - (37) ­
[(0, 0), (0, 1), (8, 1), (8, 0)]. -
(
) ( -
),
­ ( -
) (
).
() 0
. 0 -
-222
d2y d2
+
2(
+
dy 2i)
d
-
2(
+
i)2y
=0
-
222
d2y d2
+
2(
+
dy 2i)
d
-
2(
+
i)2y
+
(
+
i)2
-
1
+
2b
-
2d
= 0,
­ . , (0, 1), ­ [(0, 1), (0, 0)]. -
16
y
=
(C1
+
C2
ln
)1+
i
+
(
+
i)2 + 2(
1 + 2b + i)2
-
2d ,
(40)
­
y
=
(C1
+
C2
ln
)1+
i
,
C1, C2 ­ . , , (40) , .
[7], , (40)
y = C1
a1kk + C2 ln
a2kk
i
+
a3kk,
k=0
k=0
k=0
(41)
a1k, a2k, a3k C, a10 = a20 = 1,
( + i)2 + 1 + 2b - 2d
a30 =
2( + i)2
,
(37). -
A48
-222
d2y d2
-
2(3
-
dy 2i)
d
-
2(
-
i)2y
=0
A48
-222
d2y d2
-
2(3
-
dy 2i)
d
-
2(
-
i)2y
-
(
-
i)2
-
1
+
2b
-
2d
= 0,
­ . (8, 1), ­ [(8, 1), (8, 0)].
y
=
(C1
+
C2
ln
)-1+
i
+
(
-
i)2 + 2(
1 + 2b - i)2
-
2d ,
(42)
­
y
=
(C1
+
C2
ln
)-1+
i
,
C1, C2 ­ . ,
17
, (42) , .
(42)
y=
C1 b1k + C2 ln b2k
i
+
b3k ,
k
k
k
k=0
k=0
k=0
(43)
b1k, b2k, b3k C, b10 = b20 = 1,
( - i)2 + 1 + 2b - 2d
b30 =
2( - i)2
,
(37). (37) a1 a2,
(A - 42)2 + B + 1 = 0. (37) = + aj, j = 1, 2. . , , .
8
P2j
j +1
d2y d 2
+
P1j
j
dy d
+
P0j
jy
+
Tj
j
= 0,
(44)
j=0
P2j, P1j, P0j, Tj C, P20 = 0. - (44) ­ (-1, 1), (0, 0), (8, 0), (8, 1). , , = 0, . . , (-1, 1) [(-1, 1), (0, 0)] . -y + y = 0 -y + y = , C, (44), y = C1 = 0 y = , C1 ­ . C1, C2 (C2 C)
y = C1 c1kk + C22 c2kk + c3kk,
k=0
k=0
k=0
(45)
c1k, c2k, c3k C, c10 = c20 = 1, c30 = , (44).
18
(37) a3 a4, A2 +B+ 1 = 0. (37) = + aj, j = 3, 4. . , , .
8
S2j
j +3
d2y d 2
+
S1j
j +2
dy d
+
S0j
j +2y
+
Kj
j
= 0,
(46)
j=0
S2j, S1j, S0j, Kj C, S20 = 0. - ­ (1, 1), (0, 0), (8, 0), (8, 1). -
, ,
-
= 0, . . , (1, 1)
[(1, 1), (0, 0)] -
.
2(y + 3y ) = 0 2(y + 3y ) = , C,
(44),
y
=
C1 2
,
C1
=
0
y=
,
C1
­ . -
C1, C2 (C2 C)
y
=
C1 2
d1kk + C2
d2k k
+
1
d3k k ,
k=0
k=0
k=0
(47)
d1k, d2k, d3k C, d10 = d20 = 1, d30 = , (44). , 0, , a1, a2, a3, a4
(37) . , y = C1y1() + C2y2() + y3()
i
i
y = C1f1() + C2 ln f2() + f3(),
f1(), f2(), f3() ­ , . . y1() =
i
i
C1f1() y2() = C2 ln f2() ­ , -
,
.
19
y3() = f3() ­ , . , 1(x) = y3(Cxi). 2
k(x) (21) (19) (29).
Lk(0, 0, ¨ 0, k, k, ¨ k) - Nk(0, 0, . . . , ¨ 0, k-1, k-1, ¨ k-1) xk,
k=1
j, j, ¨ j (23), Lk 0, 0, ¨ 0, k, k, ¨ k -
(24), Nk 0, 0, ¨ 0, . . . , k-1, k-1, ¨ k-1 .
, (21) (19), k(x), k N
Lk(0, 0, ¨ 0, k, k, ¨ k) = Nk(0, 0, . . . , ¨ 0, k-1, k-1, ¨ k-1).
(48)
, (48) k, , .
x = C = Cxi, R, = 0, k(x) = ^k(), k N, (48). , (48)
Q2()
2
d2^k() d2
+
Q1k()
d^k() d
+
Q0k()^k()
=
Nk(),
(49)
Q2() = -22 20(0 - 1),
Q1k() = 2 i (i + 2k - 3)Q2() + i Q1(),
Q0k() = (k - 2)(k - 1)Q2() + (k - 1)Q1() + Q0(), Q1() = 20(320 - 30 0 - 30 + 2 0),
Q0() = -2(6a50-10a40+4a30-4c30+30-320¨ 0+30 20+20+20¨ 0- 20)1, 0, 0, ¨ 0 (36), Nk() ­ .
3. (49) ^k() ( ) .
20
3. k (49).
^k() = rk kjj,
(50)
j=0
rk Z, kj C. k = 1 1. k = 2. (49) ­
0 1, . .
N2 R2 A2jj, R2 Z,
j=0
A2j C. Q2, Q1k, Q2k, , ^2() =
r2 2jj (49),
j=0
8 B2a2
(2i
-
)242
+
O(3)
2jr2+j = R2
A2j j .
j=0
j=0
(51)
a = 0, B = 0, 2i - = 0, R. r2, R2, (51) . 2j . k = 3, N3 ­ , ( ) 0 1, ( ) 2. (50) (49)
8 B2a2
(ki
-
k
+
)242
+
O(3)
kjrk+j = Rk
Akj j ,
j=0
j=0
(52)
Rk Z, Akj C. k = 2. , ^3() . , , (49) . 2
4. k(x) = ^k() (21) (19) (29) = 0 ( ) .
4 , (49) . 2
21
5. (49) 0, , a1, a2, a3, a4 C, .
5. 0, , a1, a2, a3, a4 C. . (49) k = 2 . , (49) k = 2 , . k = 3, (49) ­ ( ), . , , k. 2
5 , (49) = 0, , a1, a2, a3, a4 C. , (19) (21) (29), Cxi = 0, , a1, a2, a3, a4.
, ^k() = a1, a2, a3, a4 C (49).
, 3(x) 4(x) (21) (29) (19) Cxi. , . .
. k(x) = ^k() ­ = Cxi .
1. Gromak I.V., Laine I., Shimomura S. Painlev´e Differential Equations in the Complex Plain. Berlin, New York: Walter de Gruyter. 2002.
2. .., .. // . 2010. . 71. . 6­118.
3. .. // . 2016. . 17. 2(58). . 64-87
22
4. Guzzetti D. Poles Distribution of PVI transcendents close to a critical point // Physica D. 2012. doi:10.1016/j.physd.2012.02.015.
5. Gontsov, R.R., Goryuchkina, I.V. On the convergence of generalized power series satisfying an algebraic ODE. Asympt. Anal. 2015. 93(4). P. 311­325.
6. Gontsov R., Goryuchkina I. An analytic proof of the Malgrange-Sibuya theorem on the convergence of formal solutions of an ODE. J. Dynam. Control Syst. 2016. V. 22(1). P. 91-100.
7. .. // . 2004. . 59. 3. . 31­80.
8. .. // . ... 2011. 15. 26 .
9. .. , , " . ", . 2015. C. 13­33.
10. .. . .:. 2009. 200 .
11. .. . .-.: . 1941. 400 .
Reference in New Issue View Git Blame Copy Permalink