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U
T
E
N
U
AI D
S
RE
VI
N
D DE
OR
I
E
*V
*
EN
EL
EZ
A
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a
1 0 0,5 0,6
a
a M= 1 A= 5
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c= G = 1
G=R 12
R g= 8 T
R g R
T
T= ( +P) U U P g+ k k
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P
U
U =dx/ds k kk = 0
= 0(1 + ) 0
R= gR
R= 8 T = 8 ( 3 P)
R= 4 [2 ( +P) U U ( P)g+ 2 k k]
T =T;= 0
x
( +P)UU; (g UU) P ;= 0
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;U + ( +P) U;+ ; k
k U+
k; k +k k;
U= 0
( k k) ; U U k k; k= 0
ds2 =e2
V
rdu2 + 2du dr
r2 d2 + sen2 d2
V u r u = x0
r = x1 (g11 = 0) = x2
= x3
g=
V
re2 e2 0 0
e2 0 0 0
0 0 r2 0
0 0 0
r2 sen2()
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g =
0 e2 0 0
e2
V
re2
0 0
0 0 r2 0
0 0 0 r2 sen2 ()
|g| = e4r4 sen2 ()
000 = 2 ,u 1
2 r(2 ,rV +V,r V /r) 022 = r e2
221 = 1/r 033 = r sen
2() e2
100
= V
2 r2 r V,uV 2 r ,u+V,r+ 2 ,rV Vr 111 = 2 ,r110 = (2 ,rV +V,r V /r) /(2 r) 122 = V e2
133 = V sen2() e2 233 = 1
2sen(2 )
332 = cot() 331 = 1/r
R
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R = R
R= , ,+
R00 =2 V ,ur
r V V,r,r
r2 V
2 ,rr3
,rrV2
r2 V V,rr
2 r2 V,u 2 ,u V
r2
R01 = 2 ,ur 2 ,rV,r+ 2 ,rrV +V,rr2 r
V ,rr2
R11 = 4 ,rr
R22 = V,r e2 1
R33 =
V,r e2 1 sen2()
8 T00=V,u 2 ,u V
r2 V
r3
e2 V,r+ 2 ,rV
8 T01= 1
r2
e2 V,r+ 2 ,rV 8 T11=
4 ,rr
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8T22 = 8T33 =e
2
2,ur 1
2 r2 [r V,rr 2,rV + 2r (,rr V +,rV,r)]
V
+
ds
2
=dt
2
dx2
dy2
dz2
T
t x y z
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dt= e Vr1/2 du+ r
V1/2 dr
dx= e r
V
1/2dr
dy= r d
dz= r sen d
p
3
T=
+
+ 3
0 0
p++ 0 00 0 p+ 00 0 0 p+
= dx/dt 1<
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>0
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T
T=x
xx
xT
x t x y z
T00=e2V
r
+P 2
1 2 +
T01= e2
1 +( P )
T11=r e2
V1 1 +
( +P)
T33 = T44 = P
T00 =
e2r ( +P) ( 1)
V (1 +)
T01 = e2 ( +P)
1 +
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T11 = [(1 2) + 2 +P] e2 V
(2 1) r
T22 = sen2() T33 = P
r2
V
U= 0 e
(1 2) 12
V
r +1 e
1 1 +
1
2
r
V
U = 0e
r
V
1 1 +
1/2+1 e
V
r
(1 2)1/2
k= 0 e
V
r
12
k =1e
V
r
12
U k UU = 1
kk= 0
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m
V =e2 [r 2 m(u, r)]
m
m
+P 2
1 2 += 1
4 r (r 2 m)m,u e2 +(r 2 m)
r m,r
1
1 +( P ) = m,r
4r2
1 1 +
( +P) =r 2m
2r2 ,r
P = ,ure2
4 + 1
81 2m
r2,rr + 42,r ,rr
+3,r(1 2m,r) m,rr
8r
P
m
= P
1 +
P =P
1 +
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P
= 0
(1 )2 ( +P) +=
m,u e24r(r 2m)
= m,r4r2
+ P =r
2m
2r2 ,r
P = ,ure2
4 +
1
8
1 2m
r
2,rr + 4
2,r
,rr
+3,r(1 2m,r) m,rr
8r
m P
e=m,r4r2
Pe=2,r(r 2m) m,r4r2
e Pe
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= e+P Pe
= ee+P
= m,u e2
4r(r 2m) (e+Pe)
(1 )2
m
P
dr
du=
V
r
1
m u
dm
du = m,u+ m,r
dr
du
dm
du =
4 r2 e2 1 2 mr P 1 +
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T1 ; = 0
e2
P+
1 2m/r
,u
+P
r +
P+
1 2m/r
4rP+m
r2
=
2
r(P P)
H(u, r)
H(u, r)
2
r
(P
P) +e2
P+
1 2m/r,u H(u, r)
H= 0
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P
r
+P+
1 2m/r 4rP+m
r2= H(u, r)
=
P =P
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= (u, r)
= ( +P)
1 + = ( + P)
1
> P > P
< P < P
e2
P+
1 2m/r
,u
+P
r +
P+
1 2m/r
4rP+m
r2
=
2
r +
= ,r+ 2m
r(r 2m) 2,ue2
1 2m/r +
4r
1 2m/r
( + 3P) 22 + 41 + 4
2e2(PtP)/r Pt
2e2
r (Pt P) =
Pt=r
2e2 +P
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= 0 = 0 = 0 Pt
Pt =P
Pt
Pt = PD+PR+P
PD
PR
PD+PR = r
2e2
= 0
r ,,,H
= P = P
,P, m,
m,r = 4 r2
,r = 2 r2
r 2 m( +P)
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P,r = P+1 2m/r
4rP+
m
r2
=
= 3 M
4 A3
m=
4
3 r
3
=1
2ln
3
1 2 M/A2
1 2 m/r 1
2
P =
1 2 m/r
1 2 M/A
3
1 2 M/A
1 2 m/r
M A
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r =
a(u) = A(u) a A
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r= A
(0 r A) ds2() (r
A)
ds2(+) u,r,,
ds2(+)=
1 2M
r
du2 + 2 dudr r2 (d2 + sen2 d2)
ds2() ds2(+)
r= A
V
(u, A) = 0
m(u, A) =M(u)
(u, r)
(u, r) =A r
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= e (2 A +V /r)1/2 (A 0 1)
= e (2 A +V /r)1/20 + ( A +V /r) 1
A dA/du u=
l= u= 0 l =e21
u=
L= l =e (2 A +V /r)1/2
r= A
() ; |a =(+) ; |a
L
L = L
L() |a =L(+) |a
,ue2 +
1 2 m
r
,r m,r
2 r
a
= 0
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r r = A
,u |a= A ,r|a
Pa= aa
a
Pa= 0
r = A A(u) F(u)
(u) E(u)
F = 1 2 MA
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= 1
1 a
E= 4 A2 a
A a = (u, a)
(1/2, ) M = m(u, a) E
A=dA
du =F( 1)
m
u r= A
M= F E
E
F >0
F
F =
2 E+ (1 F) ( 1)A
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(a)
F
F+
aa
+2F Ha
a+ (1)
4Aa
(3 1)
3 +F2 A
+F
a,ra
= 0
A M E
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r r = 0
R
R=RR
R= 64 2
2 + 3 P2 + 2 ( +P)
1 1 +
R= 64 2
2 + 3 P2 + 2 ( + P)
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P
r = 0
m
m, r(u, 0) = m, r r(u, 0) =, r(u, 0) = 0
C =C
C
C
C = 48
m
r3 4
3
2
m r3
C|r=0= 0
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m= m3(u) r3 + O(r5)
= 0(u) +2(u) r2 + O(r4)
m r
r
P(u, r) =P0(u) +P1(u) r+ O(r2)
P0 =4 2 3 m3
4
P1=
e20 2,u
2
(u, r) =0(u) +1(u) r+ O (r2)
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0=3 m34
1= e20 2,u
2
(u, r)
(u, r) =1(u) r+
O(r2)
1 = e2 0 2,u
2 2
(u, r)
(u, r) =1(u) r+ O (r2
)
1 =e2 0 (2 2,u m3,u)
4
P =P()
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(0+ 3 P0), u 61 e2 0 = 0
0+ 3 P0 6
1 e20du=
0+ 3 P0=
P1= 1
P <
P1 = 1(0+P0)
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P1
1
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u
= f(u)
f u
A(u)
P =f(u)
(1 3 a)(1 83f r2)1/2 (1 a)(1 83f A2)1/23(1 a)(1 83f A2)1/2 (1 3a)(1 83f r2)1/2
m
m=4
3f r3
= 1
2ln
(1 a)
3 (1 83f A2)1/22 (1 8
3f r2)1/2
12
+a
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= f(u) =3(1 F)
8A2
P =3(1 F)
8A2(3 2) F
3
F (3 2)
(u, r) = 1 2m/r = 1 (1 F)r2/A2
=1
2ln
1
3
F
2
1
2
+
1
r A F E
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P = 3 r16 Y2A2
(1 F)6 F r2 Y 2 3/2A2Y + 3F A2 3F3/2r24
F 3A22Y
+
F
3/2
F+
3 r
F(1 F)
2
Y + 3 (
F)
32 A2 2 Y3 3
+ 3 r (1 F)8 A3 Y2 3/2
3 r2 (1 F) 2
F Y
F4 3/2 A2 Y
+ F A
3(1 F)8 A2
3
F(1 F)[3(1 F) r2 A2 (3 +)]32 Y A4 3/2
Y =e2 = 3
2
F 1+ 1
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= 3 r
16 Y2A2(1 F)6 F r2 Y 2 3/2A2Y + 3F A2 3F3/2r2
4
F 3A22Y
+
F
3/2
F+
3 r
F(1 F)
2
Y + 3 (
F)
32 A2 2 Y3 3
+ 3 r (1 F)8 A3 Y2 3/2
3 r2 (1 F) 2 F Y F4 3/2 A2 Y
+ F A
+3(1 F)
8A2
F 3 + 2
3
F 3 + 2
+ 3
F r2 + 3 F3/2r2 +2
F A2 + 3
F A245/2A2Y
= ee+P
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e= 3 r16 Y2A2
(1 F)6 F r2 Y 2 3/2A2Y + 3F A2 3F3/2r24
F 3A22Y
+
F
3/2
F+
3 r
F(1 F)
2
Y + 3 (
F)
32 A2 2 Y3 3
+ 3 r (1 F)8 A3 Y2 3/2
3 r2 (1 F) 2
F Y
F4 3/2 A2 Y
+ F A
+3(1 F)
8A2
F 3 + 2
3
F 3 + 2 1
+ 3
F r2 + 3 F3/2r2 +2
F A2 + 3
F A245/2A2Y
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e+P = 3 r
16 Y2A2(1 F)6 F r2 Y 2 3/2A2Y + 3F A2 3F3/2r2
4
F 3A22Y
+
F
3/2
F+
3 r
F(1 F)
2
Y + 3 (
F)
32 A2 2 Y3 3
+ 3 r (1 F)8 A3 Y2 3/2
3 r2 (1 F) 2
F Y
F4 3/2 A2 Y
+ F A
+3(1 F)
8A2
3
F r2 + 3 F3/2r2 +2
F A2 + 3
F A2
45/2A2Y
= e2 r
8 A3
A F 2(1 F)A
34A2
(1 F)
F3
F 3 + 2
(1 )2
= 0
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A F
r= a
F
F + (1 F)
+
3 (1 F)2 ( 1)(2 1)2 A
= 0
m
m(u, r) = m3 r3
(u, r) = 0+ 2 r2 + 4 r
4 +O r6
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m3 = MA3
0 =1
2 ln
(1 a)
3
F
2 1
2
+a
2 = 3 (1 a)
F M
2 A3 (1 a) 3
F
1+ 2 a
4 =9 (1 a)
2 F(1 a)
F(1 3 a)
M2
4 A6
(1 3 a) 3
F(1 a)
m r
P(u, r) =P0(u) +P1(u) r +P2(u) r2 +P3(u) r
3 + O r4
P0=
3 M
4 A3 (1 3 a) (1 a)
F
3 F(1 a) (1 3 a)
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P1 =9 (1 a)
a+ 2 aF
3/2 3 aF+F 2 F3/2 + 1
F E
4 A3 (1 3 a) 3
F(1
a)
3
3 a(1 F)
18 F3/2 (1 a) F+ 27 aF 9 9 a
F
8 A3
(1 3 a) 3
F(1 a)3
P2 = 3 M2
2 A6(1 a) (3 a 1)
F
3
F(1 a) (1 3 a)
2
P3 = 9
F (1 F) (1 a) T1 E
8 A5
(1 3 a) 3
F(1 a)4
+ 9
F(1 F)2 a[3 T1+ 8 F(3 a 1)]
16 A5
(1 3 a) 3
F(1 a)4
T1 = 18 (1 a)2 F2 + (3 a 1)
18 (1 a) F3/2
+ 5 (3 a 1) F 3 (1 +a)]
(u, r) =0+P1 r+P3 r3 + O(r4)
0= 3 M
4 A3 = f(u)
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(u, r) =1 r+2 r2 +3 r3 + O (r4)
1 = P1
0+P0
2 = P10+P0
2
3 =P3 (0+P0)
2 P1 P2 (0+P0) +P31(0+P0)
3
(u, r) = e20 03 +P1 r P210+P0 r2
P3+2
9 (0 3 P0) e20 0
r3 + O r4
0 = 3 F4 A3
E+3 MA
a1 a
E
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a
A F a
C = 2 MA
=RS
A
RS= 2 M
(P0 )
A 9 RS8
(1 a)2(1
3 a/2)
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W = (1 a)2
(1 3 a/2)
A 9 Rs8
W
RS= 2 M
RB = (9 RS/8) W
RB RS= RS8
(1 3 a)2(1
3 a/2)
a = 2/3 a
1< a < 2/3
W
(1, 0) (1/2, 3/5] (0, 1/2)
(RB = RS) a = 1/3
Rs RB < A
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0.8
1
1.2
1.4
1.6
1 0.6 0.2 0.2 0.6
W
!a
a 1 0 0,5 0,6
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0.85
0.9
0.95
1
0 0.1 0.2 0.3 0.4 0.5
W
!a
a
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u
m= C(u) r3
1 +B(u) r2
=1
2ln
(1 K a)
3
F
2
1
2
+K a
= 1 2 C r2
/(1 +Br2
)
B = B(u) C = C(u)
r= A ma = M P a= aa
B=3 a(1 K)A2 (1 a)
C=MA3 (1 + 2 a 3 K a)(1 a)
m
K = 1 B = 0 C = M/A3
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= C4
(3 +B r2
)(1 +B r2)2
P =
3 +B r2
(3 +B r2)
(1 3 K a) 3
F(1 K a) (1 +B r2)
3
F(1 K a) (1 3 K a)
P
r a
a> 1
3 K 2 K= 1
K
r= A
A F
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= (2 3 + 3 K 3 K) ( K +K)(1 F) F 3 ( K +K)2 2K F
+ ( 1)
2 A
3 ( K +K)2 2K
= 3 ( 1)(42 F+ F 1 +F) K2 4 2 (6 5) F K
+(12 F 3 5 F)2
K= 0
= (2 3)3F(1 F) F+
(12F 5 F 3)6 A F
A
u
=A2(1 F)
F
(5 F+ 3)
12A2(1 F)+ b
a = 7 F 3 12 b A2 (1 F)5 F 3 12 b A2 (1 F)
b
b=12 F00 5 F0 3
12 A20(1 F0)
u = 0
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12 F 5 F 3
12 A2
(1 F) =b
P =3 r
(3 F
F
) (3 r2 C++B r2) 3 C F r2
2 3/2 (1 +B r2)3 (3
F )3 C
+3 C r3
(3 F
F
) (3 r2 C+ 2 + 2 B r2) 3 C F r2
2 3/2 (1 +B r2)4 (3 F )3 B
+ 3 C r
4
F(1 +B r2)2 (3
F )3 F
C
(
+ 3
F) B r2 + 3
F 3
4 (3
F ) (1 +B r2)3
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= 3 r (3 F F) (3 r2 C++B r2) 3 C F r22 3/2 (1 +B r2)3 (3
F )3 C
+3 C r3
(3 F
F
) (3 r2 C+ 2 + 2 B r2) 3 C F r2
2 3/2 (1 +B r2)4 (3
F )3 B
+ 3 C r
4 F(1 + B r2
)2
(3 F )3
F
+C
(2 B2 r4 + 3) (3
F ) + 3 B r2 (5
F 3 )
4 (3
F ) (1 + B r2)3
= +P
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= 3 r (3 F F) (3 r2 C++B r2) 3 C F r22 3/2 (1 +B r2)3 (3
F )3 C
+3 C r3
(3 F
F
) (3 r2 C+ 2 + 2 B r2) 3 C F r2
2 3/2 (1 +B r2)4 (3
F )3 B
+ 3 C r
4 F(1 + B r2
)2
(3 F )3
F
+B C r2
4
(3
F )Br2 5 + 3
F
(3
F ) (1 +B r2)3
+P =3 r (3 F
F
) (3 r2 C++B r2)
3 C F r2
2 3/2 (1 +B r2)3 (3
F )3 C
+3 C r3
(3 F
F
) (3 r2 C+ 2 + 2 B r2) 3 C F r2
2 3/2 (1 +B r2)4 (3
F )3 B
+ 3 C r
4
F(1 + B r2)2 (3
F
)3
F
+ C
4
(3
F )B2r4 + (5 + 9
F)B r2 + 6
F
(3
F ) (1 +B r2)3
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= C r2 B (1 +B r2) C2
(1 +B r2)2 (3
F )
3 C
F
2 (1 +B r2)2 (3
F )
(1 )2
B=4 (2 3 ) E+ ( 1)(1 F) (7 F 6 + 1)
2 A3 (1 F)
C=4 (1 +F) (2 3) E+ (1 )(1 F) (7 F+ 6 F+ 6 1)
4 A3
K= 0
Pt a
= PDa
+PRa
K = 0
Pta = 0
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PRa =
PDa
PDa = (1 F) a Z1
32 A3 F2 (1 + 2 a)3 (1 a)
PRa= Z2
2 A3 F(1 + 2 a)2 E
Z1= 192 F A2 (1 + 2 a)
2 2 F(1 a) (5 2a 46 a 31)
+ 9 (1 +a) (1 a)2 +F2 (109 3a+ 159 2 + 3 a 55)
Z2= 8 A2 (1 + 2 a)
2 + (1 + 6 F) 2a+ (1 + 3 F) a
2
E
E= a(1 F) Z116 F(1 + 2 a)(1 a) Z2
a 1/2
a 1
a M A
M= 1 A= 5
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1
0.5
0
0.5
1
0.4 0.1 0.2 0.5 0.8
E
!a
a M = 1 A = 5
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A M
A= (12 b A2 + 7) M
6 A1
3
M= F E
m
K= 0
m(u, r) = C r3 C B r5 + O r7
(u, r) =1
2 ln
3
2
F 1
2
+
3
F C
2 (3
F 1) r2
+3 CF4
3 2
F
1 C 2 3
F
1 B3
F 12 r4 + O r6
m
r r
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P(u, r) = P0(u) +P1(u) r+P2(u) r2 + O r3
P0= 34
C
F 1
3
F 1
P1= 3C
4
F
3
F 13 F 3 F
2
3
F 12 C
P2= 32
F C2
3
F 12 + 12
9 F 18
F+ 5
BC
3
F 12
P3= 3 CF
(3
F 1)2 B 3 C4
2 B (3 F 1) + 3 CF(3
F 1)4 F
+3
F
2
6
F B 9 F C 2 B+ 6 C
(3
F 1)3 C
(u, r) =0(u) +1(u) r+2(u) r2 +3(u) r3 + O r3
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0 =3 C4
1 = P1
2 = 3 C B
F3
F 1
3 = P3
P2n+1= 2n+1, n 0
(u, r) =1(u) r+2(u) r2 +3(u) r
3 + O r3
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1 = P10+P0
2 = 21 3 C2
F(18
F 9 F 5) B
8 2 (3
F 1)3 (0+P0)2
3 = 31
P2+20+P0
1+ P3
0+P0
(u, r) =1(u) r+2(u) r2 +3(u) r
3 + O r4
1=1
3 0 e2 0
P1
2= 1 P1+(18
F 9 F 5) C B4 (3
F 1)2
3=1
30 e
20 B+e20
12 [3 B 2 (0+ 3 P0)] C(0 P0) P3
(0+P0)
13
[2 (0+P0) (0+ 3 P0) + 3 (P2 1)] 1
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(P0 )
A 9 RS8
(1 K a)2(1 3 K a/2)
W = (1 K a)2(1 3 K a/2)
A 9 Rs8
W
K= 0
RB = (9 RS/8) W
RS= 2 M
RB
RS=
RS
8
(1 3 K a)2
(1 3 K a/2)
a K= 2/3
1< a K
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W K= 0
K a (K, 0) (K/2, 3K/5] (0, K/2)
(RB =RS) K a= 1/3
RS RB < A K
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L
u=
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K= 0
r
r
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