CC BY
6
ftärsi
Kywods
[Mi? ASRdkn
АНитаця
S = i и
fe-
КНХЕШПШ
MHflCER-
IiMdkn
devühßiiicfs^iEjÖESj s = 1 arS = o
асгЁгаГЬст yeczficfiE
¡ИЩсГО
(Ф, o, Фа, o, o), S = o; (o, Ф, o, Фа, o), S = 0;
(G, G, Фа, G, ф"ь), s = l; (G, G, G, Фа, ф"ь), S =1.
(Ф, G, Фа, G, ф"ь), s = G, l; (G, Ф, G, Фa, ф"ь), S = G, l.
HfetaiK^jlkri
. IFBirÉiStLíJelliiüsyíltrfcfeiiíticrBislífllcv^
-Da^a - = G,
Da^ + Db^ab - M^a
G,
Da^b - Db^a - M^ab = G, VtaßDa = da+ ieAa
rcraai
Ф = (Ф;Фо, Ф1, Ф2, Фз;^01, Ф02, Фоз, ^23, ^31, Ф12) = (H,H1,H2).
IF^tDfisyitEniznlEpiîïïrlEEintieînatiixlla^jnri
DaGaH1 + mH = g, AaDaH + KaDaH2 - mH1 = G, DaLaH1 - mH2 = G,
ír(rel^BnÍriE£ÍCpÍrrflcrtífGa)
(-Dara - м)Ф = G,
G -Ga G\ /H'
ra = I Aa G Ka I , Ф = I H1 G La G H2
A2 = (G, G, l, G)4, A3 = (G, G, G, l)4
K 0
K1
K2
K 3
L1
L2
L3
G G G G G G
-l G G G G G
G -l G G G GI
G G - l G G G
- l G G G G G\
G G G G G G
G G G G G lI
G G G G - l G
G -l G G G G\
G G G G G -l
G G G G G GI
G G G l G G
G G -l G G G\
G G G G l G
G G G -l G GI
G G G G G G
G l G G\
G G l G
L 0= G G G l
G G G G
G G G GI
G G G G
f- -l G G G \
G G G G
1 G G G G
G G G G ,
G G G -l
G G l G
G G G G\
-l G G G
2 G G G G
G G G l ,
G G G GI
G -l G G
G G G G\
G G G G
3 -l G G G
G G -l G
G l G GI
G G G G
Orten
ctctolte
cg aß (x) v®
vtae
0 gaß(x) ^ e(a)a(x), nab = diag(l,-l,-l,-l),
A0 = (l, G, G, G)4, A1 = (G, l, G, G)
[^Ц¿ + Sa(x)) - M ^(x) = G.
Q
7Q jj^ 5(55), 2Q2
4
rax)
c^ieirtlCiJßcrecttfennidvith lntiatfcrn:itieïcfe
Га (x) = e?(x)P
Aaefa)
e(a)
0 Kaea{a) a e? )
(a)
1ШпВЛПе? (x) isctírTídlyiíEltrncIcB
J
b
o o o o J1 b o o o J2 b
i
Sa (x) = 2 J abeß{a )(x)e(b)ß;a(x)
o o o
0 (Sl)a 0 ,0 0 (S2)c
дд
, i / д iBr2
G dt G dr G r ^\дф + 2
12
+ j 112 -
дz
H1 - IH = o,
дt + дr +
л21/ iBr2\ л3 д + A2^ [дф + —) +A3—
H+
+
n — x — 2 дф + Bf- + jl2
Kn — + K1 — + K2 ф + 2 +j1
д dt
д дr
+
+ kV
дz
H2 = iHl,
S1(x) = ^ J{1) ^a) (x)e(Ъ)ß;a(x),
S2(x) = 2 J(a2) ^a) (x)e(b)ß;a(x),
(x)
e?c) ¿ + i JabYabc ) - I
[ab]c. = 7[ba]c = e(b)pa epa)e1c)
*(x) = 0. Q
.IncteM
n д 1 д 2д
Ln— + L^— + L2
dt dr
hÍvvTitviil
iBr2 + ■ 12 2 + j1
+
+ dz
Hl = 1H2.
U=
-GCe<ac) da - GCJal) 2 Y abc
Ace?c)daH+
K c e(c) da + K J) 2 Y abc
Hx - iH = 0, i
H2-1H1 = 0,
U-1 =
Lce(c)da + LcJl iYabc
Hl - 1H2 = 0.
/i 0 0 0\ 0 - Л ^ 0
0 0 o i
Vo 7 72 /i 0 0 0 \
0 - 7 0 7 ° 0 0,
\0 0 i 0 /
xa = (t, r, ф, z), dS2 = dt2 - dr2 - r2 d^2 - dz2,
Br2 A* = —
tÍBctD£í£|JÜ(nÍCl€EÍíÍrnfljÉteB ^ b).
Jab = Ujabu-1, Jf = Jab ® I + I ® Jab.
letislicriiCrnfiíKfrTírciCrslDtyíiicÍCrm
/0 0 0 0N
12
o -i 00 0000 \0 0 0 ij
(-i
Го — + Г1 — + Г2 d* + iBr2/2 + J12
dt dr
+Г3 — -1
dz
+
Ф = 0.
J
12
o
o
r
r
k
r
ю
h = H, h-,
C1H1,
(C1 = U ), S2 = U ® UH2 = C2H2,
verfíivelheíüe
0 —Ga 0 Aa 0 Ka 0 L a 0
О
-GaC-1 О
C2 LaC-1
О
C1K aC-О
Fitèt vefindthencIriŒB a a = ^ai ¿зт1 g a =
GaC-1
A0
A 2
(l, 0,0,0)4, A1
(»■ Ti ■ b-n) '
0, ' 0 $
l
Ti'
a3
(0,0, l, 0)4;
G0 = (l, 0, 0,0), G1
("• Tu 0 -h)
g2
0, T., t. ), 03 = (o, -1,o).
U ® U ^ C2, H2 = C2H2 ^
О 0 0\
/__L 0
' -2 -2 0
C2 =
0 1
— — 0 -2 -2 0
ОО
ОО
00
1
'— - 0
ОО ОО
1
-— — 0
-2 -2 0
0 0 i
- 0
C
1
V2 О
О О О
л/2
i
-2 О
О О О
V2 V2 ОО ОО ОО
7 О О О
V2
О О О О -i О
ICrntheisBcftienfvcczncltfeincIlcItFtjCitsincyclii
f2 -2
K 2
-2 О i -2 О О О
О О О О i -2 О
О О О i -2 О - i -
О О О О i -2 О
О -l О О О 0 \
KT3 = О О О О О -1 ;
О О О О О 0
О О О l О О
О l О 0\
О О l О
LT 0= О О О О О О О О О l 0 0 ,
ОООО
/72
L1
V2 О
О О
1 -2 О
L2
"V2 О
О О
О О О
0
1 -2 О
О О О
0
1 -2 О
О О О
1
"-2 О
"V2 О О О
i
-О
V2
О
О О
0
1
-2 О
О
О О О
'л/2 О
L3
/0 0 0 0\
-l О О О
0 000
0 001
0 000
\ 0 -100/
' tàtic^rthevei^nicn
K 0
K1
О
^ 0
ОО ОО ОО
( 0 0 0 0 0 0\
-l О О О О О
0 -1 0 0 0 0
0 -1 0 0 0y
1
-2 О
О
О
О О
1
-2 О
0
1
-2
0
1
-2
О О
1
-2 О
= d m + Br2/i = d m +1 + Br2/i
I , am+1 I ,
dr r dr r
= d m + Br2/1 = d m - 1 + Br2/i
dr r dr r
72 Ива^ 5(555, Ш
C1Aa
О
О
itKHÈ
—ieh0 — ikh2 + bm-lhl — am+lh3 = —ßh, -ieh - ikE2 + bm-lEl - am+lE3 = |h0, —amh + am+lB2 - ikB-3 + ieEl = ßhl, ikh + ieE2 - am+lBl — bm-lB3 = |h2,
bmh + b mB2 + ikBl + ieE3 — |h3, amh0 — iehl = |El, —ikh0 — ieh2 = 1E2, -bmh0 - ieh'3 = 1E3, -bmh2 + ikh3 = |Bl, bm-lhl + am+lhä = 1B2, —ikhl - amh2 = |B3.
ikh + ieE2 - am+lBl - bm-lBz = |h2, —ikh0 — ieh-2 = 1E2,
bm-lhl + am+l h-3 = 1B2 ■
Pl = 1 У(У - 1), P2 = 1 У(У + 1), P3 = 1 - У2
vjHrtl^^:f^^P02 = РоР p+l = p+l, p-l = p-l, Ро + P±t+ P=1 = lj
uiuiicuc-s
Ф = Фо + Ф+1 + Ф-1, Фа = Pa Ф, а = 0, +1,-1.
ф1(г) = (0,0,h1, О, 0,E1,0,0,0, 0,B3)tf1(r), Ф2(r) = (0,0,0,0,h3, 0,0,E3,Bl, 0, 0)tf2(r),
ф3(г) = (h,h0,0,h2,0,0,E2,0,0,B2,0)tf3(r).
pi(Al0xl0 Ф) = 0,
for
Pl
—amh + amB2 — ikB3 + ieEl = ßhl, amh0 — iehl = |El, —ikhl — amh2 = 1B3;
fCTP2
bmh + b mB2 + ikBl + ieE3 — |h3,
-bmh0 - ieh'3 = 1E3,
—bm h2 + ikh-3 = |Bi;
fcrP3
—ieh0 — ikh2 + bm-lhl — am+h = ßh,
-ieh - ikE2 + bm-lEl - am+lE3 = ßh-о,
- amfä(r)h + amfä(r)B2 - ikfl(r)Bä+
+ iefl(r)El = |fl(r)hl ^ amf3 = Clfl,
amfä(r)h0 - iefl(r)hi =
= ßfl(r)El ^ amf'3 = Clfl,
- ikfl (r)hl - amfä(r)h2 =
= |fl(r)B3 ^ amf3 = Clfl, firP2
bmf'3 (r)h + bmf3(r)B2 + ikf2 (r)Bl +
+ ief2(r)Eä = |f2(r)h3 ^ bmf'3 = C2f2,
- bmf'3(r)h0 - ief2(r)h3 =
= |f2(r)E3 ^ bmf3 = C2/2,
- bmh(r)h2 + ikh(r)h3 =
= |f2(r)Bl ^ bmh = C2f2, fcrP3
- ief3(r)h0 - ikfä(r)h2 + bm-lfl (r)hl-
- bm-lfl(r)h3 = |l/з(r)h ^ bm-lfl = C3f3,
- ief3(r)h - ikf3(r)E2 + bm-1 fl(r)El-
- am+lf2(r)E3 = |f3(r)h0 ^
^ bm-lfl = C3f3, am+lf2 = C4f3,
ikf3(r)h = ief3(r)E2 - am+lf2(r)Bl-
- bm-lfl(r)Bä = |f3(r)h2 ^
^ bm-1f 1 = C3f3, am+1 f2 = C4f3,
- ikf3(r)h0 - iefä(r)h2 = |f3(r)E2, bm-lfl (r)hl + am+lf2(r)h3 = |f3(r)B2 ^
^ bm-lfl = C3f3, am+lf2 = C4f3.
!Fu?velBectiivEdllEElgfi)cjce£i£liob
—C1h + C^2 - ikB3 + ieE1 = ßh1, Cl h0 — iehl = |El, —ikhl — Clh2 = 1B3, C2h + C2B2 + ikBl + ieE3 = |h3,
—C2hn — ieh3 = 1E3, —C2h2 + ikh3 = 1B1, —iehn — ikh2 + C3hl — C3h3 = ih, —ieh — ikE2 + C3 El — C4E3 = ihn, ikh + ieE2 — C4B1 — C3B3 = ih2, —ikhn — ieh2 = 1E2, C3hl + C^h3 = 1B2,
f d^ 1 ( ) \dr2 r dr 4
(m - 1)2
- Bm - C2 fl = 0,
^cf^e^BiC^virxjan^^^
С d2 1
() I dr2 r dr 4
bm-lfl(r) = Cfr), amf3(r) = Clfl(r), M am+lf2(r) = Cf3(r), bmf3(r) = C2f2(r). ^
iciuuiiK
bm-1amf3 = C1C3 f3, ambm-1f1 = C1C3f1, am+lbmf3 = C2C4 f3, bmam+lf2 = C2C^f2.
C1tC4 = C2.
(m + 1)2
- Bm - C2 + 2B)f2 = 0.
lit
B
X1
_ / d2 1 d B2r2 m2 \ n
(1) (d^ + rdr - 72 - Bm + X)f3 = 0,
(2) (Ü 1 ±_B2r2 (2) 1Z2 + Г12 —
dr2 r dr 4
(m - 1)2
bm-l fl(r) = Cl f3, amf3 = Clfl, am+lf2(r) = C2f3, bmf3 = C2f2;
[bm-lam - C]f3 = 0, [ambm-1 - C¡]fl =0, (J) [am+lbm — C¡ ]f3 = 0, [bmam+1 — C%]f2 = 0.
Irex^dtfcrr^Qnic]
( d2 1 d B2r2 m2
+ --T -
- в (m +1)+ X\fl = 0,
/ d2 1 d B2r2 () l dr2 r dr 4
(m + 1)2
- B(m - 1)+ X)f2 = 0
)
vüníbívvcíícügx = bff2 Fth^ci€CntheiCrnn
dr2 r dr 4 r2
-Bm + B - f3 = 0,
d2 1 d B2r2 (m - 1)2 + --r -
(1)
d2 1 d 1 (m/2)2
+------
dx2 x dx 4
x
2
+
1 /_ m
+ i \ ~2 + 2B
f3 = 0,
dr2 r dr 4
-Bm - C2^j fl =0 ( d2 1 d B2r2
(2)
d2 1 d 1 [(m - 1)/2]2
dx2 x dx 4 x2
+----
dr2 r dr 4
-Bm - B - C2J f3 = 0,
'dL 1 2r2 (m + i)2
dr2 r dr
(3) -7-2
if m + 1 X
+x ( + 2B
d2 1 d 1 [(m + 1)/2]2
dx2 x dx 4 x2
1 / m - 1 X
+x \ + 2B
fl = 0,
-Bm - C22 )f2 = 0.
f2 = 0.
ràeqitmrcëâ
3 (x) = xAeCxF (x),
vi
C2 = C2 - 2B crdoly
(1) f il 1 d B2r2
I dr2 r dr 4
m
- Bm + B - Cl )f3 = 0,
xF'' + (2A+1 + 2Cx)F ' + (m/2)2
x
, . .. (c2 - 4)
IdiBin^etuctiC^b
A2 - (m/2)2 = 0 ^ A = ±\m/2\
m X 2 1
+C - -2+2B + nC 2 - 4
+ 2AC+ F = 0.
2
r
2
r
2
m
2
r
2
r
C2 - T = 0 ^ C + ±-.
— 1{П2,Пз.
hacjl^^
П — 1«n
tmite
BüICvvevüL á^^tíBiin^ сиаИш-
A = ±\m/2\, C = - 2.
|ь«ailSinlÍi!jClffa1cfaccгtóÍWcniiIic
X = 2BN > 0, N =(JH+m + 2 + ^
N = 1, 3,... . 2' 2'
xF '' + (\m\ + 1 — x)F ' -
vühpmtes
-
\m\ + m 1 X
+ 2 - 2B F = 0
)
\m\ + m 1 X
a = 2 +2 - 2B,
c = \m\ + 1, F = Ф^,с^).
lÍБpciyíIniяlCDlCilicnk = -m IfHÖD
(3) ^ X = +2B
\m\ + m 1 2 + 2
+ - + nH > 0,
nl =0,1, 2,....
lííUIviгl]lIУicrшrrjï|
iiiiiiíííiiíh
im
—iehn — ikh2 + C-3hl — C3h3 = —ßh, —ieh — ikE2 + C3E1 — C4E3 = ihn, -C1h + CxB2 - ikB3 + ieE1 = ih1, ikh + ieE2 — C4B1 — C3 Ьз = ih2, C2h + C2B2 + ikBl + ieE3 = h, Clhn — iehl = iEl, —ikhn — ieh2 = iE2, —C2hn — ieh'3 = iE3, —C2 h2 + ikh-з = цвх, C3hl + C4h-3 = iB2, —ikhl — Clh2 = цВз .
R&rn
(3) f3(x) = x^^^ x—xX/ Fl(x),
Fl(x) = Ф(—nl, \m\ + 1,x).
lvIIl^(ClíllcrБcj\filiniitrllïlIfc5líll?v«íвe
Cl = C3 = VX - в, C2 = C4 = VXTB.
cn
ф = o.
i3(k2 +12 - 2X - e2) —2B (5k + ц2 - 2X - 5e2)+
(3)
f3(x) = x+ т x-x/2Fl (x),
F3(x) = ф(-пь \m\ + 1,x),
X = 2В( ^^ + 1 + nl)
+B(—k2-i2 + 2X+e2 ){2\[x2—B2-k2-ц2 + e2)--(k2 + i2 - 2X - e2)2 (/X2 - B2 - k2 +
n3 = 0,1,2,...
+i2 + X + e2
(6) IraiA-ítorac)
0.
Ря = P2P6
(1) fl(x)
x) = x+ 2 x ~''F2(x Fx(x) = Ф(—n2, \m — 1\ + 1, x), \m — 1\ + m +1 1
C/2f2(x),
X = 2B
n3 = 0,1,2,...
2
+ « + n2 \ > B,
2
k2 +12 - 2X - e2 = 0 ^ e2 - ц2 = k2 - 2X.
k2 гсвс
-W3 + W2 (-VX2 - B2 + B + i2 - 5X) +
e2
(2)
f2(x) = x+im+i x-x/2F3(x), F2(x) = Ф(—n3, \m + 1\ + 1, x), \m +1\ + m — 1 1
0 +W
— (2X -12
2B
- B2 - ц2 + x) -2 - B2 + i2 + 4X) + 10B2
sm
X = 2B
n2 = 0,1,2,..
2
+ 2+ пз) > B,
+(2X - i2)
B
2 - B2 -
— (2X - i2
Vilhcinf^гteE^cîali5
B2 + i2 + X + 2B2
+
OR
W X B , i
—2 ^ w, —2 ^ x, —2 ^ b, — ^ 1, i2 i2 i2 i
2
1l
fttctecnlhEföm
w3 - w2^-\Jx2 - b2 + b +1 - 5xj --w ib^Vx2 - b2 - 1 + x)-
+
0. lltl
— (2x - 1)(2л/x2 - b2 + 1 + 4x) + 10b2
+(1 - 2x) b(2\Jx2 - b2 - l) + +(1 - x2 - b2 + 1 + x) +2Ь2 =
iriaMosae дотажл
that x = ibN.
ncci
b = 0.01(0.98,0.94,0.9,0.86, 0.82, 0.78,0.74, 0.7/ b = 0.05(0.9,0.7,0.5,0.3,0.1, -0.1, -0.3, -0.5)4, b = 0.1(0.8,0.4,0., -0.4, -0.8, -1.2, -1.6, -1)4.
1
0.78919 0.576651 0.361149 0.138565 -0.0934233 + 0.0312522í —0.2929 + 0.0548385i —0.49238 + 0.0709556i /
Qnkfkri
—4bN „1 - 2x> _0
w
rypTí"
Rforas
b = 0.001
b = 0.01
í -1 0.99600i 1.
-1.00483 0.992007 0.995996
-1.0089 0.988011 0.99199i
-1.01i93 0.984015 0.987988
-1.01695 0.980019 0.983984
-1.02096 0.976023 0.97998
-1.02496 0.972027 0.975976
V -1.02897 0.968031 0.971972У
(- - 1.0002 0.960i04 1. \
1.04858 0.920701 0.959595
-1.0893 0.88113 0.919181
-1.1296 0.841564 0.878757
1.16977 0.802008 0.8383i4
-1.20988 0.76i461 0.797879
-1.24996 0.722926 0.757423
- 1.29002 0.683403 0.716955
b = 0.05
-1.00554 0.805539
—1.25012 0.619508
-1.45486 0.433263
-1.65743 0.i49873
-1.85931 0.0735313
—2.06088 —0.0934233 - 0.0312522í
—226217 —0.2929 - 0.0548385i
V-2.46357 —0.49238 - 0.0709556i
c^lvtvc^íBíiveíccts. In ш-
iiscriiiicciiiiMi
6
7
ШзйКм^^ 77
