SCIENTIFIC ABSTRACT KHAPAYEV, M.P. - KHARA, I.S

• Document Type: CREST
• Collection: Scientific Abstracts
• Document Number (FOIA) /ESDN (CREST): CIA-RDP86-00513R000721810002-0
• Release Decision: RIF
• Original Classification: S
• Document Page Count: 100
• Document Creation Date: November 2, 2016
• Document Release Date: September 17, 2001
• Sequence Number: 2
• Publication Date: December 31, 1967
• Content Type: SCIENTIFIC ABSTRACT

87392
8/020/(;0/135/Oo6/oc6/037
C 111/ C 333
Asymptotic Expansions of Solutions to Ordinary Linear DiffErential
Equationn Having Small Coe:'ficients With Their Higher DeriNatives
in the Neighborhood of an ::rregular Singular Point
~ irtko if 2 -~ k i9 m
"M tyk( go) ~o -1
?40 ~A K 0 - Jo m + 1
Let TO ;0 to + 2Tr . Let
(13) 1 z C- Gk( -)) if - 2/3 af I'k( < arg z <
4 21/ Tk(T, 0)
2.,) G
k( Sd' if
0) < arg z < 2,/ 5 JF
-2/33r - Af/k(
Yk( E 0)
Card 4/ a

87392
S/020/60/135/006/006/037
G 111/ G 333
Asymptotic Expansions of Solutions to Ordinary Linear Difj'erential
Equationa Having Small Coerficients With Their Higher Der.-.vatives
in the Noighborhood of an Erregular Singular Point
5.) z C- Gk0 1 if -2/33r - I*ko < arg z < 2/37r - A+ko.
Let G be the intersection of the G 0 ; G( S ) intersection of the
0 k 0
ak( ~c G(,~ 0 interse3tion of the G k( 9 , ~.). Let G,,, ( So) be
narrower than G( 6 ) and lot it be contained in G(S,, for all
sufficie.-atly small T . By the transformation
(14) W(z. P- ) - e k, (0,)z , (r, (F,) U(
lot (1) pass over into
r1 (k) M
IE] - 2: F_ ' Pk(z Z Pk(z E u
where Karoij-1 00 00
s
Pn (z' Pk (z Eaak,s(z) Z- bk,s(
SV0 S.0
Card 5/6

87392
S/020/60/135/006/'(jo6/037
C 111/ C 333
Asymptotic Expansions of Sclutions to Ordinary Linear Dif:~erential
Equatione Having Small Coefficiento With Their Higher De:7-..vatives
In the N(ilghborhood of an Irregular Singular Point
Let L C LI,E,] be representod as
00
(23) u,
Z cs~, I U1
5=0
where 11A - 01 n
~ C U'l U(M) + 7 ak,o(z) u (k), [u1 ak,o-k+m (Z)U (k)+
0 ic C 0
+ L ak,s (Z) u(k) , rhere aky 0 for s < 0.
K%. 0
Theorem: Lqt u(Z,E ) be th~.3 solution of (15) and have th,(!
asymptotic expansion
00
(28) U(Z~ + L C19S Z-0 in G, ( io) -
Card 6/Ei Sal

87392
S/020/60/135/006,/006/037
C ill/ C 333
AsymptotLc Expanaions of SAutione to Ordinary Linear Differential
Equations Having Small Coefficients With Their Higher Derivatives
in the Neighborhood of an Irregular Singular Point
Aseume that the fuiction u 0(z) satisfies
(25) 7 [u0j - 0
and has the asymptotic expansion
00
(27) 11 U (Z) = 1 + C 0 Z_s in G09
0 1 i's
while the functions a S(z) are determined by the equations
(26) L 0 [U 8] US-1-
as well as by the conditor. that they decrease at infinity as 1/Z
in G0 . Then the formal exj,unsion of u(Z, in terms of C -powers
Card 7/ a

87392
3/020/60/135/0o6/006/037
C Ill/ C 333
Asymptot:ic Expansions of Solutions to Ordinary Linear DJfferential
Equationi-. Having Small Coefficient,,i With Thoir Higher D(!rivEttive3
in tile Nt:,ighborhood of an .,,'rregular Singular Point
00
(24) U(z~ & ) - ~- 0 E "LB(z)
is asymp-.otic in G"( SO ) :.or E -* 0 (arg E 0 ) so Via t
lim u(z, u (Z) -
;-40 C
The author thanks Yu,. L. Ribinovich and D. P, Kostomarov for
as:~istance.
There are 6 references: 5 3oviet and 1 Belgian.
ASSOCIATIONx Moskovskiy goaudarstvennyy universitet imeni M. V,
LomonoBova (M:)scow State7-Unlversity imeni M. V.
Lomonosov)
PPESEFTED: July 79 1960, by J. G, Petrovskiy, Academician
SUBMITTED: July 7, 1960
Card 6/8

Z883-1
3/140/61/000/005/007/007
1, 1 11 1 / Q' ? 2 2
TITLEi Tlv~ atym,,01)~- Irrveloymout (,[ hypergeometric and
fulle ti ons
PERIODICIJ': I i vyushikh u:~haijnykh zn~iodeniy. Matematika,
19611 98-101
TEXT: The author -obtains asymptotic developments of the hypergeometric.
funotion F(a,b,c,.) and thG degeneratod hYF,-,rK,3ometric function
F(a,c...z) i7or the eai3e that n and c are large and have the same ordo".
The author starts from the oquations
d it du 0
- (z-c) -
2 TZ
and 2
+ (.,b,l),] du - abu - 0 (13)
7z
dz"
I
respectively, where a I , a -X'l, 1 large, c4j 0, he
Card 1/*

28811
5/14 61/000/005/007/007
The aeymptotio dovolopment 0111%222
intrclu,,as the now variable t. z vn,I obtains equal.'.on, r,'-h -t
small parE,meter for the higbest derivativs-), e. g.~ (i) changv~ 1_",
6t VIt + [F, t(d+1) + vI + F_tdv 0 (2)
- t L
with v(t,) - e
u t d-1 - Th e a s ymj t o tc d c, prr

c,f the sol ution of (2) regular in 0 cor respo nds to F(a, :_~, Z)
Thlis the author obtains the developments

F (u, 1
y ~' 75) z I +
y
2 2
and
Card 2/1~

M12
S/140/61/000/01)5/007/007
The asymptotic development C111/C222
b(b+l) 2
b ) .1
1 ,- 6 + + (17)
2 tt (,_t)2
For large m,n and I M-n I - 0
UDC: 517. 934

ACC NR: ',.~60M47
and t an-I x-,!' DJXY. X). IV (t), aid for any finite interval 1~1 t;21 the folio.4ing b1old
N (1) ill,< V, (t,
c; there exists a summable function 11(t) and a constant HO, and also a non-vanishing
f-Iliction ~W, lim ~W :-- 0, such that for t > 0 and xQD
".0
X (t, X') - X (t, e) I