ivane javaxiSvilis saxelobis Tbilisis saxelmwifo universiteti zust da sabunebismetyvelo mecnierebaTa fakulteti maTematikis departamenti Salva zviadaZe furies SeuRlebuli trigonometriuli mwkrivebis zogierTi Tvisebis Sesaxeb sadoqtoro disertacia samecniero xelmZRvaneli Tsu asocirebuli profesori fiz.-maT. mec. doqtori Teimuraz axobaZe Tbilisi 2012 avtoris stili daculia
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ivane javaxiSvilis saxelobis Tbilisis saxelmwifo universiteti
zust da sabunebismetyvelo mecnierebaTa fakulteti
maTematikis departamenti
Salva zviadaZe
furies SeuRlebuli trigonometriuli mwkrivebis zogierTi Tvisebis Sesaxeb
sadoqtoro disertacia
samecniero xelmZRvaneli Tsu asocirebuli profesori fiz.-maT. mec. doqtori Teimuraz axobaZe
Tbilisi 2012
avtoris stili daculia
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2
s a r C e v i
Sesavali . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Tavi I
erTi cvladis funqciebis furies SeuRlebuli trigonometriuli mwkrivebi
1.1. aucilebeli aRniSvnebi, gansazRvrebebi da ZiriTadi Teoremebi erTi cvladis perioduli funqciebisaTvis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2. aucilebeli aRniSvnebi, gansazRvrebebi da ZiriTadi Teoremebi erTi cvladis lokalurad jamebadi funqciebisaTvis . . . . . . . . . . . . . . . . . . . . 8 1.3. perioduli funqciebis furies SeuRlebuli trigonometriuli mwkriveb-is yofaqcevis Sesaxeb ZiriTadi Teoremebis damtkiceba . . . . . . . . . . . . . . 11 1.4. lokalurad jamebadi funqciebis furies SeuRlebuli trigonometriu-li mwkrivebis yofaqcevis Sesaxeb ZiriTadi Teoremebis damtkiceba . . . . . 20
Tavi II
ori cvladis funqciebis furies SeuRlebuli trigonometriuli mwkrivebi
2.1. aucilebeli aRniSvnebi, gansazRvrebebi da ZiriTadi Teoremebi ori cvladis perioduli funqciebisaTvis . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.2. ori cvladis perioduli funqciebis furies SeuRlebuli trigonometr-iuli mwkrivebis yofaqcevis Sesaxeb ZiriTadi Teoremebis damtkiceba . . . 57 literatura . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
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3
Sesavali funqciaTa Teoriis bevri ZiriTadi cneba da Sedegi miRebul iqna
aproqsimaciis TeoriasTan dakavSirebiT. am Teoriis ganviTarebam warmoaCina misi mWidro kavSiri funqciaTa Teoriisa da funqcionaluri analizis bevr problemasTan. aproqsimaciis Teoriis mniSvelovani mimarTulebaa funqciaTa mwkrivebis Teoria. funqciaTa mwkrivebis TeoriaSi erT-erTi ZiriTadi problemaa funqciisaTvis im minimaluri pirobebis dadgena, rac uzrunvelyofs am funqciis warmodgenas funqciaTa mwkriviT. funqciaTa mwkriviT funqciis warmodgena niSnavs mis krebadobas an romelime meTodiT Sejamebadobas mocemuli funqciiisken.
funqciaTa Teoriis, da kerZod, klasikuri harmoniuli analizis amocanebisadmi interesi gaZlierda mas Semdeg, rac maTematikis gamoyenebebisaTvis friad aqtualuri sakiTxebis gadaWra klasikur harmoniul analizs daeyrdno.
axlandel droSi jeradi trigonometriuli da orTogonaluri mwkrivebis Teoria swrafad viTardeba. is gadmocemulia mraval monografiasa da naSromSi, romelTa CamoTvla Sors wagviyvans.
f. lukaCma [1] daamtkica, rom lebegis azriT integrebadi 2 -perioduli funqciis furies SeuRlebuli trigonometriuli mwkrivis kerZo jamebis mimdevroba funqciis pirveli gvaris wyvetis wertilebSi ganSladia logariTmis rigiT.
r. riadma [2] ganixila lukaCis Teoremis analogi SeuRlebuli uolSis mwkrivebisaTvis.
f. morisma ([3], [4]) ganazogada lukaCis es debuleba abel _ puasonis saSuloebisaTvis, kerZod, man aCvena, rom lebegis azriT integrebadi, 2 -perioduli funqciis furies SeuRlebuli trigonometriuli mwkrivis abel _ puasonis saSualoebi funqciis pirveli gvaris wyvetis wertilebSi kvlav ganSladia logariTmis rigiT. agreTve man ganixila Sesabamisi analogi formalurad gawarmoebuli abelis saSualoebisTvis.
m. pinskim [5] Seiswavla anlogiuri sakiTxebi RerZze integrebadi da arsebiTad SemosazRvruli funqciebisaTvis; agreTve, ganixila aRniSnuli Teoremis analogebi garkveuli tipis kenti gulebisTvis.
Cven ([6], [7]) mier ganxilul iqna lukaCis aRniSnuli Teoremis analogi
T. axobaZis ([7]-[10]) mier SemoRebuli Cezaros ganzogadebuli ( , )nC -
saSualoebisTvis da dadebiTi regularuli matriculi SejamebadobisTvis. naCvenebi iqna, rom lebegis azriT integrebadi, 2 -perioduli funqciis
furies SeuRlebuli trigonometriuli mwkrivis ( , )nC saSualoebisTvis
ganSladobis logariTmuli rigi SenarCunebulia, xolo am funqciis furies SeuRlebuli trigonometriuli mwkrivis wrfivi saSualoebisTvis Sedegi arsebiTad damokidebulia gasaSualoebis matricze.
p. Joum da s. Joum [11] daamtkices lukaCis Teoremis analogi logariTmuli rigis mqone kenti gulebisTvis.
i. danSengma, p. Joum da s. Joum [12] ganixiles analogiuri debulebebi maRali rigis formaluri warmoebulebisTvis da SeuRlebuli puasonis gulis tipis kenti gulebisaTvis.
r. taberskim ([13], [14]) ganixila lokalurad integrebadi funqciebi, rolebic akmayofileben Semdeg pirobebs:
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4
1lim ( ) 0
L C
LL
f x dxL
,
1lim ( ) 0
L
LL C
f x dxL
.
man gansazRvra aseTi funqciebisaTvis furies koeficientebi da Sesabamisi furies trigonometriuli mwkrivi. manve Seiswavla am funqciebis trigonometriuli, SeuRlebuli trigonometriuli mwkrivebisa da maTi kerZo jamebis ( ,1)C -saSualoebis yofaqcevis zogierTi sakiTxi.
Cven [15] mier Seswavlil iqna lukaCis aRniSnuli Teoremis analogi taberskis mier ganxiluli funqciebisa da mwkrivebisaTvis; agreTve, - am
Sedegis analogi Cezaros ganzogadebuli ( , )nC -saSualosTvis da dadebiTi
regularuli matriculi SejamebadobisTvis. f. morisma ([16], [17]) ganazogada lukaCis aRniSnuli debuleba ori
cvladis funqciebisaTvis, kerZod, man aCvena, rom lebegis azriT integrebadi, cal-calke cvladebis mimarT 2 -perioduli funqciis, furies SeuRlebuli orjeradi trigonometriuli mwkrivi, garkveul pirobebSi kvlav ganSladia logariTmebis namravlis rigiT; manve ganazogada aRniSnuli debuleba abel-puasonis saSualoebisTvis da ganixila SeuRlebuli orjeradi trigonometriuli mwkrivis apel_puasonis saSualos Sefaseba funqciis meore rigis Sereuli kerZowarmoebulis sigluvis wertilebSi.
i. danSengma, p. Joum da s. Joum [18] ganixiles morisis debulebebis analogebi maRali rigis Sereuli kerZo warmoebulebisTvis da SeuRlebuli puasonis gulis tipis kenti gulebisaTvis.
Cven [19] mier ganzogadebuli iqna f. morisis aRniSnuli Sedegi;
agreTve miRebulia am Sedegis analogi Cezaros ganzogadebuli ( , , )n nC -
saSualosTvis da dadebiTi regularuli matriculi SejamebadobisTvis. f. morisma da v. veidma [20] ganixiles aRniSnuli Teoremis analogi
orjeradi furie – uolSis mwkrivebisaTvis.
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5
Tavi I
erTi cvladis funqciebis furies SeuRlebuli trigonometriuli mwkrivebi
1.1. aucilebeli aRniSvnebi, gansazRvrebebi da ZiriTadi Teoremebi erTi cvladis perioduli funqciebisaTvis
vTqvaT f aris 2 perioduli, namdvili cvladis namdvili funqcia da
periodze lebegis azriT integrebadi. am funqciis furies trigonometriul mwkrivs aqvs Semdegi saxe:
(1.1) 0
1
( cos sin )2
k k
k
aa kx b kx
,
sadac
1( ) cos
ka f x kxdx
da 1
( )sink
b f x kxdx
,
aris f funqciis furies koeficientebi. (1.1) mwkrivis SeuRlebuli mwkrivi ase moicema:
(1.2) 1
( sin cos )k k
k
a kx b kx
.
( ; )n
S f x -iT aRvniSnoT (1.2) mwkrivis n -uri kerZo jami.
lukaCma daamtkica Semdegi Teorema. Teorema A. Tu [ ; ]f L da raime x wertilSi arsebobs sasruli
zRvari
(1.3) 0
( ) lim[ ( ) ( )]x
td f f x t f x t
,
maSin
( ; ) ( )lim
ln
n x
n
S f x d f
n .
vTqvaT n aris raime mimdevroba [0; ]b intervalidan, sadac b aris
sasruli ricxvi. SeuRlebuli trigonometriuli mwkrivis (1.2) kerZo jamis Cezaros ganzogadebul saSualos eqneba Semdegi saxe:
(1.4) 1
0
1 1( ; ) ( ; ) ( ) ( )n n n
n
n
n n k k n
kn
t f x A S f x f x t t dtA
,
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6
sadac
1
0
1( ) ( )n n
n
n
n n k k
kn
t A D tA
da
(1.5) (1 )(2 ) ... ( )
!n n n n
n
nA
n
.
Tu ( )n
aris mudmivi mimdevroba, anu n
, n , maSin es saSualo
daemTxeva Cezaros ( , )C saSualos. davuSvaT
( , ) ( ) ( ) ( )x
x t f x t f x t d f ,
sadac ( )x
d f raime ricxvia damokidebuli funqciaze da wertilze.
samarTliania Semdegi: Teorema 1.1.1. vTqvaT [ ; ]f L da raime x wertilisTvis moiZebneba
iseTi ( )x
d f ricxvi, rom sruldeba toloba:
(1.6) 0
0
1lim | ( , ) | 0.
h
hx t dt
h
maSin
(1.7) ( ; ) ( )
limln
n
n x
n
t f x d f
n
.
daisva sakiTxi: arsebobs, Tu ara am debulebis analogi ufro zogadi
wrfivi saSualoebisaTvis. vTqvaT ( )q n aris mimdevroba, romelis mniSvnelobebi naturaluri
ricxvebia, amasTan 2 ( )q n , n da lim ( )n
q n
. aviRoT gasaSualoebis
matrici ( )nka ise, rom Tu ( )k q n , maSin 0nka .
furies SeuRlebuli trigonometriuli mwkrivis kerZo jamebis wrfiv saSualos aRniSnuli matricis mimarT eqneba Semdegi saxe:
( ) ( )
0 0
1( ; ) ( ; ) ( ) ( )
q n q n
n nk k nk k
k k
f x a S f x f x t a D t dt
.
matrics ewodeba regularuli Tu sruldeba Semdegi sami piroba
(I) l 0im nkn
a
, {1,2,...}k ;
(II) nN -SemosazRvrulia, sadac 1
| |n nk
k
N a
;
(III) l 1im nn
A
, sadac 1
n nk
k
A a
.
vTqvaT mocemulia raime s sasruli zRvrisken krebadi ( )ks mimdevroba,
maSin ( )ks mimdevrobis regularuli matricis mimarT k saSualoc
krebadia igive s zRvrisken. aRsaniSnavia, rom moyvanili sami piroba aris aucilebeli da sakmarisi (ix. [21] Tavi III Teorema (1.2)).
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7
samarTliania Semdegi: Teorema 1.1.2. vTqvaT [ ; ]f L , maSin yovel x wertilSi, romelSic
sruldeba (1.6) gvaqvs:
a) Tu ( ) 0x
d f , maSin, nebismieri dadebiTi regularuli matricisTvis
( ; )lim 1
( ) ln ( )
n
nx
f x
d f q n
;
b) nebismieri [0;1] -Tvis moiZebneba iseTi regularuli dadebiTi matrici, rom
( ; ) ( )l
ln ( )im n x
n
f x d f
q n
.
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8
1.2. aucilebeli aRniSvnebi, gansazRvrebebi da ZiriTadi Teoremebi erTi cvladis lokalurad jamebadi funqciebisaTvis
Cven interess warmoadgens ganvixiloT aRniSnuli Sedegebis analogebi
r. taberskis mier ganxiluli mwkrivebisTvis; agreTve am Sedegebis analogi abelis saSualosaTvis, romelic perioduli funqciebisaTvis mocemuli hqonda f. moriss (ix. [3], [4]).
SemoviRoT aRniSvnebi: vTqvaT :f , f E . es niSnavs, rom nebismieri
dadebiTi fiqsirebuli C -Tvis, L -is mimarT sruldeba
(1.8) 1
( ) (1)
L C
L
f x dx OL
, 1
( ) (1)
L
L C
f x dx OL
, L .
cxadia, nebismieri perioduli funqcia akmayofilebs ukanasknel Sefasebebs.
ganvixiloT mwkrivi /
( )i kx L
k
f k e
,
sadac
/1( ) ( )
L
i ku L
L
f k f u e duL
.
mocemuli mwkrivis SeuRlebuli mwkrivis kerZo jams eqneba saxe
(1.9) /1
( ; ) ( sig ) ( ) ( ) ( )
L
L i kx L L
n n
k n L
S f x i k f k e f t D t x dtL
,
sadac
(1.10) 1
( ) sin( / )n
L
n
k
D t kt L
.
davuSvaT ( ) nL n L namdvil ricxvTa iseTi mimdevrobaa, rom lim nn
L
.
samarTliania Semdegi: Teorema 1.2.1. ( )i vTqvaT f E , maSin yovel x wertilSi,
romlisTvisac sruldeba (1.6) da
(1.11) 1
( , )(ln )
nL x tdt o n
t
, n ,
adgili eqneba tolobas
(1.12) ( ; ) ( )
limln
nL
n x
n
S f x d f
n .
( )ii arsebobs nL mimdevroba da perioduli f funqcia, iseTi, rom
adgili aqvs Sefasebas
(1.13) 1
( , )(ln )
nL x tdt O n
t
,
magram ar sruldeba (1.12).
iseve rogorc wina paragrafSi, daisva sakiTxi Teorema 1.2.1-is analogis arsebobis Sesaxeb Cezaros ganzogadebuli saSualoebisTvis da dadebiTi regularuli wrfivi saSualoebisaTvis.
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9
(1.9) kerZo jamebis Cezaros ganzogadebul saSualos eqneba saxe
1
,
0
1( ; ) ( ; )n
n n
nL L
n n k k
kn
t f x A S f xA
,
1( ) ( )
n
L
L
n
L
f t t x dtL
,
sadac
1
,
0
1( ; ) ( )n
n n
nL L
n n k k
kn
f x A D tA
,
xolo [0; ]n
b yoveli n -Tvis, b sasruli ricxvia da nn
A gasazRvrulia
(1.5)-iT.
samarTliania Semdegi: Teorema 1.2.2. ( )i vTqvaT f E , maSin yovel x wertilSi, romelSic
sruldeba (1.6) da (1.11) gvaqvs
(1.14) ,
( ; ) ( )lim
ln
n
n
L
n x
n
t f x d f
n
.
( )ii arsebobs nL mimdevroba da perioduli f funqcia, rom sruldeba
(1.13), magram ar arsebobs n mimdevroba, romlisTvisac Sesruldeba (1.14).
Teorema 1.2.1-is analogi matriculi saSualoebisaTvis Camoyalibdeba
Semdegnairad. vTqvaT ( )q n naturalur ricxvTa araklebadi mimdevroba,
amasTan 2 ( )q n , n da lim ( )n
q n
. aviRoT gasaSualoebis matrici ( )nka
ise, rom Tu ( )k q n , maSin 0nka .
(1.9) kerZo jamebis wrfiv saSualos aRniSnuli matricis mimarT aqvs saxe ( ) ( )
0 0
1( ; ) ( ; ) ( ) ( )
Lq n q nL L L
n nk k nk k
k kL
f x a S f x f t a D t x dtL
.
samarTliania Semdegi: Teorema 1.2.3. ( )i vTqvaT f E , maSin yovel x wertilSi, romelSic
sruldeba (1.6) da
(1.15) 1
( , )(ln ( ))
nL x tdt o q n
t
,
adgili aqvs Semdegs:
a) Tu ( ) 0xd f , maSin nebismieri dadebiTi regularuli matricisTvis gvaqvs
(1.16) ( ; )
lim 1( ) ln ( )
nL
n
nx
f x
d f q n
;
b) [0;1] -Tvis moiZebneba iseTi dadebiTi, regularuli matrici, rom
(1.17) ( ; ) ( )
lln ( )
imnL
n x
n
f x d f
q n
.
( )ii arsebobs nL , ( )q n mimdevrobebi da perioduli f funqcia iseTi, rom
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10
(1.18) 1
( , )(ln ( ))
nL x tdt O q n
t
.
magram rogoric ar unda iyos [0;1] da misiSesabamisi dadebiTi, regularuli matrici ar Sersuldeba (1.17).
bunebrivia daisva sakiTxi Teorema 1.2.1.-is analogis arsebobis Sesaxeb abel-puasonis saSualoebisaTvis. SemoviRoT aRniSvnebi.
taberskis mier ganxiluli SeuRlebuli trigonometriuli mwkrivebis abel-pusonis saSualos eqneba saxe:
(1.19) / | |1
( , ) ( sig ) ( ) ( ) ( , )
L
L i kx L k
L
k L
f r x i k f k e r f t Q r t x dtL
,
sadac
(1.20) ( , ) sin( / )kL
k
Q r t r kt L
.
vTqvaT mocemulia funqcia :[0;1)L iseTi, rom
(1.21) 1
lim ( )r
L r
.
samarTliania Semdegi
Teorema 1.2.4. ( )i vTqvaT f E , maSin yovel x wertilSi sruldeba (1.6) da
(1.22) ( )
1
( , )(ln(1 ))
L rx t
dt o rt
, 1r .
adgili aqvs tolobas
(1.23) ( )
1
( )( , )lim
ln(1 )
L r
x
r
d ff r x
r
.
( )ii arsebobs perioduli f funqcia da funqcia L iseTi, rom
(1.24) ( )
1
( , )(ln(1 ))
L rx t
dt O rt
, 1r ,
magram ar sruldeba (1.23).
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11
1.3. perioduli funqciebis furies SeuRlebuli trigonome-triuli mwkrivebis yofaqcevis Sesaxeb ZiriTadi Teoremebis damtkiceba
Teorema 1.1.1-is damtkiceba
ganvixiloT ( ; )nn
t f x . vinaidan ( )n
nt
da ( ) ( )f x t f x t t -s mimarT kenti
fuqnciebia, integralis adiciurobidan gveqneba:
1( ; ) [ ( ) ( )] ( )n n
n nt f x f x t f x t t dt
0
1( ) ( ) ( ) ( )n
x nf x t f x t d f t dt
1 2
0
( )( ) ( ) ( )nx
n
d ft dt A n A n
.
ganvixiloT 1( )A n , (1.6)-is ZaliT, nebismieri dadebiTi ricxvisTvis
moiZebneba iseTi dadebiTi ( ) ricxvi, rom
0
1| ( , ) |x t du
.
aviRoT n imdenad didi rom 1/ n ,
1
0
1( ) ( , ) ( )n
nA n x t t dt
0 1/
1/1 1
( , ) ( ) ( , ) ( )n nn n
n
n
x t t dt x t t dt
1 2 3
1( , ) ( )n
nx t t dt B B B
.
(1.5) gansazRvrebidan advili dasanaxia, rom 1nk
A ricxvebi dadebiTia, roca
0n
, amasTan Tu gaviTvaliswinebT faqts, rom ( )k
D t k (ix. [21], (5.11)),
gveqneba
1
1
00
1/1 1
| | | ( , ) | | ( ) |nn
n
n k k
kn
n
B x t A D t dtA
0
1/
| ( , ) |
nn
x t dt
.
(ix. [21], (5.11)), ( ) 2 /k
D t t . nawilobiTi integrebiT gveqneba:
1
2
01/
1 1| | | ( , ) | | ( ) |n
n
n
n k k
knn
B x t A D t dtA
1/ 1/ 0
2 1 2 1( , ) ( , )
t
n n
x t dt d x u dut t
2
0 1/ 01/
2 1 2 1( , ) ( , )
t t
nn
x u du x u dut t
-
12
1/
0 0
2 2( , ) ( , )
nn
x u du x u du
1/
12 2 ln
n
dt nt
,
amasTan f -is integrebadobidan miviRebT
1
3
0
1 1| | | ( , ) | | ( ) |n
n
n
n k k
kn
B x t A D t dtA
1 22 ( , ) ( , )x t dt x t dt
t
0
2( , ) (1)x t dt O
,
amrigad,
(1.25) 1( )
lim 0lnn
A n
n .
SevafasoT 2 ( )A n . n mimdevroba davyoT or qvemimdevrobad Semdegnairad:
[0;1)im
da [1; ]ik
b , i . 1 2 1 2{ , ,...} { , ,...}m m k k , 1 2 1 2{ , ,...} { , ,...}m m k k .
jer ganvixiloT im
qvemimdevroba:
1
00 0
1( ) ( )m mi i
mi
i
i i k k
ki
t dt A D t dtA
1 1
0 00
1 1( )m mi i
m mi i
i i
i k k i k k
k ki i
A D t dt A UA A
.
gvaqvs
10 0
( ) sinn
n
i
D t dt itdt
1 10
cos cos cos 0n n
i i
it i
i i i
1 1
1 ( 1) 1 12 1 ...
3 2 1
in n
i ii i s
,
sadac [( 1) / 2]s n . cxadia,
1 12 1 ...
3 2 1s
1 1 1 1 1 1 12 1 ... 1 ...
2 3 2 1 2 2 3s s
12 ln ln ln
2s s s
.
advili dasanaxia, rom ln lns n , amitom nebismieri 0 arsebobs iseTi ( )N N , rom yoveli k N , gvaqvs
(1.26) 1 1ln
kU
k .
ganvixiloT
-
13
/1 1
0 0
1 1 1 1
ln ln
m mi i
m mi i
i Mi
i k k i k k
k ki i
A U A Ui iA A
1
1 2
/ 1
1 1
ln
mi
mi
i
i k k
k i Mi
A U D Di A
.
ukanaskneli gamosaxuleba SevafasoT qvemodan. (1.26)-is ZaliT gvaqvs
1
2
/ 1
1 1
ln
mi
mi
i
i k k
k i Mi
D A Ui A
1
/ 1
1 1(1 ) ln
ln
mi
mi
i
i k
k i Mi
A ki A
1
/ 1
(1 )(ln ln )
ln
mi
mi
i
i k
k i Mi
i MA
i A
/1 1
0 0
(1 )(ln ln )
ln
m mi i
mi
i Mi
i k i k
k ki
i MA A
i A
/1
0
ln ln 1(1 ) 1 (1 ) 1
ln ln
mi
mi
i M
i k
ki
M MA
i i A
1 2E E .
Tu 0 1im
, maSin 1 1 0im
. aqedan gamomdinareobs 1mi
i kA
-s klebadoba.
amgvarad, gveqneba /
1
2
0
ln 1(1 ) 1
ln
mi
mi
i M
i k
ki
ME A
i A
/1
/
0
ln 1(1 ) 1
ln
mi
mi
i M
i i M
ki
MA
i A
.
Tu ( )M M -s imdenad dids aviRebT, rom 1/ M da miiA
ricxvebis
Sefasebebs gaviTvaliswinebT (ix. [9] lema 2), miviRebT
/1
/
0
ln 1(1 ) 1
ln
mi
mi
i M
i i M
ki
MA
i A
1 1
(1)/
m mi i
i i
m mi i
m mi ii i
O O oM Mii i M
.
anu
1
ln(1 ) 1
ln
ME
i
da
2(1)E o .
amis garda, (1.26) ZaliT da miiA
mimdevrobis klebadobis gamo i -s mimarT
samarTliania 1D -is Semdegi Sefaseba: /
1 1
1
0 1
1 1
ln ln
m mi i
m mi i
i MN
i k k i k k
k k Ni i
D A U A Ui A i A
-
14
/1 11
0 1
( ) 1(1 ) ln
ln ln
m mi i
m mi i
i MN
i k i k
k k Ni i
C NA A k
i A i A
/1 11
0 1
( ) (1 ) ln
ln ln
m mi i
m mi i
i MN
i k i k
k k Ni i
C N NA A
i A i A
/1 11
0 1
( ) (1 ) ln
ln ln
m mi i
m mi i
i MN
i i
k k Ni i
C N NA A
i A i A
1 1ln ln
m mi i
i i
m mi i
m mi i i
O O NMi i i i
1
( )1
ln ln
mi
i
mi
mi MN i
O Oi i i M i
1 1
ln lnO O
i i i
,
sadac
10
( ) mink
k iC N U
.
maSasadame,
1 2
0
1lim ( ) lim ( )
ln
mi
ii i
t dt D Di
1 1 2 1lim ( ) lim 1i i
D E E E
.
2( )A n wevrSi mocemuli integrali SevafasoT zemodan. (1.26)-is ZaliT gvaqvs
1
00 0
1 1 1( ) ( )
ln ln
m mi i
mi
i
i i k k
ki
t dt A D t dti i A
1 1
0 00
1 1( )
ln ln
m mi i
m mi i
i i
i k k i k k
k ki i
A D t dt A Ui A i A
1 1
0 1
1 1
ln ln
m mi i
m mi i
N i
i k k i k k
k k Ni i
A U A Ui A i A
1 1
0 1
1 1(1 ) ln
ln ln
m mi i
m mi i
N i
i k k i k
k k Ni i
A U A ki A i A
1 12
0 1
( ) (1 ) ln
ln ln
m mi i
m mi i
N i
i k i k
k k Ni i
C N iA A
i A i A
.
mi
iA
mimdevrobis i -s mimarT klebadobis gamo da miiA
ricxvebis
warmodgenidan (ix. [9] lema 2) miviRebT:
1 12
0 1
( ) (1 ) ln
ln ln
m mi i
m mi i
N i
i k i k
k k Ni i
C N iA A
i A i A
1 1 12
0 0 0
( ) (1 )
ln
m m mi i i
m mi i
N i N
i N i k i k
k k ki i
C NA A A
i A A
1 12
0 0
( ) (1 )(1 )
ln
m mi i
m mi i
N N
i N i
k ki i
C NA A
i A A
-
15
1 1( )
1ln ln
m mi i
i i
m mi i
m mi N i
O Oi i i i
1 11 1
( ) ln ln
miNO O
i i N i i i
11
lnO
i i
,
sadac
20
( ) maxk
k NC N U
,
e. i.
0
1lim ( ) 1
ln
mi
ii
t dti
.
sabolood miviRebT:
0
1lim ( ) 1
ln
mi
ii
t dti
.
ganvixiloT ik
qvemimdevroba. gvaqvs
/1 1
0 0
1 1 1 1
ln ln
k ki i
k ii
i Mi
i j j i j j
j jii
A U A Ui i AA
1
1 2
/ 1
1 1
ln
ki
ki
i
i j j
j i Mi
A U F Fi A
.
Tu 1ik
b , 1ki
i jA
klebadia j -s mimarT, maSin (1.26) ZaliT gveqneba:
1
2
/ 1
1 1
ln
ki
ki
i
i j j
j i Mi
F A Ui A
1
/ 1
1(1 ) ln
ln
ki
ki
i
i j
j i Mi
A ji A
1
/ 1
(1 )(ln ln )
ln
ki
ki
i
i j
j i Mi
i MA
i A
/1 1
0 0
(1 )(ln ln )
ln
k ki i
ki
i Mi
i j i j
j ji
i MA A
i A
/1
1 2
0
ln 1(1 ) 1 1
ln
ki
ki
i M
i j
ji
MA G G
i A
.
radgan 1ki
iA
zrdadia rogorc i -s funqcia, amitom
1
ln(1 ) 1
ln
MG
i
,
xolo /
1
2
0
ln 1(1 ) 1
ln
ki
ki
i M
i j
ki
MG A
i A
-
16
/1
0
ln 1(1 ) 1
ln
ki
ki
i M
i
ki
MA
i A
1(1)
ki
i
ki
ki i
O oMi
.
(1.26)-ZaliT da 1ki
iA
zrdadobidan gvaqvs: /
1 1
1
0 1
1 1 1 1
ln ln
k ki i
k ki i
i MN
i j j i j j
j j Ni i
F A U A Ui iA A
/1 11
0 1
( ) 1(1 ) ln
ln ln
k ki i
k ki i
i MN
i j i j
j j Ni i
C NA A j
i A i A
/1 11
0 1
( ) (1 ) ln
ln ln
k ki i
k ki i
i MN
i j i j
j j Ni i
C N NA A
i A i A
/1 11
/
0 1
( ) (1 ) ln
ln ln
k ki i
k ki i
i MN
i N i i M
k k Ni i
C N NA A
i A i A
1 1
ln ln /
k ki i
i i
k ki i
k ki i i
O O NMi N i i i i M
( )1
ln ln
i
ki
ki MN
O Oi i i M i
1 1
ln lnO O
i i i
.
aqedan davaskvniT, rom
1 2
0
1lim ( ) lim ( )
ln
ki
ii i
t dt F Fi
1 1 2 1lim ( ) lim 1i i
F G G G
.
axla SevafasoT igive gamosaxuleba zemodan:
1
00 0
1 1 1( ) ( )
ln ln
k ki i
ki
i
i i j j
ji
t dt A D t dti i A
1 1
0 00
1 1( )
ln ln
k ki i
k ki i
i i
i j j i j j
j ji i
A D t dt A Ui A i A
1 1
0 1
1 1
ln ln
k ki i
k ki i
N i
i j j i j j
j j Ni i
A U A Ui A i A
1 12
0 1
( ) 1ln
ln ln
k ki i
k ki i
N i
i i j
j j Ni i
C NA A j
i A i A
1 1 12
0 0 1
( ) (1 ) ln
ln ln
k k ki i i
k ki i
N i N
i i j i j
k j j Ni i
C N iA A A
i A i A
1 12
0 0
( ) (1 )1
ln
k ki i
k ki i
N N
i i N
j ji i
C NA A
i A A
-
17
1 11
ln ( )
k ki i
k ki i
i iO O
i i i N
1 11
ln lnO O
i i i
;
e. i.
0
1lim ( ) 1
ln
ki
ii
t dti
,
amgvarad,
0
1lim ( ) 1
ln
ki
ii
t dti
.
sabolood gvaqvs
2( )( )
limln
x
n
d fA n
n ,
bolo toloba (1.25)-SefasebasTan erTad amtkicebs Teorema 1.1.1-s.
Teorema 1.1.2.-is damtkiceba
ganvixiloT ( ; )n f x , integralSi cvladis SecvliT da ( )kD t -kentobidan
gvaqvs ( )
0
1( ; ) ( ) ( )
q n
n nk k
k
f x f x t a D t dt
=( )
00
1( ( ) ( )) ( )
q n
nk k
k
f x t f x t a D t dt
( )
00
1( ( ) ( ) ( )) ( )
q n
x nk k
k
f x t f x t d f a D t dt
( )
1 2
00
( )( ) ( ) ( )
q n
x
nk k
k
d fa D t dt A n A n
.
jer davamtkicoT Teoremis a) nawili. (1.6)-is ZaliT, nebismieri dadebiTi 0 ricxvisTvis moiZebneba iseTi dadebiTi ( ) 0 ricxvi, rom
0
1| ( , ) |x t du
.
maSin vinaidan lim ( )n
q n
aviRoT n imdenad didi, rom 1/ ( )q n . gvaqvs
( )
1
00
1( ) ( , ) ( )
q n
nk k
k
A n x t a D t dt
( )
00
1/ ( )1
( , ) ( )q n
nk k
k
q n
x t a D t dt
( )
01/ ( )
1( , ) ( )
q n
nk k
kq n
x t a D t dt
-
18
( )
1 2 3
0
1( , ) ( )
q n
nk k
k
x t a D t dt B B B
.
radgan ( )k
D t k (ix. [21], (5.11)), amitom
1
0
1/ ( )( )
( , ) nn
q nAq n
B A x t dt
,
sadac 1
n nk
k
A a
da l 1im nn
A
.
radgan ( ) 2 /k
D t t (ix. [21], (5.11)) amitom nawilobiTi integrebiT gveqneba
2
1/ ( ) 1/ ( ) 0
2 21 1( , ) ( , )
t
n n
q n q n
A AB x t dt d x u du
t t
2
0 1/ ( ) 01/ ( )
2 21 1( , ) ( , )
t t
n n
q nq n
A Ax u du x u du
t t
1/ ( )
0 0
2 2 ( )( , ) ( , )
q n
n nA A q n
x u du x u du
1/ ( )
12 2 ln ( )
n n
q n
A dt A q nt
,
amasTan, funqciis integrebadobidan gamomdinareobs
3
12 ( , )
nB A x t dt
t
0
2 2( , ) ( , ) (1)n n
A Ax t dt x t dt O
.
amrigad,
(1.27) 1( )
lim 0ln ( )n
A n
q n .
ganvixiloT ( )
2
00
( )( ) ( )
q n
x
nk k
k
d fA n a D t dt
( ) ( )
0 00
( ) ( )( )
q n q n
x x
nk k nk k
k k
d f d fa D t dt a U
.
Semdeg (1.26)-is ZaliT gveqneba ( ) ( )
0 0 1
q n q nN
nk k nk k nk k
k k k N
a U a U a U
( )
2
0 1
( ) (1 ) ln ( )q nN
nk nk
k k N
C N a q n a
2( ) (1 ) ln ( )
n nC N A A q n .
miviRebT
2( )
lim 1( ) ln ( )n
x
A n
d f q n
,
-
19
ukanaskneli, (1.27) da ( ; )n
f x -is warmodgena amtkicebs Teorema 1.1.2-is a)
nawils. b) ganvixiloT nebismieri [0;1] da avagoT ( )n mimdevroba ise, rom
roca n , maSin ( ) ( )nq n . ( )nka matrici ganvsazRvroT Semdegnairad ( )
( )
1, [ ( )],
0, [ ( )].
n
nk n
k q na
k q n
Tu
Tu
cxadia, ( )nka regularuli matricia.
vinaidan ( ) ( )nq n , amitom (1.26) ZaliT gvaqvs
( )( )( )
( )
0
(1 ) ln ( )1
ln ( ) ln ( ) ln ( )
nnq n
q n
nk k
k
U q na U
q n q n q n
( )(1 ) ln ( )
(1 ) ( )ln ( )
nq n
nq n
,
e. i. ( )
00
1lim ( )
ln ( )
q n
nk kn
k
a D t dtq n
.
meore mxriv,
( )( )( )
( )
0
(1 ) ln ( )1
ln ( ) ln ( ) ln ( )
nnq n
q n
nk k
k
U q na U
q n q n q n
( )(1 ) ln( ( ) / 2) (1 ) ln 2
(1 ) ( )ln ( ) ln ( )
nq n
nq n q n
.
vinaidan lim ( )n
q n
, amitom
( )
00
1lim ( )
ln ( )
q n
nk kn k
a D t dtq n
;
e. i. ( )
00
1lim ( )
ln ( )
q n
nk kn
k
a D t dtq n
.
sabolood ukanaskneli da (1.27) Sefasebebidan, ( )
( ; )q n
f x -is da 2 ( )A n -is
warmodgenidan gvaqvs
( ) 1 2( ; ) ( )( ) ( )
l lln ( ) ln ( )
im imq n x
n n
f x d fA n A n
q n q n
.
Teorema 1.1.2 damtkicebulia.
-
20
1.4. lokalurad jamebadi funqciebis furies SeuRlebuli trigonometriuli mwkrivebis yofaqcevis Sesaxeb ZiriTadi Teoremebis damtkiceba
Teorema 1.2.1-is damtkiceba.
ganvixiloT ( ; )nLn
S f x ,
1( ; ) ( ) ( )
n
n n
n
L
L L
n n
n L
S f x f t D t x dtL
integralSi cvladis SecvliT u t x miviRebT
1( ; ) ( ) ( )
n
n n
n
L x
L L
n n
n L x
S f x f x u D u duL
,
meores mxriv Tu cvlads SevcvliT u t x da gaviTvaliswinebT, rom ( )Lk
D t
kenti funqciaa t -s mimarT, miviRebT
1( ; ) ( ) ( )
n
n n
n
L x
L L
n n
n L x
S f x f x u D u duL
,
sabolood integralis adiciurobis ZaliT gveqneba
1( ; ) ( ) ( )
2
n
n n
n
L x
L L
n n
n L x
S f x f x u D u duL
1( ) ( )
2
n
n
n
L x
L
n
n L x
f x u D u duL
1( ( ) ( )) ( )
2
n
n
n
L x
L
n
n L x
f x u f x u D u duL
1( ) ( )
2
n
n
n
L x
L
n
n L x
f x u D u duL
(1.28) 1 2 3
1( ) ( )
2
n
n
n
L x
L
n
n L x
f x u D u du A A AL
.
ganvixiloT 1A . vinaidan ( ) ( )f x u f x u da ( )nL
nD u u -s mimarT kenti
funqciebia, amitom gveqneba
1
1( ( ) ( )) ( )
2
n
n
n
L x
L
n
n L x
A f x u f x u D u duL
0
1( ( ) ( )) ( )
n
n
L x
L
n
n
f x u f x u D u duL
0
1( ( ) ( )) ( )
n
n
L
L
n
n
f x u f x u D u duL
(1.29) 1 2
1( ( ) ( )) ( )
n
n
n
L
L
n
n L x
f x u f x u D u du B BL
.
ganvixiloT ( )nLn
D u . ( )n
D u -is warmodgenaSi (ix. [21], Tavi II, (5.6)) cvladis
SecvliT miviRebT Sefasebas ( )nLn
D u -Tvis:
-
21
1| ( ) |
sin( / 2 )nL
n
n
D uu L
.
ganvixiloT 2
B . advili dasanaxia, rom (1.8)-Si, integralSi cvladis
SecvliT da bolos miRebuli Sefasebis ZaliT davadgenT, rom
2
1| ( ) ( ) | | ( ) |
n
n
n
L
L
n
n L x
B f x u f x u D u duL
1| ( ) ( ) |
sin[ ( ) / 2 ]
n
n
L
n n n L x
f x u f x u duL L x L
1
| ( ) | | ( ) |sin[ ( ) / 2 ]
n
n
L
n n n L x
f x u f x u duL L x L
(1.30) 2
1| ( ) | | ( ) | (1)
sin[ ( ) / 2 ]
n n
n n
L x L x
n n n L L x
f t dt f t dt OL L x L
.
ganvixiloT 2A da 3A . maSin 2B -is msgavsad, (1.8)-isa da integralSi cvladis
Secvlis ZaliT miviRebT Sefasebebs:
2
1| | | ( ) |
2 sin[ ( ) / 2 ]
n
n
L x
n n n L x
A f x u duL L x L
(1.31) 2
1| ( ) | (1)
2 sin[ ( ) / 2 ]
n
n
L x
n n n L
f t dt OL L x L
,
da
3
1| | | ( ) |
2 sin[ ( ) / 2 ]
n
n
L x
n n n L x
A f x u duL L x L
(1.32) 2
1| ( ) | (1)
2 sin[ ( ) / 2 ]
n
n
L
n n n L x
f t dt OL L x L
.
ganvixiloT 1B ,
1
0
1( ( ) ( ) ( )) ( )
n
n
L
L
x n
n
B f x u f x u d f D u duL
1 2
0
( )( )
n
n
L
Lx
n
n
d fD u du C C
L .
SevafasoT 1C . (1.6)-is ZaliT nebismieri dadebiTi 0 ricxvisTvis,
moiZebneba dadebiTi ricxvi (sazogadod damokidebuli -ze) ( ) iseTi,
rom
(1.33) 0
1| ( , ) |x t du
.
aviRoT n imdenad didi, rom 1/ n , maSin miviRebT 1/
1
0 1/
1 1( , ) ( ) ( , ) ( )n n
n
L L
n n
n n n
C x u D u du x u D u duL L
(1.34) 1 2 3
1( , ) ( )
n
n
L
L
n
n
x u D u du D D DL
.
-
22
(1.10)-is ZaliT advilad davaskvniT, rom
(1.35) | ( ) |nLn
D u n .
amitom yoveli u -Tvis. (1.33)-is ZaliT gveqneba
(1.36) 1/
1
0
| | | ( , ) |
n
n n
nD x u du
L L
.
( ) 2 /k
D t t (ix. [21], (5.11)), amitom (1.10)-is gamo da cvladis SecvliT, advilad
davaskvniT, rom
(1.37) 2
| ( ) |nL n
n
LD u
u , 0 nu L .
maSasadame nawilobiTi integrebiT miviRebT Semdeg Sefasebas
2
1/ 1/
21 2 | ( , ) || | | ( , ) | n
n n n
L x uD x u du du
L u u
1/ 0 0 1/
2 1 2 1| ( , ) | | ( , ) |
u u
n n
d x t dt x t dtu u
2
1/ 0 0
2 1 2 1| ( , ) | | ( , ) |
u
n
x t dtdu x t dtu
1/
2
0 1/ 0
2 2 1| ( , ) | | ( , ) |
n u
n
nx t dt x t dtdu
u
(1.38) 1/
2 22 (2 ln ln1 ln )) (ln )
n
dun o n
u
.
amasTan, (1.11) da (1.37)-is ZaliT gveqneba
3
21 2 | ( , ) || | | ( , ) |
n nL L
n
n
L x uD x u du du
L u u
1 1
1
2 | ( , ) | 2 | ( , ) | 2| ( , ) |
nLx u x udu du x u du
u u
(1.39) 1
2 | ( , ) |(1) (ln )
nL x udu O o n
u
.
(1.36)-(1.39)-dan gamomdinareobs Semdegi Sefaseba
(1.40) 1lim 0lnn
C
n .
ganvixiloT 2C . maSin integralSi cvladis / nu t L SecvliT
10 0
( ) sin( / )n n
n
L LnL
n n
i
D t dt it L dt
1 1 00
cossin
n nn n
i i
L L ititdt
i
1 1
21 ( 1) 1 11 ...
3 2 1
in nn n
i i
L L
i i k
,
sadac [( 1) / 2]k n . gvaqvs
-
23
2 1 11 ...
3 2 1
nL
k
2 1 1 1 1 1 1 11 ... 1 ...
2 3 2 1 2 2 3
nL
k k
2 1ln ln ln
2
n nL L
k k k
.
vinaidan ln lnk n , miviReT, rom
(1.41) 0
( ) lnn
n
L
L n
n
LD t dt n
.
ukanasknelis ZaliT gveqneba
(1.42) 2( )
limln
x
n
d fC
n .
sabolood (1.28)-(1.32), (1.40) da (1.42) Sefasebebis ZaliT gvaqvs (1.12) rac amtkicebs Teoremis ( )i nawils.
( )ii ganvixiloT funqcia
(1.43) 0
2, [ 1;0];( )
2, (0;1).
xf x
x
Tu
Tu
es funqcia gavagrZeloT mTels RerZze periodiT 2.
nL mimdevroba avagoT Semdegnairad 4
nL n .
cxadia, rom 0
f funqcia 0-wertilSi ganicdis pirveli gvaris wyvetas da
0 0( ) 4d f . vinaidan
0f fuqcia 2-iT periodulia, (0, )t funqciac 2-iT
periodulia da
0, [0;1);(0, )
8, [1;2).
tt
t
Tu
Tu
aRniSnulidan gamomdinare advili saCvenebelia, rom sruldeba (1.6) piroba. sakmarisad mcire h -ebisTvis 0 1h gvaqvs
(1.44) 0 0
0 0
1 1lim | (0, ) | lim 0 0h h
h h
t dt dth h
.
vaCvenoT, rom amgvarad gansazRvruli nL mimdevrobisTvis da (1.43)
funqciisTvis sruldeba (1.13), vinaidan (0, )t periodulia periodiT 2.
amitom nawilobiTi integrebiT gveqneba:
1 1 0 0 1
(0, ) 1 1(0, ) (0, )
nn n
LL L t tt
dt d s ds s dst t t
1
2 2
1 0 0 0 1 0
1 1 1(0, ) (0, ) (0, ) (0, )
n n nL L Lt t
n
s dsdt s ds s ds s dsdtt L t
2 1[ / 2]
1 2 2 2[ / 2] 0
1(0, ) (0, ) (0, )
nn
n
LkL
kn k L
s ds s ds s dsL
2 2
1 0 2 0
21 1
(0, ) (0, )nLt t
s dsdt s dsdtt t
-
24
2[ / 2]
21 0 2 0
1 1(0, ) (1) (0, )
nnL tL
kn
s ds O s dsdtL t
2[ / 2]
212 2 2 2[ / 2]
1(1) (0, ) (0, )
nL k tt
k k t
O s ds s ds dtt
2[ / 2]
212 0
1(1) (0, ) (1)
n t
k
L
O s ds O dtt
2 2
2 2
1 1(1) 8 [ / 2] (1) 4 2[ / 2]
n nL L
O t dt O t dtt t
2
2
1(1) 4 ( ( 2[ / 2])
nL
O t t t dtt
2
2 2
1(1) 4 ( (1)) (1) 4
n nL L dtO t O dt O
t t
(1.45) 2
(1) 4 ln (1) 4 ln lnnL
nO t O L n .
ganvixiloT 0
( ;0)nL
nS f . SevniSnoT, rom aRniSnul SemTxvevaSi
samarTliania (1.28)-(1.38) warmodgenebi da Sefasebebi. Sesafasebeli rCeba 3
D
Sesakrebi (1.34) warmodgenidan.
3
1(0, ) ( )
n
n
L
L
n
n
D u D u duL
(1.46) 2
1 2
2
1 1(0, ) ( ) (0, ) ( )
n
n n
L
L L
n n
n n
u D u du u D u du E EL L
.
ganvixiloT 1E , (1.37) Sefasebis Tanaxmad
(1.47) 2 2
1
21 2| | | (0, ) | | (0, ) | (1)n
n
LE u du u du O
L u
.
SevafasoT 2E .
*
2
2
1(0, ) ( )
n
n
L
L
n
n
E u D u duL
(1.48) 1 2
2
1(0, ) sin( / )
2
nL
n
n
u nu L du F FL
,
sadac
* 1( ) ( ) sin( / )
2n nL L
n n nD u D u nu L .
kargadaa cnobili ([21], (5.2)), rom
* 1 1 cos( ) ( ) sin( )
2 2 tan( / 2)n n
nuD u D u nu
u
.
amitom (1.10)-is ZaliT da cvladis SecvliT, advilad davaskvniT, rom
(1.49) *1 cos( / )
( )2 tan( / 2 )
nL n
n
n
nu LD u
u L
.
SevafasoT 2F . (0, )u -is aradadebiTobis da periodulobis gamo gveqneba
-
25
(1.50) 2
2 2
1 1| | | (0, ) | 8 4
2 2
n nL L
n n
F u du duL L
.
vinaidan (0, )u aradadebiTia, ganvixiloT 1F . (1.49) warmodgenis ZaliT da
nawilobiTi integrebiT gvaqvs
1
2
1 cos( / )1(0, )
2 tan( / 2 )
nL
n
n n
nu LF u du
L u L
2
1 cos( / )1(0, ) cos( / 2 )
2 sin( / 2 )
nL
n
n
n n
nu Lu u L du
L u L
/ 2
2
1 cos( / )1(0, ) cos( / 2 )
2 sin( / 2 )
nL
n
n
n n
nu Lu u L du
L u L
/ 2
2
1 cos( / )2(0, )
4 sin( / 2 )
nL
n
n n
nu Lu du
L u L
/ 2
2
1 cos( / )2(0, )
2
nL
nnu L
u duu
/ 2
2 0
2 1(0, )(1 cos( / ))
2
nL u
nd s ns L ds
u
/ 2
0 2
2 1(0, )(1 cos( / ))
2
nLu
ns ns L ds
u
/ 2
2
2 0
2 1(0, )(1 cos( / ))
2
nL u
ns ns L dsdu
u
/ 2
0
2(0, )(1 cos( / ))
nL
n
n
s ns L dsL
2
0
2(0, )(1 cos( / ))
4n
s ns L ds
(1.51) / 2
1 2 32
2 0
2 1(0, )(1 cos( / ))
2
nL u
ns ns L dsdu G G G
u
.
ganvixiloT 1G . (0, )u -is aradadebiTobis, periodulobisa da martivi
utolobis ( |1 cos( / ) | 2nns L ) ZaliT davandgenT, rom 1 (1)G O . msgavsi
msjelobiT davaskvniT, rom 2 (1)G O .
SevafasoT 3G : / 2 / 2
3 2 2
2 0 2 0
2 1 2 1(0, )(1 cos( / )) (0, )
2 2
n nL Lu u
nG s ns L ds s dsdu
u u
(1.52) / 2
1 22
2 0
2 1(0, ) cos( / ))
2
nL u
ns ns L dsdu H H
u
.
adgili aqvs 2H Sesakrebis Semdeg warmodgenas / 2 2[ / 2]
2 212 2 2
2 1(0, ) cos( / ))
2
nL ku
n
k k
H s ns L dsduu
-
26
(1.53) / 2
1 22
2 2[ / 2]
2 1(0, ) cos( / ))
2
nL u
n
u
s ns L dsdu I Iu
.
ganvixiloT 2
I . gvaqvs
(1.54) / 2 / 2
2 2 2
2 2[ / 2] 2
2 1 4 2| | | (0, ) | (1)
2
n nL Lu
u
duI s dsdu O
u u
.
SevafasoT 1I . jamSi TiToeul SesakrebSi SevcvaloT cvladiT
2 2s t k da gamoviyenoT funqciis perioduloba periodiT 2. maSin / 2 2[ / 2]
1 212 0
2 1(0, ) cos( ( 2 2) / )
2
nL u
n
k
I t n t k L dtduu
.
radgan /n
n L , amitom Tanabrad k -s mimarT 2
0
(0, ) cos( ( 2 2) / ) 0n
t n t k L dt , n .
amitom, yoveli 0 -Tvis movZebniT iseT 0
n nomers, rom roca 0
n n da
yoveli k -Tvis 2
0
(0, ) cos( ( 2 2) / )n
t n t k L dt .
maSasadame, gveqneba / 2 2[ / 2]
212 0
1(0, ) cos( ( 2 2) / )
nL u
n
k
t n t k L dtduu
/ 2 / 2 / 2[ / 2]
2 2 212 2 2
1 1 12
2 2 2
n n nL L Lu
k
u udu du du
u u u
/ 2 / 2
2 2
2 2
1 12 (1)
2 2 2
n nL Luu u du u O du
u u
/ 2 / 2
2
2 2
(1)(ln )
2
n nL L
n
du Odu o L
u u
.
Tu gaviTvaliswinebT nL -mimdevrobis gansazRvrebas, miviRebT 2H Sesakrebis
Sefasebas:
(1.55) 2 (ln )H o n .
ganvixiloT 1H Sesakrebi. gamoviyenoT funqciis perioduloba periodiT
2, maSin gveqneba / 2 / 22 2[ / 2] [ / 2]
1 2 21 12 2 2 2 0
2 1 2 1(0, ) (0, )
2 2
n nL Lku u
k kk
H s dsdu s dsduu u
/ 2 / 2
2 2
2 2
4 2 1 2 2 12
2 2
n nL Lu udu du
u u
/ 2 / 2
2 2
2 2
2 2 1 2 2 12 (1)
2
n nL Luu u du u O du
u u
/ 2 / 2
2
2 2
2 2 1 2 2(1) ln (1)
n nL L
n
duO du L O
u u .
-
27
sabolood, Tu gaviTvaliswinebT n
L -mimdevrobis gansazRvrebas, maSin
sakmarisad didi n -ebisTvis gveqneba
(1.56) 1
2ln (1)
2H n O
.
ese igi sabolood (1.52)-(1.56) Sefasebebis ZaliT gveqneba
3
2ln (1)
2G n O
.
maSin am ukanaskneli Sefasebis da (1.46)-(1.48), (1.50) da (1.51) Sefasebebis ZaliT gveqneba
(1.57) 3| | 2
limln 2n
D
n .
meore mxriv, (1.46) warmodgenidan ganvixiloT 2E -is Sefaseba. (1.37)-is ZaliT
gvaqvs:
2
2 2 2
21 1 2 | (0, ) || | | (0, ) || ( ) | | (0, ) |
n n n
n
L L L
L n
n
n n
L uE u D u du u du du
L L u u
.
(1.45)-Sefasebaze dayrdnobiT ioli saCvenebelia, rom
2
2 | (0, ) | 2ln
nL udu n
u
,
anu
3| | 2
limlnn
D
n .
am ukanasknel da (1.57) Sefasebebze dayrdnobiT da im faqtis
gaTvaliswinebiT, rom 3 0D gvaqvs: (1.43)-iT gansazRvruli funqcia
periodulia da misTvis sruldeba (1.8). amitom Teoremis mtkicebidan advili dasanaxia, rom aRniSnuli funqciisTvis agreTve ZalaSi rCeba (1.28)-(1.38), (1.41) da (1.42). ganvixiloT
0 1 2 3
0 0
( ;0)lim lim
( ) ln 4 ln
nL
n
n n
S f A A A
d f n n
1 1 2 1lim lim lim4 ln 4 ln 4 lnn n n
A B B B
n n n
1 2 31 2lim lim 14 ln 4 lnn n
D D DC C
n n
3lim 1 14 lnn
D
n
.
es ukanaskneli Sefaseba ki amtkicebs Teorema 1.2.1-is ( )ii nawils da asrulebs Teoremis mtkicebas.
Teorema 1.2.2-is damtkiceba. cxadia Teorema 1.2.2-is mtkicebisas gamogvadgeba Teorema 1.2.1-is
mtkicebisas gamoyenebuli Sefasebebis, warmodgenebis da meTodebis nawili, romelTa ganxilva Teorema 1.2.2-is mtkicebaSi aucilebeli ar aris da
-
28
sakmarisi iqneboda migveTiTebina Sesabamisi formulis nomeri, Tumca mtkicebis sicxadisaTvis damtkicebas moviyvanT srulad.
ganvixiloT ,
( ; )nn
L
nt f x ,
1
,
0
1( ; ) ( ; )n n n
n n
nL L
n n k k
kn
t f x A S f xA
,
1( ) ( )
n
n
n
n
L
L
n
n L
f t t x dtL
,
sadac
(1.58) 1,
0
1( ; ) ( )n n n
n n
nL L
n n k k
kn
f x A D tA
,
sadac nn
A gasazRvrulia (1.5)-iT.
integralSi cvladis SecvliT u t x miviRebT
, ,
1( ; ) ( ) ( )
n
n n
n n
n
L x
L L
n n
n L x
t f x f x u u duL
,
meore mxriv, Tu cvlads SevcvliT u t x da gaviTvaliswinebT, rom ( )Lk
D t
kenti funqciaa t -s mimarT, miviRebT
, ,
1( ; ) ( ) ( )
n
n n
n n
n
L x
L L
n n
n L x
t f x f x u u duL
.
sabolood integralis adiciurobis ZaliT gveqneba
, ,
1( ; ) ( ) ( )
2
n
n n
n n
n
L x
L L
n n
n L x
t f x f x u u duL
,
1( ) ( )
2
n
n
n
n
L x
L
n
n L x
f x u u duL
,
1( ( ) ( )) ( )
2
n
n
n
n
L x
L
n
n L x
f x u f x u u duL
,
1( ) ( )
2
n
n
n
n
L x
L
n
n L x
f x u u duL
(1.59) , 1 2 3
1( ) ( )
2
n
n
n
n
L x
L
n
n L x
f x u u du A A AL
.
ganvixiloT 1A . vinaidan ( ) ( )f x u f x u da ( )nL
nD u u -s mimarT kenti
funqciebia, amitom gveqneba
1 ,
1( ( ) ( )) ( )
2
n
n
n
n
L x
L
n
n L x
A f x u f x u u duL
,
0
1( ( ) ( )) ( )
n
n
n
L x
L
n
n
f x u f x u u duL
,
0
1( ( ) ( )) ( )
n
n
n
L
L
n
n
f x u f x u u duL
-
29
(1.60) , 1 2
1( ( ) ( )) ( )
n
n
n
n
L
L
n
n L x
f x u f x u u du B BL
.
ganvixiloT ,
( )nn
L
nu . ( )nD u -Si (ix. [21], Tavi II, (5.6)) cvladis SecvliT da
,( )n
n
L
nu -is warmodgenis (ix. Teorema 1.2.2) gamoyenebiT miviRebT Sefasebas
,( )n
n
L
nu -Tvis:
,
1| ( ) |
sin( / 2 )n
n
L
n
n
uu L
.
ganvixiloT 2
B . maSin bolos miRebuli Sefasebidan advili dasanxia,
rom (1.8), (1.58)-is da integralSi cvladis SecvliT davadgenT, rom
2 ,
1| ( ) ( ) | | ( ) |
n
n
n
n
L
L
n
n L x
B f x u f x u u duL
1
0
1 1| ( ) ( ) | | ( ) |
n
n n
n
n
L nL
n k k
kn nL x
f x u f x u A D t duL A
1
0
1 1| ( ) ( ) |
sin[ ( ) / 2 ]
n
n
n
n
L n
n k
kn n n nL x
f x u f x u A duL L x L A
1| ( ) ( ) |
sin[ ( ) / 2 ]
n
n
L
n n n L x
f x u f x u duL L x L
1
| ( ) | | ( ) |sin[ ( ) / 2 ]
n
n
L
n n n L x
f x u f x u duL L x L
(1.61) 2
1| ( ) | | ( ) | (1)
sin[ ( ) / 2 ]
n n
n n
L x L x
n n n L L x
f t dt f t dt OL L x L
.
ganvixiloT 2A da 3A . 2B -is msgavsad advili dasanxia, rom (1.8), (1.58)-is da
integralSi cvladis SecvliT miviRebT Sefasebebs:
2
1| | | ( ) |
2 sin[ ( ) / 2 ]
n
n
L x
n n n L x
A f x u duL L x L
(1.62) 2
1| ( ) | (1)
2 sin[ ( ) / 2 ]
n
n
L x
n n n L
f t dt OL L x L
,
da
3
1| | | ( ) |
2 sin[ ( ) / 2 ]
n
n
L x
n n n L x
A f x u duL L x L
(1.63) 2
1| ( ) | (1)
2 sin[ ( ) / 2 ]
n
n
L
n n n L x
f t dt OL L x L
.
ganvixiloT 1B ,
1 ,
0
1( ( ) ( ) ( )) ( )
n
n
n
L
L
x n
n
B f x u f x u d f u duL
, 1 2
0
( )( )
n
n
n
L
Lx
n
n
d fu du C C
L .
-
30
SevafasoT 1
C . (1.6)-is ZaliT nebismieri dadebiTi 0 ricxvisTvis,
moiZebneba dadebiTi ricxvi ( ) (sazogadod damokidebuli -ze) iseTi, rom adgili eqneba (1.33).
aviRoT n imdenad didi, rom 1/ n , maSin miviRebT 1/
1 , ,
0 1/
1 1( , ) ( ) ( , ) ( )n n
n n
n
L L
n n
n n n
C x u u u x u u duL L
(1.64) , 1 2 3
1( , ) ( )
n
n
n
L
L
n
n
x u u du D D DL
.
(1.10), (1.35) da (1.58)-is ZaliT yoveli u -Tvis advilad davaskvniT, rom
1
,
0
1| ( ) | | ( ) |n n n
n n
nL L
n n k k
kn
u A D tA
(1.65) 1 1
0 0
1n n
n n
n n
n k n k
k kn n
nA k A n
A A
.
amitom (1.33) da (1.65)-is safuZvelze davaskvniT
(1.66) 1/
1
0
| | | ( , ) |
n
n n
nD x u du
L L
.
(1.37) da (1.58)-is gaTvaliswinebiT miviRebT
1
,
0
1| ( ) | | ( ) |n n n
n n
nL L
n n k k
kn
u A D tA
(1.67) 1
0
2 21n
n
nn n
n k
kn
L LA
u A u
, 0 nu L .
maSin (1.67) Sefasebidan gamomdinare nawilobiTi integrebiT davadgenT Semdegi Sefasebis samarTlianobas
2
1/ 1/
21 2 | ( , ) || | | ( , ) | n
n n n
L x uD x u du du
L u u
1/ 0 0 1/
2 1 2 1| ( , ) | | ( , ) |
u u
n n
d x t dt x t dtu u
2
1/ 0 0
2 1 2 1| ( , ) | | ( , ) |
u
n
x t dtdu x t dtu
1/
2
0 1/ 0
2 2 1| ( , ) | | ( , ) |
n u
n
nx t dt x t dtdu
u
(1.68) 1/
2 22 (2 ln ln1 ln )) (ln )
n
dun o n
u
.
amasTan (1.11) da (1.67)-is ZaliT gveqneba
3
21 2 | ( , ) || | | ( , ) |
n nL L
n
n
L x uD x u du du
L u u
1 1
1
2 | ( , ) | 2 | ( , ) | 2| ( , ) |
nLx u x udu du x u du
u u
-
31
(1.69) 1
2 | ( , ) |(1) (ln )
nL x udu O o n
u
.
(1.59)-(1.69)-dan gamomdinareobs Semdegi Sefaseba
(1.70) 1lim 0lnn
C
n .
ganvixiloT 2
C .
1
2 ,
00 0
( ) ( ) 1( ) ( )
n n
n n n
n n
L L nL Lx x
n n k k
kn n n
d f d fC u du A D t du
L L A
1 1
0 00
( ) ( )1 1( )
n
n n n n
n n
Ln nL Lx x
n k k n k k
k kn n n n
d f d fA D t du A U
L A L A
.
SevafasoT
1
0lnn n
n
nL
n k k
kn n
A UL A n
.
(1.41)-is ZaliT yoveli 0 , arsebobs iseTi ( )N N , rom yoveli k N , gvaqvs
(1.71) 1 1ln
nL
k
n
U
L k
.
n mimdevroba davyoT or qvemimdevrobad Semdegnairad [0;1)
im da
[1; ]ik
b , i . 1 2 1 2{ , ,...} { , ,...}m m k k , 1 2 1 2{ , ,...} { , ,...}m m k k .
jer ganvixiloT im
qvemimdevroba:
/1 1
0 0ln ln
m mi i i i
m mi i
i MiL L
i k k i k k
k ki i i i
A U A UL A i L A i
1
1 2
/ 1ln
mi i
mi
iL
i k k
k i Mi i
A U E EL A i
.
2E gamosaxuleba SevafasoT qvemodan. (1.71)-is ZaliT gvaqvs
1
2
/ 1ln
mi i
mi
iL
i k k
k i Mi i
E A UL A i
1
/ 1
1 1(1 ) ln
ln
mi
mi
i
i k
k i Mi
A ki A
1
/ 1
(1 )(ln ln )
ln
mi
mi
i
i k
k i Mi
i MA
i A
/1 1
0 0
(1 )(ln ln )
ln
m mi i
mi
i Mi
i k i k
k ki
i MA A
i A
/1
0
ln ln 1(1 ) 1 (1 ) 1
ln ln
mi
mi
i M
i k
ki
M MA
i i A
1 2F F .
Tu 0 1im
, maSin 1 1 0im
. aqedan gamomdinareobs 1mi
i kA
-is klebadoba.
amgvarad, gveqneba
-
32
/1
2
0
ln 1(1 ) 1
ln
mi
mi
i M
i k
ki
MF A
i A
/1
/
0
ln 1(1 ) 1
ln
mi
mi
i M
i i M
ki
MA
i A
.
Tu ( )M M -s imdenad dids aviRebT, rom 1/ M da miiA
ricxvebis
Sefasebebs gaviTvaliswinebT (ix. [9] lema 2) miviRebT
/1
/
0
ln 1(1 ) 1
ln
mi
mi
i M
i i M
ki
MA
i A
1 1
(1)/
m mi i
i i
m mi i
m mi ii i
O O oM Mii i M
.
amgvarad,
1
ln(1 ) 1
ln
MF
i
,
maSasadame,
2(1)F o .
amis garda, (1.71) ZaliT da miiA
mimdevrobis klebadobis gamo i -s mimarT
samarTliania Semdegi Sefaseba: /
1 1
1
0 1ln ln
m mi i i i
m mi i
i MNL L
i k k i k k
k k Ni i i i
E A U A UL A i L A i
/1 11
0 1
( ) 1(1 ) ln
ln ln
m mi i
m mi i
i MN
i k i k
k k Ni i i
C NA A k
L A i i A
/1 11
0 1
( ) (1 ) ln
ln ln
m mi i
m mi i
i MN
i k i k
k k Ni i i
C N NA A
L A i i A
/1 11
0 1
( ) (1 ) ln
ln ln
m mi i
m mi i
i MN
i i
k k Ni i i
C N NA A
L A i i A
1 1ln ln
m mi i
i i
m mi i
m m
i
i i iO O N
Mi L i i i
1
( )1
ln ln
mi
i
mi
m
i
i MN iO O
i L i i M i
1 1
ln lni
O Oi L i i
,
sadac
10
( ) min iL
kk N
C N U
.
maSasadame,
, 1 2
0
lim ( ) lim ( )ln
i
i
mi
L
ii i
i
L
t dt E EL i
1 1 2 1lim ( ) lim 1i i
E F F F
.
-
33
2C wevrSi mocemuli integrali
im mimdevrobisTvis SevafasoT zemodan.
(1.71)-is ZaliT gvaqvs
1
,
00 0
1( ) ( )
ln ln
i i
mi i i
m mi i
iL L
i i k k
ki i i
L L
u dt A D t dtL i L i A
1 1
0 00
( )ln ln
i
m mi i i i
m mi i
i iL L
i k k i k k
k ki i i i
L
A D t dt A UL i A L i A
1 1
0 1ln ln
m mi i i i
m mi i
N iL L
i k k i k k
k k Ni i i i
A U A UL i A L i A
1 1
0 1
1(1 ) ln
ln ln
m mi i i
m mi i
N iL
i k k i k
k k Ni i i
A U A kL i A i A
1 12
0 1
( ) (1 ) ln
ln ln
m mi i
m mi i
N i
i k i k
k k Ni i i
C N iA A
L i A i A
.
mi
iA
mimdevrobis i -s mimarT klebadobis gamo da miiA
ricxvebis
Sefasebebidan (ix. [9] lema 2) miviRebT:
1 12
0 1
( ) (1 ) ln
ln ln
m mi i
m mi i
N i
i k i k
k k Ni i i
C N iA A
L i A i A
1 1 12
0 0 0
( ) (1 )
ln
m m mi i i
m mi i
N i N
i N i k i k
k k ki i i
C NA A A
L i A A
1 12
0 0
( ) (1 )(1 )
ln
m mi i
m mi i
N N
i N i
k ki i i
C NA A
L i A A
1 11 1
( ) ln ln
mi
i
NO O
i i N L i i i
11
lnO
i i
,
sadac
20
( ) max iL
kk N
C N U
,
e. i.
,
0
lim ( ) 1ln
i
i
mi
L
ii
i
L
t dtL i
.
sabolood miviRebT:
,
0
lim ( ) 1ln
i
i
mi
L
ii
i
L
t dtL i
.
2C -Si mocemuli integrali
ik mimdevrobisTvis SevafasoT qvemodan.
(1.71)-is ZaliT gvaqvs
1
,
00 0
1( ) ( )
ln ln
i i
ki i i
k ki i
iL L
i i j j
ji i i
L L
u dt A D t dtL i L i A
/1 1
0 00
( )ln ln
i
k ki i i i
k ki i
i MiL L
i j j i j j
j ji i i i
L
A D t A UL A i L A i
-
34
1
1 2
/ 1ln
ki i
ki
iL
i j j
j i Mi i
A U G GL A i
.
ganvixiloT 2G . vinaidan 1 ik b , maSin 1ki
i jA
klebadia j -s mimarT, maSin
(1.71) ZaliT gveqneba:
1
2
/ 1ln
ki i
ki
iL
i j j
j i Mi i
G A UL A i
1
/ 1
1(1 ) ln
ln
ki
ki
i
i j
j i Mi
A ji A
1
/ 1
(1 )(ln ln )
ln
ki
ki
i
i j
j i Mi
i MA
i A
/1 1
0 0
(1 )(ln ln )
ln
k ki i
ki
i Mi
i j i j
j ji
i MA A
i A
/1
1 2
0
ln 1(1 ) 1 1
ln
ki
ki
i M
i j
ji
MA H H
i A
.
radgan 1ki
iA
zrdadia rogorc i -s funqcia, amitom
1
ln(1 ) 1
ln
MH
i
,
xolo /
1
2
0
ln 1(1 ) 1
ln
ki
ki
i M
i j
ki
MH A
i A
/1
0
ln 1(1 ) 1
ln
ki
ki
i M
i
ki
MA
i A
1
1ki
i
ki
ki i
O OM Mi
.
( )M M -is xarjze SegviZlia miviRoT Sefaseba:
2(1)H o .
(1.71)-ZaliT da 1ki
iA
zrdadobidan gvaqvs: /
1 1
1
0 1ln ln
k ki i i i
k ki i
i MNL L
i j j i j j
j j Ni i i i
G A U A UL A i L A i
/1 11
0 1
( ) 1(1 ) ln
ln ln
k ki i
k ki i
i MN
i j i j
j j Ni i i
C NA A j
L i A i A
/1 11
0 1
( ) (1 ) ln
ln ln
k ki i
k ki i
i MN
i j i j
j j Ni i i
C N NA A
L i A i A
/1 11
/
0 1
( ) (1 ) ln
ln ln
k ki i
k ki i
i MN
i N i i M
k k Ni i i
C N NA A
L i A i A
1 1
ln ln /
k ki i
i i
k ki i
k k
i
i i iO O N
Mi N L i i i i M
-
35
( )1
ln ln
i
ki
k
i
i MNO O
i L i i M i
1 1
ln lni
O Oi L i i
.
aqedan davaskvniT, rom
, 1 2
0
lim ( ) lim ( )ln
i
i
ki
L
ii i
i
L
u dt G GL i
1 1 2 1lim ( ) lim 1i i
G H H H
.
axla SevafasoT igive gamosaxuleba zemodan. (1.71)-is ZaliT gvaqvs:
1
,
00 0
1( ) ( )
ln ln
i i
ki i i
k ki i
iL L
i i j j
ji i i
L L
u dt A D t dtL i L i A
1 1
0 00
( )ln ln
i
k ki i i i
k ki i
i iL L
i j j i j j
j ji i i i
L
A D t A UL A i L A i
1 1
0 1ln ln
k ki i i i
k ki i
N iL L
i j j i j j
j j Ni i i i
A U A UL A i L A i
1 12
0 1
( ) 1ln
ln ln
k ki i
k ki i
N i
i i j
j j Ni i i
C NA A j
L A i i A
1 1 12
0 0 1
( ) (1 ) ln
ln ln
k k ki i i
k ki i
N i N
i i j i j
k j j Ni i i
C N iA A A
L A i i A
1 12
0 0
( ) (1 )1
ln
k ki i
k ki i
N N
i i N
j ji i i
C NA A
L A i A
1 11
ln ( )
k ki i
k ki i
i
i iO O
i L i i N
1 11
ln lni
O Oi L i i
;
e. i.
,
0
lim ( ) 1ln
i
i
ki
L
ii
i
L
u dtL i
,
anu gvaqvs
,
0
lim ( ) 1ln
i
i
ki
L
ii
i
L
u dtL i
.
maSasadame,
,
0
lim ( ) 1ln
n
n
n
L
nn
n
L
u dtL n
,
saidanac davaskvniT, rom
(1.72) 2( )
limln
x
n
d fC
n .
es ukanaskneli ki (1.70) erTad amtkicebs (1.14). amiT Teorema 1.2.2-is ( )i nawili damtkicebulia.
-
36
davamtkicoT ( )ii .
ganvixiloT (1.43) formuliT gansazRvruli funqcia, vTqvaT n
L n .
cxadia am funqciisTvis da n
L mimdevrobisTvis 0-wertilSi sruldeba (1.6)
da (1.13), rasac amtkicebs (1.44) da (1.45). Teorema 1.2.2-is ( )i nawilis mtkicebidan gamomdinare am SemTxvevaSic
samarTliani iqneba (1.59)-(1.68) da (1.72) Sefasebebis analogebi. gasaxilveli
gvrCeba 3
D (ix. (1.64)). ganvixiloT warmodgena (ix. [21], Tavi. III, Teorema (1.22)-
is damtkiceba)
1 2 1
1 1
1 1( ) ( )n n n n
n n
n nL L
n k k n k k k
k kn n
A D t A A K tA A
,
sadac ( )nLk
K t dirixles SeuRlebuli gulis feieris saSualoa, anu
0
1( ) ( )
1n n
kL L
k s
s
K t D tk
.
ganvixiloT 3D .
2 1
3
1
1 1(0, ) ( )
n
n n
n
L nL
n k k k
kn n
D u A A K t duL A
2
2 1
1
1 1(0, ) ( )n n
n
nL
n k k k
kn n
u A A K t duL A
2 1
1 2
12
1 1(0, ) ( )
n
n n
n
L nL
n k k k
kn n
u A A K t du I IL A
.
SevafasoT zemodan 1
I . (1.67)-is ZaliT gvaqvs 2 2
2 1
1
1
1 1(0, ) ( ) 2 | (0, ) |n n
n
nL
n k k k
kn n
duI u A A K u du u
L A u
2 2
0
2 2 16| (0, ) | | (0, ) |u du u du
.
dirixles SeuRlebuli gulis feieris ( )n
K t saSualos cnobili
warmodgenaSi cvladis Secvlis ZaliT miviRebT ( )nLk
K t -sTvis analogiur
warmodgenas
2
sin(( 1) / )1( ) cot( / 2 )
2 ( 1)(2sin( / 2 ))nL n
k n
n
k t LK t t L
k t L
.
amgvarad, 2I wevrisTvis gvaqvs
2 1
2
12
1 1 1(0, ) cot( / 2 )
2
n
n
n
L n
n k k n
kn n
I t A A t LL A
1 22
sin(( 1) / )
( 1)(2sin( / 2 ))
n
n
k t Ldt J J
k t L
.
ganvixiloT 1J , vinaidan (0, ) 0t yvela t -sTvis, maSin nawilobiTi
integrebiT miviRebT Semdeg Sefasebas:
2 1
1
12
1 1(0, ) cot( / 2 )
2
n
n
n
L n
n k k n
kn n
J t A A t L dtL A
-
37
/ 2
2 2
1 1(0, ) cot( / 2 ) (0, ) cot( / 2 )
2 2
n nL L
n n
n n
t t L dt t t L dtL L
/ 2 / 2
2 2
2 (0, ) 2 (0, )
4 sin( / 2 ) 2
n nL L
n n
t tdt dt
L t L t
/ 2/ 2
2 0 0 2
2 1 2 1(0, ) (0, )
2 2
nn
LL t t
d s ds s dst t
/ 2 / 2
2
2 0 0
2 1 2(0, ) (0, )
2
n nL Lt
n
s dsdt s dst L
/ 22
1 2 32
0 2 0
2 2 1(0, ) (0, )
4 2
nL t
s ds s dsdt K K Kt
.
SevafasoT 2
K , 2
2
0
2 2 2(0, )
4K s ds
.
1K wevris Sesafaseblad sakmarisia gaviTvaliswinoT Sefaseba | (0, ) | 8t .
amitom martivad davaskvniT, rom / 2
1
0
2| | 8 (1)
nL
n
K ds OL
.
ganvixiloT 3K : / 2 2[ / 2]
3 1 2212 2 2 2[ / 2]
2 1(0, ) (0, )
2
nL k tt
k k t
K s ds s ds dt M Mt
.
SevafasoT 2M zemodan, gvaqvs / 2
2 2
2 2[ / 2]
2 1| | | (0, ) |
2
nL t
t
M s dsdtt
/ 2 / 22
2 2 2
2 0 2 2
2 1 4 2 4 2| (0, ) | (1)
2
n nL L dt dts dsdt O
t t t
.
1M -Tvis miviRebT:
/ 2 2[ / 2]
1 212 2 2
2 1(0, )
2
nL kt
k k
M s dsdtt
/ 2 2[ / 2]
212 0
2 1(0, )
2
nL t
k
s dsdtt
/ 2 / 2
2 2
2 2
4 2 1 2 2 12
2 2
n nL Lt tdt dt
t t
/ 2 / 2
2 2
2 2
2 2 1 2 2 12 (1)
2
n nL Ltt t dt t O dt
t t
/ 2/ 2
2
2
2 2 1 2 2(1) ln
n
n
LL
dt O tt
-
38
2 2 2 2 2 2ln( / 2) (1) ln (1) ln (1)
n n nL O L O L O
.
amitom n
L mimdevrobis gansazRvris ZaliT gvaqvs
1
2ln
2M n
.
ganvixiloT 2
J , vinaidan 1 1k
A k gveqneba
2 1
2 21
sin(( 1) / )1 1| | (0, )
( 1)(2sin( / 2 ))
n
n
n
L nn
n k k
kn n n
k t LJ t A A dt
L A k t L
2
21
sin(( 1) / )1 1(0, )
(2sin( / 2 ))
n
n
n
L nn
n k
kn n n
k t Lt A dt
L A t L
2
21
1 1 1| (0, ) |
(2sin( / 2 ))
n
n
n
Ln
n k
kn n n
A t dtA L t L
1 1
2 2
82
sin ( / 2 )
n nn n
n n
L L
n n n
n n n n
A L Adt dt
A L t L A t
1 1 4
2 2 3/ 4
8 10
n n
n n
n n
n
A L dt n nO O
A t n n
,
n . maSasadame, gveqneba
3| | 2
limln 2n
D
n .
meore mxriv 3D -is warmodgenidan ganvixiloT 2I wevris Sefaseba. (1.67)-is
ZaliT gvaqvs:
2 ,
2 2 2
21 1 2 | (0, ) || | | (0, ) || ( ) | | (0, ) |
n n n
n
n
L L L
L n
n
n n
L uI u u du u du du
L L u u
.
(1.45)-Sefasebebis gaTvaliswinebiT ioli saCvenebelia, rom
2
2 | (0, ) | 2ln
nL udu n
u
,
anu
3| | 2
limlnn
D
n .
am ukanasknel Sefasebebze dayrdnobiT gvaqvs:
1 2 31
0
lim lim( ) ln 4 lnn n
D D DC
d f n n
1 2 1 2lim lim4 ln 4 lnn n
I I J J
n n
1 2 3 1 2lim lim4 ln 4 lnn n
K K K M MP
n n
.
3| |D -is miRebuli Sefasebebidan gvaqvs, rom 0P da | | 1P , amitom bolo
Sefasebidan da (1.72)-dan sabolood miviRebT, rom
, 0 1 2
0 0
( ;0)lim lim | 1| 1
( ) ln 4 ln
n
n
L
n
n n
t f C CP
d f n n
.
-
39
Teorema 1.2.2 damtkicebulia.
Teorema 1.2.3-is damtkiceba.
ganvixiloT ( ; )nLn
f x , ( )
0
( ; ) ( ; )n nq n
L L
n nk k
k
f x a S f x
( )
0
1( ) ( )
n
n
n
L q nL
nk k
kn L
f t a D t x dtL
,
sadac ( )nLk
D t gansazRvrulia (1.10),
integralSi cvladis SecvliT u t x miviRebT ( )
0
1( ; ) ( ) ( )
n
n n
n
L x q nL L
n nk k
kn L x
f x f x u a D u duL
.
meore mxriv, Tu cvlads SevcvliT u t x da gaviTvaliswinebT, rom ( )LkD t
kenti funqciaa t -s mimarT, miviRebT ( )
0
1( ; ) ( ) ( )
n
n n
n
L x q nL L
n nk k
kn L x
f x f x u a D u duL
.
sabolood integralis adiciurobis ZaliT gveqneba ( )
0
1( ; ) ( ) ( )
2
n
n n
n
L x q nL L
n nk k
kn L x
f x f x u a D u duL
( )
0
1( ) ( )
2
n
n
n
L x q nL
nk k
kn L x
f x u a D u duL
( )
0
1( ( ) ( )) ( )
2
n
n
n
L x q nL
nk k
kn L x
f x u f x u a D u duL
( )
0
1( ) ( )
2
n
n
n
L x q nL
nk k
kn L x
f x u a D u duL
(1.73) ( )
1 2 3
0
1( ) ( )
2
n
n
n
L x q nL
nk k
kn L x
f x u a D u du A A AL
.
ganvixiloT 1A . vinaidan ( ) ( )f x u f x u da ( )nL
nD u u -s mimarT kenti
funqciebia, amitom gveqneba ( )
1
0
1( ( ) ( )) ( )
2
n
n
n
L x q nL
nk k
kn L x
A f x u f x u a D u duL
( )
00
1( ( ) ( )) ( )
n
n
L x q nL
nk k
kn
f x u f x u a D u duL
( )
00
1( ( ) ( )) ( )
n
n
L q nL
nk k
kn
f x u f x u a D u duL
-
40
(1.74) ( )
1 2
0
1( ( ) ( )) ( )
n
n
n
L q nL
nk k
kn L x
f x u f x u a D u du B BL
.
ganvixiloT ( )
0
( )nq n
L
nk k
k
a D u
, ( )nD u -is warmodgenaSi (ix. [21], Tavi II, (5.6))
cvladis SecvliT miviRebT Sefasebas ( )nLn
D u -Tvis:
1| ( ) |
sin( / 2 )nL
n
n
D uu L
;
xolo ukanasknelidan miviRebT ( )
0
| ( ) |sin( / 2 )
n
q nL n
nk k
k n
Aa D u
u L ,
sadac ( )
0
q n
n nk
k
A a
da 1nA , n .
ganvixiloT 2B . bolo Sefasebidan (1.8)-is gaTvaliswinebiT da
integralSi cvladis Secvlis ZaliT davadgenT, rom ( )
2
0
1| ( ) ( ) | | ( ) |
n
n
n
L q nL
nk k
kn L x
B f x u f x u a D u duL
( )
0
1| ( ) ( ) | | ( ) |
n
n
n
L q nL
nk k
kn L x
f x u f x u a D u duL
( )
0
1| ( ) ( ) |
sin[ ( ) / 2 ]
n
n
L q n
nk
kn n n L x
f x u f x u a duL L x L
| ( ) ( ) |sin[ ( ) / 2 ]
n
n
L
n
n n n L x
Af x u f x u du
L L x L
| ( ) | | ( ) |sin[ ( ) / 2 ]
n
n
L
n
n n n L x
Af x u f x u du
L L x L
(1.75) 2
| ( ) | | ( ) | (1)sin[ ( ) / 2 ]
n n
n n
L x L x
n
n n n L L x
Af t dt f t dt O
L L x L
.
vinaidan 1nA , n , Semdeg SefasebebSi gamovtovebT aRniSnul
gamosaxulebas.
ganvixiloT 2A , 3A . maSin 2B -is msgavsad advili dasanxia, rom (1.8)-is da
integralSi cvladis Secvlis ZaliT miviRebT Sefasebebs:
2
1| | | ( ) |
2 sin[ ( ) / 2 ]
n
n
L x
n n n L x
A f x u duL L x L
(1.76) 2
1| ( ) | (1)
2 sin[ ( ) / 2 ]
n
n
L x
n n n L
f t dt OL L x L
,
da
3
1| | | ( ) |
2 sin[ ( ) / 2 ]
n
n
L x
n n n L x
A f x u duL L x L
-
41
(1.77) 2
1| ( ) | (1)
2 sin[ ( ) / 2 ]
n
n
L
n n n L x
f t dt OL L x L
.
ganvixiloT 1
B , ( )
1
00
1( ( ) ( ) ( )) ( )
n
n
L q nL
x nk k
kn
B f x u f x u d f a D u duL
( )
1 2
00
( )( )
n
n
L q nLx
nk k
kn
d fa D u du C C
L .
SevafasoT 1
C . (1.6)-is ZaliT nebismieri dadebiTi 0 ricxvisTvis,
moiZebneba dadebiTi ricxvi (sazogadod damokidebuli -ze) ( ) iseTi, rom adgili eqneba (1.33).
vinaidan
lim ( )n
q n
,
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