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558Funksional ketma-ketliklar va qatorlarning tekis yaqinlashishi. Koshi kriteriysi
Maxsud Tulqin o'g'li Usmonov maqsudu32@gmail.com Mirzo Ulug'bek nomidagi O'zbekiston Milliy universiteti
Annotatsiya: Ushbu maqolada Oliy matematikaning qiziqarli mavzularidan biri bo'lgan Funksional ketma-ketliklar va qatorlarning tekis yaqinlashishi hamda Koshi kriteriysi haqida ma'lumotlar keltirildi va quyidagi muammolar xal etildi. Funksional ketma-ketlik va limit funksiya tushunchalari. Funksional ketma-ketlikning tekis yaqinlashuvchiligi. Tekis yaqinlashuvchi funksional ketma-ketlikning xossalari. Funksional qatorning tekis yaqinlashuvchiligi. Bu hollarda qo'yilgan masalalarni yechishda quyida biz o'rganadigan qatorlar nazariyasi katta ahamiyatga ega.
Kalit so'zlar: Tekis yaqinlashuvchi funksional ketma-ketlikning xossalari. Funksional qatorning tekis yaqinlashuvchiligi, funksional ketma-ketlikning yaqinlashish to'plami.
Smooth approximation of functional sequences and series.
Cauchy criterion
Maxsud Tulqin o'g'li Usmonov maqsudu32@gmail.com National University of Uzbekistan named after Mirzo Ulugbek
Abstract: In this article, one of the interesting topics of Higher Mathematics, information about functional sequences and smooth convergence of series and the Cauchy criterion was presented, and the following problems were solved. Concepts of functional sequence and limit function. Smooth approximation of a functional sequence. Properties of linearly converging functional sequences. Smooth approximation of a functional series. The theory of series, which we will study below, is of great importance in solving the problems posed in these cases.
Keywords: Properties of a smooth converging functional sequence. Convergence of a functional series, set of convergence of a functional sequence.
10. Funksional ketma-ketlik va limit funkstiya tushunchalari. Aytaylik, har bir
natural n songa E œ r to'plamda aniqlangan bitta fn ) funkstiyani mos qo'yuvchi qoida berilgan bo'lsin. Bu qoidaga ko'ra
f (x) , /2 (x) , fn (x),... (1)
to'plam hosil bo'ladi. Uni funksional ketma-ketlik deyi-ladi. E to'plam (1) funksional ketma-ketlikning aniqlanish to'plami deyiladi.
Odatda, (1) funksional ketma-ketlik, uning n -hadi yordamida {f (x)} yoki
fn (x) kabi belgilanadi. Masalan,
.. / \ . 4x . sfx . Vx . Vx /n (x) = sin— : sin—,sin—,...,sin—,... n 12 n
lar funksional ketma-ketliklar bo'ladi va ularning aniqlanish to'plami mos
ravishda
E — R , E — [0,+ro)
bo'ladi. Ravshanki, x o'zgaruvchining biror tayinlangan x = Xo e E qiymatida ushbu
{/n (x0 )} : f1 (x0 ),f2 (x0 fn (x0 V"
sonlar ketma-ketligiga ega bo'lamiz.
1-ta'rif. Agar {fn (x°)} sonli ketma-ketlik yaqinlashuvchi (uzoqlashuvchi)
bo'lsa, {f„ (x)} funksional ketma-ketlik x x° nuqtada yaqinlashuvchi
(uzoqlashuvchi) deyiladi. x° nuqta esa bu funksional ketma-ketlikning yaqinlashish (uzoqlashish) nuqtasi deyiladi.
2-ta'rif.
{fn(x)}
funksional ketma-ketlikning barcha yaqinlashish nuqtalarida iborat E° ^E to'plam,
{fn(x)}
funksional ketma-ketlikning yaqinlashish to'plami
deyiladi.
Masalan, ushbu
/^ (x) — x • x, x , x x
funksional ketma-ketlik aniqlashish to'plami E — R bo'lib, u g(- 1,l] nuqtada yaqinlashuvchi, x e R \ (-1,1] da uzoqlashuvchi bo'ladi. Demak, ketma-ketlikning yaqinlashish to'plami E° — (- 1,l] bo'ladi. Faraz qilaylik,
{fn(x)}
funksional ketma-ketlikning yaqinlashish to'plami E°(e° c r) bo'lsin. Ravshanki, bu holda har bir x e E° da
fi(x)> f2 (x),-> fn (x),~
ketma-ketlik yaqinlashuvchi, ya'ni
lim fn (x)
x — x
(x)
E lim fn(x) . , ,
mavjud bo ladi. Endi har bir ga n—ro ni mos qo ysak, ushbu
f:x — lim fn (x)
n—ro
funkstiya hosil bo'ladi. Bu
f (x) funkstiya i/(x)i funksional ketma-ketlikning
limit funkstiyasi deyiladi:
lim fn (x)= f (x) (x £ Eo )
n—>CO
Bu munosabat quyidagini anglatadi: ixtiyoriy s > 0 son va har bir x £ uchun shunday natural n = n° (s x) son topiladiki, ixtiyoriy n > n da
\fn (x)- f (x )<S
ya ni
Vs>0 , Eln0 = n0(s,x)eN , Vn>n0 : |fn(x)- f (x)<s bo'ladi.
1-misol. Ushbu
fn (x)= n Sin~
n
funksional ketma-ketlikning limit funkstiyasi topilsin.
Berilgan funksional ketma-ketlik E = [0,+ro) da aniqlangan. Uning limit funkstiyasi
. vx
Sin—
f (x )= lim fn (x )= lim n sin— = lim—• V x = V x
n—ro n—ro n n—ro x
n
va
bo'ladi. Demak, funksional ketma-ketlik E [°,+ro) da yaqinlashuvchi
lim n sin—— = Vx
n—ro n
2-misol. Ushbu
fn (x)= xn
funksional ketma-ketlikning limit funkstiyasi topilsin. Bu funksional ketma-ketlik E = ^ da aniqlangan. Ravshanki w lim fn (x)= lim xn = +ro ,
Vx £ (I,+ro) da n—ro n—ro
w f-1 A lim f (x)= lim xn = 0
Vx £ ( I,I) da w—ro> n—ro
lim fn (x)= lim1 = 1
x — i da n—^ro n—^ro
Vv £(-Ю-1) , lim fn (x)
va £ ^ ^ da п—ю mavjud emas.
Demak, berilgan funksional ketma-ketlik E° =(- -1'1] yaqinla-shuvchi bo'lib, uning limit funkstiyasi
f0 , агар -1 < x < 1 булса,
f (x )= lim
п—ю 11 , агар x = 1 булса
bo'ladi.
3-misol. Ushbu
f (x)= n2 »x - n+yx) (x > 0)
funksional ketma-ketlikning limit funkstiyasi topilsin.
Berilgan funksional ketma-ketlikning limit funkstiyasi quyidagicha topiladi:
f 1 1 Л
f (x)= lim(f (x))= lim n2 "x - п+Ух )= lim п
n—Ю n—ю n—ю
у" _ Vn+1
х х
V 1
1 f 1 1 Л 2 1 2
„2 ,„ п
= lim п х
п—
2 V п+1
хп п+1 -1
V
п - хп +п -1
= lim—--хп+1----= ln х.
п—ю п + п 1
п2 + п
20. Funksional ketma-ketlikning tekis yaqinlashuv-chiligi. Faraz qilaylik, f (4
f1 (x) , f2 (x),. . ., fn (-XU
funksional ketma-ketlik E° to'plamda yaqinlashuvchi (ya'ni yaqinlashish to'plami Eo) bo'lib, uning limit funkstiyasi f (x) bo'lsin:
lim fn (x) = f (x)
п—ю
Ma'lumki, bu munosabat
lo = по (s,x) £ > по
Vs> 0 , Eln0 = п0(s,x)e N, Vn > п0 : \fn(x)- f (x)|<s
bo'lishini anglatadi. Shuni ta'kidlash lozimki, yuqoridagi natural n° son
ixtiyoriy olingan s >0 son bilan birga qaralayotgan x e Eo nuqtaga ham bo^liq
bo'ladi (chunki, x e Eo ning turli qiymatlarida ularga mos ketma-ketlik, umuman aytganda turlicha bo'ladi).
3-ta'rif. Agar Vs >0 son olinganda ham shu s>0 gagina bo^liq bo'lgan
natural n0 = n0(s) son topilsaki, V n > n0 va ixtiyoriy x e E0 da
\fn (x)- f (x )<s
tengsizlik bajarilsa, ya'ni
Vs > 0 , 3n0 = n0 (s) £ N, Vn > n0, Vx e : \fn (x) - f (x) < s
bo'lsa, f (x)) funksional ketma-ketlik E° to'plamda f (x) ga tekis
E
yaqinlashadi (funksional ketma-ketlik E° to'plamda tekis yaqinlashuvchi) deyiladi.
Shunday qilib, f W) funksional ketma-ketlik E° to'plamda f (x) limit funkstiyaga ega bo'lsa, uning shu limit funkstiyasiga yaqinalishish ikki xil bo'lar ekan:
^Vs>0, Eln° = n°(s,x)eN ,Vn>n° : \fn(x)- f (x)<s
bo'lsa, f (x)) funksional ketma-ketlik E° da f (x) ga yaqinlashadi (oddiy yaqinlashadi). Bu holda
fn (x)^ f (x) (x e E0 )
kabi belgilanadi.
Vs > 0 , Eln0 = n0 (s)e N, Vn > n0, Vx e E0 : \fn (x) - f (x) < s bo'lsa
{fn (x))
E
f (x)
holda
funksional ketma-ketlik E° da f (x) ga tekis yaqinlashadi. Bu
fn (x f (x) (x e E0 )
kabi belgilanadi.
Ravshanki,
{fn W)
E
funksional ketma-ketlik E0 to'plamda
f (x)
funkstiyaga
tekis yaqinlashsa u shu to'plamda
f (x)
ga yaqinlashadi:
fn (x)-f (x) ^ fn (xf (x)
Aytaylik,
fn (x£f (x)
bo'lsin . Bu holda Vn > n° va Vx e E° da
(x e Eo ).
(x e Eo )
\fn (x) - f (x} < s, ya'ni f (x) - s < fn (x) < f (x) + s bo'ladi. Bu esa {/n (x)) funksional ketama-ketlikning biror hadidan boshlab,
keyingi barcha hadlari f (x) funkstiyaning "s -orali^i"da butunlay joylashishini bildiradi (29-chizma)
29-chizma
4-misol. Ushbu
sin nx
fn (X ) =
n
funksional ketma-ketlikning R da tekis yaqinlashuvchiligi ko'rsatilsin. Ravshanki,
sin nx
Demak, limit funkstiya
lim fn (x) = lim-= 0
f (x) = 0
no =
s
deyilsa, unda Vn > no va Vx e R uchun sin nx
sin nx
1 1
<-<-< s
n n0 + 1
Agar Vs > 0 son olinganda
\fn (x )- f (x ) =
nn
bo'lishini topamiz. Demak ta'rifga binoan
sin nx ^ ^ 0
n
bo'ladi.
Faraz qilaylik, {fn (*)} funksional ketma-ketlik E° to'plamda f (x) limit funkstiyaga ega bo'lsin.
1-teorema. {fn (x)}
funksional ketma-ketlik E° to'plamda f (x) funkstiyaga
tekis yaqilashishi uchun
limsup| fn(x)- f (x } = 0
xeE0
bo'lishi zarur va etarli.
Zarurligi. Aytaylik,
fn (x (x) (x e E0 )
bo'lsin. Ta'rifga binoan
Vs > 0, 3n0 = n0(s)e N, Vn > n0, Vx e E0 : \ fn(x)- f (x) < s
bo'ladi. Bu tengsizlikdan
sup fn (x )- f (x
xeE0
bo'lib, undan
limsup fn (x)- f (x } = 0
xeEn
bo'lishi kelib chiqadi.
Etarliligi. Aytaylik
limsup fn (x)- f (x } = 0
xeE0
bo'lsin. Limit ta'rifga ko'ra
Vs > 0, 3n0 e N Vn > n0,: Supfn(x)- f (x) < s
bo'ladi. Ravshanki
xeEn
\fn (x)- f (x)< sup|fn (x)- f (x)
xeEo
U holda Vx e Eo uchun
bo'ladi. Bundan
bo'lishi kelib chiqadi. 5-misol. Ushbu
\fn (x)- f (x )<s
fn (x )—— f (x) (x e E0 )
fn (x ) =
V
2 1
x2 + —
n
E = R
funksional ketama-ketlikning E° R da tekis yaqinlashuv-chiligi ko'rsatilsin. Berilgan funksional ketma-ketlikning limit funkstiyasi
lJ
f (x )= lim fn (x )= lim.
n——X n—X
2 _1-l I x ~ — x
n
(x e R)
bo'ladi. Endi
ni topamiz:
SUP| fn (x)-f (x )|
Sup
xeR
2 1 I I x I I x
n
Sup
xeR
n
2 1 I I x + — + x n2
Sup-1
xeR n
21
x + — + x
n2
n
Demak,
lim Sup
n—X xeR
2 ±_| I
x + * x
n
— lim— — 0
n—X n
bo'lib,
bo'ladi.
i
x + 9 —— x n
(x e R)
Eslatma. Agar 1/(4 funksional ketma-ketligi uchun E c R to'plamda
limsup fn (x)- f (x 0
n——X
xeE
x
1
1
1
bo'lsa, {/n(x)} funksional ketma-ketlik E da tekis yaniqla-shishi shart emas. Endi funksional ketma-ketlikning limit funkstiyaga ega bo'lishi va unga tekis ya^inlashishini ifodalovchi teoremani keltiramiz:
2-teorema (Koshi teoremasi). {fn (x)}
funksional ketma-ketlik E to'plamda limit funkstiyaga ega bo'lishi va unga tekis yaqinlashishi uchun Vs>0 son
olinganda ham shunday n° = n°(s) e N topilib, Vn > n°, Vp e N va Vx e E da
fn+p (x)- fn (x)<s
ya'ni
Vs > 0 , 3n0 = n0 (s)e N, Vn > n0, Vp e N va Vx e E^a
\fn+p (x )- fn (x }<s (2)
bo'lishi zarur va etarli.
Zarurligi. Aytaylik, E to'plamda {fn (x)} funksional ketma-ketlik limit funkstiya f (x) ga ega bo'lib, unga tekis yaqinlashsin:
fn (x f (x) . (x e E0 )
Tekis yaqinlashish ta'rifiga ko'ra
Vs> 0, 3n0 = n0(s)e N, Vk > n0, Vx e E :| fk(x)- f (x)< —
2 bo'ladi. Xususan,
k = n, n > n° va k = n + p, p e N da
\fn (x )- f (x)<s, \fn+p (x)- f (x)<s
2
tengsizliklar bajarilib, ulardan
fn+p (x) - fn (x) = |fn+p (x) - f (x) - (fn (x) - f (x)) <
<
fn+p (x) - f (x^ + | fn (x) - (x) < s + s = s
bo'lishi kelib chiqadi. Demak, (2) shart o'rinli.
Etarliligi. f (x)} funksional ketma-ketlik uchun (2) shart bajarilsin. Uni quyidagicha yozamiz:
Vs > 0, 3nQ = n (s) e N, Vn > n0, Vp e N, Vx e E
f„+p (x)- f„ (x }<s
bo'ladi.
Ravshanki, tayin xo e E da {n (x0 )} sonlar ketma-ketligi uchun (3) shartning bajarilishidan uning fundamental ketma-ketlik ekanligi kelib chiqadi. Koshi
teoremasiga ko'ra {fn (xo )} yaqinlashuvchi bo'ladi. Binobarin, chekli
limfn (x0)
n -—ro (4)
limit mavjud.
Modomiki, har bir x e E da (4) limit mavjud bo'lar ekan, unda avval ayganimizdek, E to'plamda aniqlangan
x -— lim fn (x) (x e E)
n-—ro
funkstiya hosil bo'ladi Uni f (x) bilan belgilaymiz. Bu funkstiya {f. (x)} funksional ketma-ketlikning limit funkstiyasi bo'ladi:
fn (x) —f(x) (x e e)
Endi (3) tengsizlikda, n va x larni tayinlab o'tamiz. Natijada
(n > n0, x e E) p —>
da limitga
If (x )- fn (x )<s<s
hosil bo'ladi. Bu
bo'lishini bildiradi. 6-misol. Ushbu
fn (x )— f (x )
fn (x )
(x e Eo )
ln nx
nx
funksional ketma-ketlik E = (0,1) to'plamda tekis yaqinlashuv-chilikka tekshirilsin.
Agar ixtiyoriy k e N uchun
V v * 11
n = k , p = k = n , x = — = —
k n
deyilsa,
fn--p (x)- f (x) =
(11 ( f2 n fn
V n y
V n y
ln2
V2
- ln1
ln2
V2
= s
bo'ladi. Demak ln2
3so = V2
Vk e N, 3n > k, 3p e N, 3x* = 1 e (0,1): | fn+p (x*)- fn (x*)> s,
Bu esa yuqoridagi teoremaning shartini bajarilmasligini ko'rsatadi. Demak, berilgan funksional ketma-ketlik E = (0,1) da tekis yaqinlashuvchi emas.
Aytaylik, f (x)} funksional ketma-ketlik E to'plamda yaqinlashuvchi bo'lib,
f (x) funkstiya uning limit funkstiyasi bo'lsin:
fn (x)^ f(x) (x e E).
Agar
3s > 0 , Vk e N , 3n > k , 3x* e E : fn (x*)- f (x*) > s0
bo'lsa, {f„ (x)} funksional ketma-ketlik E to'plamda f (x) funkstiyaga notekis yaqinlashadi deyiladi. 7-misol. Ushbu
fn (x) = n
nx
funksional ketma-ketlik E da tekis yaqinlashishiga tekshirilsin.
Ravshanki,
1 sin 1
lim fn (x )= lim n sin— = lim nx = -
n^-ro n^-ro nx n^-ro 1 x
1
sin -
nx
1
- • x
nx
f (x ) =1
Demak, berilgan funksional ketma-ketlikning limit funkstiyasi x
bo'ladi.
* 1
x = —
Aytaylik, n bo'lsin. Unda
fn (x*)- f (x *) = |n sin1 - n > 1 - sin1 = s0
munosabat ixtiyoriy n e N da o'rinli bo'ladi.
fn (x) = n f (x)= ^
Demak, nx funksional ketma-ketlik limit funkstiya x ga
E = (°,1) da tekis yaqinlashmaydi.
30. Tekis yaqinlashuvchi funksional ketma-ketlikning xossalari. Tekis
yaqinlashuvchi funkstiyaonal ketma-ketliklar qator xossalarga ega. Bu xossalarni keltiramiz.
Aytaylik. {fn (x)} :
f1 (x) , f2 (x),..., fn (x),...
funksional ketma-ketlik E ^ R to'plamda yaqinlashuvchi bo'lib, f (x) uning limit funkstiyasi bo'lsin:
fn (x)^ f(x) (x e e)
1-xossa. Agar {fn W) funksional ketma-ketlikning har bir fn (x) (n = 1,2,3,...) hadi E to'plamda uzluksiz bo'lib,
fn (x)—— f (x) (x e E)
bo'lsa, limit funkstiya f (x) shu E to'plamda uzluksiz bo'ladi. Demak, bu holda
lim( lim fn (t ))= lim(lim fn (t))
t—x yn—yxi ' n—yx v t—x '
munosabat o'rinli bo'ladi.
2-xossa. Agar f (x)) funksional ketma-ketlikning har bir fn (x) (n = 1,2,3,...) hadi E = [a, b] da uzluksiz bo'lib,
fn (x)— f (x) (x e[a,b])
bo'lsa,
b
li^fl f fn (x)^x — f f (x )dx
n—<» J J
a a
bo'ladi.
Demak, bu holda
lim f fn (x)dx — f (lim fn (x) )dx
ii—* * Mi—'
n —xn ——<» a
munosabat o'rinli bo'ladi.
3-xossa. Agar f (x)) funksional ketma-ketlikning har bir fn (x) (n = 1,2,3, . ) hadi E = [a,b] da uzluksiz fn(x) (n = 1,2,3,.) hosilalarga ega bo'lib,
fn(x )—V(x) (x e[a,b])
bo'lsa,
(p(x)— f'(x)
bo'ladi.
Shu kabi xossalarga keyinroq o'rganiladigan tekis yaqinlashuvchi funksional qatorlar ham ega bo'ladi. Ayni paytda, ular bir mulohaza asosida isbotlanadi Mazkur xossalarning isbotini funksional qatorlarga nisbatan keltiramiz.
Foydalanilgan adabiyotlar
1. Usmonov, M. T. o'g'li. (2021). Matritsa rangi. Matritsa rangini tuzatish usullari. Fan va ta'lim, 2(8), 280-291.
http://openscience.uz/index.php/sciedu/article/view/1758 dan olindi.
b
2. Usmonov, M. T. o'g'li. (2021). Matritsalar va ular ustida amallar. Fan va ta'lim, 2(8), 226-238. http://openscience.uz/index.php/sciedu/article/view/1752 dan olindi.
3. Usmonov, M. T. o'g'li. (2021). Vektorlar. Fan va ta'lim, 2(8), 173-182. https://openscience.uz/index.php/sciedu/article/view/1747 dan olindi.
4. Usmonov, M. T. o'g'li. (2021). Chiziqli algebraik tenglamalar tizimini echishning matritsa, Gauss va Gauss-Jordan usullari. Fan va ta'lim, 2(8), 312-322. http://openscience.uz/index.php/sciedu/article/view/1761 dan olindi.
5. Usmonov, M. T. o'g'li. (2021). Chiziqli operatorlar va komissiya xossalari. Fan va ta'lim, 2(8), 133-145. http://openscience.uz/index.php/sciedu/article/view/1744 dan olindi.
6. Usmonov, M. T. o'g'li. (2021). Chiziqli operatorlar va komissiya xossalari. Fan va ta'lim, 2(8), 146-152. http://openscience.uz/index.php/sciedu/article/view/1744 dan olindi.
7. Usmonov, M. T. o'g'li. (2021). Kvadratik forma va uni kanonik korinishga keltirish. Fan va ta'lim, 2(8), 153-172. https://www.openscience.uz/index.php/sciedu/article/view/1746 dan olindi.
8. Usmonov, M. T. o'g'li. (2021). Arifmetik vektor fazo va unga misollar. Fan va ta'lim, 2(8), 109-120. https://www.openscience.uz/index.php/sciedu/article/view/1742 dan olindi.
9. Usmonov, M. T. o'g'li. (2021). Vektorlarning skalyar ko'paytmasi. Fan va ta'lim, 2(8), 183-191. https://www.openscience.uz/index.php/sciedu/article/view/1748 dan olindi.
10. Usmonov, M. T. o'g'li. (2021). Vektorlarning vektor va aralash ko'paytmalari. Fan va ta'lim, 2(8), 271-279. http://openscience.uz/index.php/sciedu/article/view/1757 dan olindi.
11. Usmonov, M.T. & Shokirov.,Sh.H, (2022). Teylor formulasini matematik masalalarni echishdagi ahamiyati. "«Science and Education» Scientific Journal" Scientific Journal, Tom-3, 19-23.
12. Usmonov, M.T. & Shokirov.,Sh.H, (2022). Darajali qatorlarning taqribiy hisoblashlarga tatbiqi. «Science and Education» Scientific Journal, Tom-3, 29-32.
13. Usmonov, M.T. & Shokirov.,Sh.H, (2022). Ishoralari almashinib keluvchi qatorlar. Leybnits alomati. «Science and Education» Scientific Journal, Tom-3, 2428.
14. Usmonov, M.T. & Shokirov.,Sh.H, (2022). Teylor qatori va uning tadbiqlari. «Science and Education» Scientific Journal, Tom-3, 33-38.
15. Усмонов, М.Т. (2021). Вычисление центра тяжести плоской ограниченной фигуры с помощью двойного интеграла. «Science and Education» Scientific Journal, Tom-2, 64-71.
16. Усмонов, М.Т. (2021). Биномиальное распределение вероятностей. «Science and Education» Scientific Journal, Tom-2, 81-85.
17. Усмонов,М.Т. (2021). Поток векторного поля. Поток через замкнутую поверхность. «Science and Education» Scientific Journal, Tom-2, 52-63.
18. Усмонов,М.Т. (2021). Вычисление определенного интеграла по формуле трапеций и методом Симпсона. «Science and Education» Scientific Journal, Tom-2, 213-225.
19. Усмонов,М.Т. (2021). Метод касательных. «Science and Education» Scientific Journal, Tom-2, 25-34.
20. Усмонов,МТ. (2021). Вычисление предела функции с помощью ряда. «Science and Education» Scientific Journal, Tom-2, 92-96.
21. Усмонов,МТ. (2021). Примеры решений произвольных тройных интегралов. Физические приложения тройного интеграла. «Science and Education» Scientific Journal, Tom-2, 39-51.
22. Усмонов,МТ. (2021). Вычисление двойного интеграла в полярной системе координат. «Science and Education» Scientific Journal, Tom-2, 97-108.
23. Усмонов,МТ. (2021). Криволинейный интеграл по замкнутому контуру. Формула Грина. Работа векторного поля. «Science and Education» Scientific Journal, Tom-2, 72-80.
24. Усмонов,МТ. (2021). Правило Крамера. Метод обратной матрицы. «Science and Education» Scientific Journal, Tom-2, 249-255.
25. Усмонов,МТ. (2021). Теоремы сложения и умножения вероятностей. Зависимые и независимые события. «Science and Education» Scientific Journal, Tom-2, 202-212.
26. Усмонов,МТ. (2021). Распределение и формула Пуассона. «Science and Education» Scientific Journal, Tom-2, 86-91.
27. Усмонов,МТ. (2021). Геометрическое распределение вероятностей. «Science and Education» Scientific Journal, Tom-2, 18-24.
28. Усмонов,М.Т. (2021). Вычисление площади поверхности вращения. «Science and Education» Scientific Journal, Tom-2, 97-104.
29. Усмонов,МТ. (2021). Нахождение обратной матрицы. «Science and Education» Scientific Journal, Tom-2, 123-130.
30. Усмонов,МТ. (2021). Вычисление двойного интеграла. Примеры решений. «Science and Education» Scientific Journal, Tom-2, 192-201.
31. Усмонов,МТ. (2021). Метод прямоугольников. «Science and Education» Scientific Journal, Tom-2, 105-112.
32. Усмонов,МТ. (2021). Как вычислить длину дуги кривой?. «Science and Education» Scientific Journal, Tom-2, 86-96.
33. Усмонов,МТ. (2021). Вычисление площади фигуры в полярных координатах с помощью интеграла. «Science and Education» Scientific Journal, Tom-2, 77-85.
34. Усмонов,МТ. (2021). Повторные пределы. «Science and Education» Scientific Journal, Tom-2, 35-43.
35. Усмонов,МТ. (2021). Дифференциальные уравнения второго порядка и высших порядков. Линейные дифференциальные уравнения второго порядка с постоянными коэффициентами. «Science and Education» Scientific Journal, Tom-2, 113-122.
36. Усмонов,МТ. (2021). Пределы функций. Примеры решений. «Science and Education» Scientific Journal, Tom-2, 139-150.
37. Усмонов,МТ. (2021). Метод наименьших квадратов. «Science and Education» Scientific Journal, Tom-2, 54-65.
38. Усмонов,МТ. (2021). Непрерывность функции двух переменных. «Science and Education» Scientific Journal, Tom-2, 44-53.
39. Усмонов,МТ. (2021). Интегрирование корней (иррациональных функций). Примеры решений. «Science and Education» Scientific Journal, Tom-2, 239-248.
40. Усмонов,МТ. (2021). Криволинейные интегралы. Понятие и примеры решений. «Science and Education» Scientific Journal, Tom-2, 26-38.
41. Усмонов,МТ. (2021). Гипергеометрическое распределение вероятностей. «Science and Education» Scientific Journal, Tom-2, 19-25.
42. Усмонов,МТ. (2021). Абсолютная и условная сходимость несобственного интеграла. Признак Дирихле. Признак Абеля. «Science and Education» Scientific Journal, Tom-2, 66-76.
43. Усмонов,МТ. (2021). Решение систем линейных уравнений. «Science and Education» Scientific Journal, Tom-2, 131-138.
44. Usmonov, M.T. (2021). Matritsalar va ular ustida amallar. «Science and Education» Scientific Journal, Tom-2, 226-238.
45. Usmonov, M.T. (2021). Teskari matritsa. Teskari matritsani hisoblash usullari. «Science and Education» Scientific Journal, Tom-2, 292-302.
46. Usmonov, M.T. (2021). Bir jinsli chiziqli algebraik tenglamalar sistemasi. «Science and Education» Scientific Journal, Tom-2, 323-331.
47. Usmonov, M.T. (2021). Chiziqli fazo. Yevklid fazosi. «Science and Education» Scientific Journal, Tom-2, 121-132.
48. Usmonov, M.T. (2021). Vektorlarning skalyar ko 'paytmasi. «Science and Education» Scientific Journal, Tom-2, 183-191.
49. Usmonov, M.T. (2021). Xos vektorlari bazis tashkil qiluvchi chiziqli operatorlar. «Science and Education» Scientific Journal, Tom-2, 146-152.
50. Usmonov, M.T. (2021). Chiziqli algebraik tenglamalar sistemasi va ularni еchish usullari. «Science and Education» Scientific Journal, Tom-2, 303-311.
51. Usmonov, M.T. (2021). Vektorlar. «Science and Education» Scientific Journal, Tom-2, 173-182.
52. Usmonov, M.T. (2021). Kvadratik forma va uni kanonik korinishga keltirish. «Science and Education» Scientific Journal, Tom-2, 153-172.
53. Usmonov, M.T. (2021). Arifmetik vektor fazo va unga misollar. «Science and Education» Scientific Journal, Tom-2, 109-120.
54. Usmonov, M.T. (2021). Chiziqli operatorlar va ularning xossalari. «Science and Education» Scientific Journal, Tom-2, 133-145.
55. Usmonov, M.T. (2021). Determinantlar nazariyasi. «Science and Education» Scientific Journal, Tom-2, 256-270.
56. Usmonov, M.T. (2021). Matritsa rangi. Matritsa rangini hisoblash usullari. «Science and Education» Scientific Journal, Tom-2, 280-291.
57. Usmonov, M.T. (2021). Autentification, authorization and administration. «Science and Education» Scientific Journal, Tom-2, 233-242.
58. Usmonov, M.T. (2021). Vektorlar nazariyasi elementlari. «Science and Education» Scientific Journal, Tom-2, 332-339.
59. Usmonov, M.T. (2021). EHTIMOLLAR NAZARIYASI. «Science and Education» Scientific Journal, Tom-1, 10-15.
60. Usmonov, M.T. (2021). Chiziqli algebraik tenglamalar sistemasi va ularni еchish usullari. «Science and Education» Scientific Journal, Tom-2, 333-311.
61. Usmonov, M.T. (2021). Bir jinsli chiziqli algebraik tenglamalar sistemasi. «Science and Education» Scientific Journal, Tom-21, 323-331.
62. Usmonov, M.T. (2021). Vektorlar nazariyasi elementlari. «Science and Education» Scientific Journal, Tom-2, 332-339.
63. Usmonov, M.T. (2021). Chiziqli fazo. Yevklid fazosi. «Science and Education» Scientific Journal, Tom-2, 121-132.
64. Usmonov M. T. & Qodirov F. E, BIR JINSLI VA BIR JINSLIGA OLIB KELINADIGAN DIFFERENSIAL TENGLAMALAR. AMALIY MASALALARGA TADBIQI (KO'ZGU MASALASI) , BARQARORLIK VA YETAKCHI TADQIQOTLAR ONLAYN ILMIY JURNALI: Vol. 2 No. 1 (2022): БАРКДРОРЛИК ВА ЕТАКЧИ ТАД^ЩОТЛАР ОНЛАЙН ИЛМИЙ ЖУРНАЛИ
65. Usmonov Maxsud Tulqin o'g'li, Sayifov Botirali Zokir o'g'li, Negmatova Nilufar Ergash qizi, Qodirov Farrux Ergash o'g'li, BIRINCHI VA IKKINCHI TARTIBLI HUSUSIY HOSILALAR. TO'LA DIFFERENSIAL. TAQRIBIY HISOBLASH , BARQARORLIK VA YETAKCHI TADQIQOTLAR ONLAYN ILMIY JURNALI: 2022: SPECIAL ISSUE: ZAMONAVIY UZLUKSIZ TA'LIM SIFATINI OSHIRISH ISTIQBOLLARI
66. Usmonov Maxsud Tulqin o'g'li, Sayifov Botirali Zokir o'g'li, Negmatova Nilufar Ergash qizi, Qodirov Farrux Ergash o'g'li, IKKI ARGUMENTLI FUNKSIYANING ANIQLANISH SOHASI, GRAFIGI, LIMITI VA UZLUKSIZLIGI , BARQARORLIK VA YETAKCHI TADQIQOTLAR ONLAYN ILMIY JURNALI: 2022: SPECIAL ISSUE: ZAMONAVIY UZLUKSIZ TA'LIM SIFATINI OSHIRISH ISTIQBOLLARI
67. Usmonov Maxsud Tulqin o'g'li. (2022). FURYE QATORI. FUNKSIYALARNI FURYE QATORIGA YOYISH. https://doi.org/10.5281/zenodo.6055125
68. Usmonov. M. T. ., & Qodirov. F. E. . (2022). DARAJALI QATORLAR. DARAJALI QATORLARNING YAQINLASHISH RADIUSI VA SOHASI. TEYLOR FORMULASI VA QATORI. IJTIMOIY FANLARDA INNOVASIYA ONLAYN ILMIY JURNALI, 8-20. Retrieved from http://www.sciencebox.uz/index.php/jis/article/view/1151
69. Usmonov. M. T. ., & Qodirov. F. E.. (2022). FURE QATORI VA UNING TADBIQLARI. IJTIMOIY FANLARDA INNOVASIYA ONLAYN ILMIY JURNALI, 21-33. Retrieved from http://www.sciencebox.uz/index.php/jis/article/view/1152
70. M.T Usmonov, M.A Turdiyeva, Y.Q Shoniyozova, (2021). SAMPLE POWER. SELECTION METHODS (SAMPLE ORGANIZATION METHODS). ООО НАУЧНАЯ ЭЛЕКТРОННАЯ БИБЛИОТЕКА , 59-60.
71. Усмонов,МТ, М.А.Турдиева (2021). ГЛАВА 9. МЕТОДЫ И СРЕДСТВА СОВРЕМЕННОЙ ЗАЩИТЫ КОМПЬЮТЕРНЫХ СЕТЕЙ. РИСКИ И ПРИНЦИПЫ ЗАЩИТЫ ИНФОРМАЦИИ В ЭЛЕКТРОННОЙ ПОЧТЕ. ББК 60 С69, Ст-99.
72. Усмонов,М.Т, J.M.Saipnazarov, K.B. Ablaqulov (2021 SOLUTION OF MATHEMATICAL PROBLEMS IN LOWER CLASSES. Книга: АКТУАЛЬНЫЕ ВОПРОСЫ СОВРЕМЕННОЙ НАУКИ И ОБРАЗОВАНИЯ, 167-177.
73. Усмонов М.Т. (2022). E-LEARNING И ЕГО РОЛЬ В СОВРЕМЕННОЙ СИСТЕМЕ ОБРАЗОВАНИЯ. : Special Issue_Ta'limni modernizatsiyalash jarayonlari muammolar va echimlar». 168-171.
74. Usmonov. M. T. ., & Qodirov. F. E.. (2022). STOKS FORMULASI. SIRT INTEGRALLARI TADBIQLARI. IJTIMOIY FANLARDA INNOVASIYA ONLAYN ILMIY JURNALI, 34-45. Retrieved from https://sciencebox.uz/index.php/jis/article/view/1153
75. Usmonov M. T. The Concept of Compatibility, Actions on Compatibility. International Journal of Academic Multidisciplinary Research (IJAMR), Vol. 5 Issue 1, January - 2021, Pages: 10-13.
76. Usmonov M. T. The Concept of Number. The Establishment of the Concept of Natural Number and Zero. International Journal of Academic Information Systems Research (IJAISR), Vol. 4 Issue 12, December - 2020, Pages: 7-9.
77. Usmonov M. T. The Concept of Compatibility, Actions on Compatibility. International Journal of Engineering and Information Systems (IJEAIS), Vol. 4 Issue 12, December - 2020, Pages: 66-68.
78. Usmonov M. T. General Concept of Mathematics and Its History. International Journal of Academic Multidisciplinary Research (IJAMR). Vol. 4 Issue 12, December - 2020, Pages: 38-42
79. Usmonov M. T. Asymmetric Cryptosystems. International Journal of Academic Engineering Research (IJAER) ISSN: 2643-9085 Vol. 5 Issue 1, January -2021, Pages: 6-9.
80. Usmonov M. T. Basic Concepts of Information Security. International Journal of Academic and Applied Research (IJAAR) ISSN: 2643-9603 Vol. 5 Issue 1, January - 2021, Pages: 5-8.
81. Usmonov M. T. Communication Control Systems, Methodology. International Journal of Academic Engineering Research (IJAER) ISSN: 2643-9085 Vol. 5 Issue 1, January - 2021, Pages: 47-50.
82. Usmonov M. T. Compatibility between the Two Package Elements. Binar Relations and Their Properties. International Journal of Academic Multidisciplinary Research (IJAMR) ISSN: 2643-9670 Vol. 5 Issue 1, January - 2021, Pages: 52-54.
83. Usmonov M. T. Cryptographic Protection of Information. International Journal of Academic and Applied Research (IJAAR) ISSN: 2643-9603 Vol. 5 Issue 1, January - 2021, Pages: 24-26.
84. Usmonov M. T. Electronic Digital Signature. International Journal of Academic Pedagogical Research (IJAPR) ISSN: 2643-9123 Vol. 5 Issue 1, January -2021, Pages: 30-34.
85. Usmonov M. T. "Equal" And "Small" Relations. Add. Laws Of Addition. International Journal of Academic Information Systems Research (IJAISR) ISSN: 2643-9026 Vol. 5 Issue 1, January - 2021, Pages: 27-29.
86. Usmonov M. T. Establish Network Protection. International Journal of Academic Engineering Research (IJAER) ISSN: 2643-9085 Vol. 5 Issue 1, January -2021, Pages: 14-21.
87. Usmonov M. T. Fundamentals of Symmetric Cryptosystem. International Journal of Academic and Applied Research (IJAAR) ISSN: 2643-9603 Vol. 5 Issue 1, January - 2021, Pages: 36-40.
88. Usmonov M. T. General Concepts of Mathematics. International Journal of Academic Information Systems Research (IJAISR) ISSN: 2643-9026 Vol. 5 Issue 1, January - 2021, Pages: 14-16.
89. Usmonov M. T. Identification and Authentication. International Journal of Academic Pedagogical Research (IJAPR) ISSN: 2643-9123 Vol. 5 Issue 1, January -2021, Pages: 39-47.
90. Usmonov M. T. Information Protection and Its Types. International Journal of Academic and Applied Research (IJAAR) ISSN: 2643-9603 Vol. 5 Issue 1, January - 2021, Pages: 1-4.
91. Usmonov M. T. Information Protection in Wireless Communication Systems. International Journal of Academic Pedagogical Research (IJAPR) ISSN: 2643-9123 Vol. 5 Issue 1, January - 2021, Pages: 61-64.
92. Usmonov M. T. Information protection supply. International Journal of Academic and Applied Research (IJAAR) ISSN: 2643-9603 Vol. 5 Issue 1, January -2021, Pages: 12-15.
93. Usmonov M. T. Information Security Policy. International Journal of Academic Pedagogical Research (IJAPR) ISSN: 2643-9123 Vol. 5 Issue 1, January -2021, Pages: 70-73.
94. Usmonov M. T. Information War. International Journal of Academic and Applied Research (IJAAR) ISSN: 2643-9603 Vol. 5 Issue 1, January - 2021, Pages: 79-82.
95. Usmonov M. T. International and National Legal Base in the Field Of Information Security. International Journal of Academic Pedagogical Research (IJAPR) ISSN: 2643-9123 Vol. 5 Issue 1, January - 2021, Pages: 7-14.
96. Usmonov M. T. Legal Legislative Basis for Detection of Information Crime. International Journal of Academic Engineering Research (IJAER) ISSN: 2643-9085 Vol. 5 Issue 1, January - 2021, Pages: 80-87.
97. Usmonov M. T. Mathematical Proofs. Incomplete Induction, Deduction, Analogy. The Concept Of Algorithm And Its Properties. International Journal of Academic Multidisciplinary Research (IJAMR) ISSN: 2643-9670 Vol. 5 Issue 1, January - 2021, Pages: 26-29.
98. Usmonov M. T. Means of Information Protection. International Journal of Academic and Applied Research (IJAAR) ISSN: 2643-9603 Vol. 5 Issue 1, January -2021, Pages: 27-30.
99. Usmonov M. T. Organization of E-Mail Protection. International Journal of Academic Engineering Research (IJAER) ISSN: 2643-9085 Vol. 5 Issue 1, January -2021, Pages: 36-40.
100. Usmonov M. T. Organizing Internet Protection. International Journal of Academic Engineering Research (IJAER) ISSN: 2643-9085 Vol. 5 Issue 1, January -2021, Pages: 24-28.
101. Usmonov M. T. Origin and Equal Strength Relationships between Sentences. Necessary and Sufficient Conditions. Structure of Theorem and Their
Types. International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2643-640X Vol. 5 Issue 1, January - 2021, Pages: 45-47.
102. Usmonov M. T. PhysicalSecurity. International Journal of Academic and Applied Research (IJAAR) ISSN: 2643-9603 Vol. 5 Issue 1, January - 2021, Pages: 58-61.
103. Usmonov M. T. Practical Security Management. International Journal of Academic Engineering Research (IJAER) ISSN: 2643-9085 Vol. 5 Issue 1, January -2021, Pages: 71-74.
104. Usmonov M. T. Problem Solving In Primary Schools. International Journal of Academic Information Systems Research (IJAISR) ISSN: 2643-9026 Vol. 5 Issue 1, January - 2021, Pages: 72-83.
105. Usmonov M. T. Reproduction. The Laws of Reproduction. International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2643-640X Vol. 5 Issue 1, January - 2021, Pages: 36-40.
106. Usmonov M. T. Security Models. International Journal of Academic Pedagogical Research (IJAPR) ISSN: 2643-9123 Vol. 5 Issue 1, January - 2021, Pages: 18-23.
107. Usmonov M. T. Solving Problems In Arithmetic Methods. International Journal of Academic Information Systems Research (IJAISR) ISSN: 2643-9026 Vol. 5 Issue 1, January - 2021, Pages: 58-61.
108. Usmonov M. T. Stenographic Protection of Information. International Journal of Academic and Applied Research (IJAAR) ISSN: 2643-9603 Vol. 5 Issue 1, January - 2021, Pages: 31-35.
109. Usmonov M. T. Telecommunications and Network Security. International Journal of Academic Engineering Research (IJAER) ISSN: 2643-9085 Vol. 5 Issue 1, January - 2021, Pages: 57-61.
110. Usmonov M. T. The Concept of Compatibility, Actions on Compatibility. International Journal of Academic Multidisciplinary Research (IJAMR) ISSN: 26439670 Vol. 5 Issue 1, January - 2021, Pages: 10-13.
111. Usmonov M. T. The Concept Of National Security. International Journal of Academic and Applied Research (IJAAR) ISSN: 2643-9603 Vol. 5 Issue 1, January -2021, Pages: 73-75.
112. Usmonov M. T. The Concept of Number. The Establishment of the Concept of Natural Number and Zero. International Journal of Academic Multidisciplinary Research (IJAMR) ISSN: 2643-9670 Vol. 5 Issue 1, January -2021, Pages: 18-21.
113. Usmonov M. T. The Concept of Relationship. Characteristics of Relationships. International Journal of Academic Multidisciplinary Research (IJAMR) ISSN: 2643-9670 Vol. 5 Issue 1, January - 2021, Pages: 38-40.
114. Usmonov M. T. The Concept of Size and Measurement. International Journal of Academic Information Systems Research (IJAISR) ISSN: 2643-9026 Vol. 5 Issue 1, January - 2021, Pages: 36-40.
115. Usmonov M. T. The Emergence and Development of Methods of Writing All Negative Numbers. International Journal of Academic Information Systems Research (IJAISR) ISSN: 2643-9026 Vol. 5 Issue 1, January - 2021, Pages: 48-50.
116. Usmonov M. T. The Purpose, Function and History Of The Development Of Mathematical Science. International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2643-640X Vol. 5 Issue 1, January - 2021, Pages: 8-17.
117. Usmonov M. T. True and False Thoughts, Quantities. International Journal of Academic Information Systems Research (IJAISR) ISSN: 2643-9026 Vol. 5 Issue 1, January - 2021, Pages: 1-5.
118. Usmonov M. T. Virtual Protected Networks. International Journal of Academic Pedagogical Research (IJAPR) ISSN: 2643-9123 Vol. 5 Issue 1, January -2021, Pages: 55-57.
119. Usmonov M. T. What Is Solving The Problem? Methods of Solving Text Problems. International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2643-640X Vol. 5 Issue 1, January - 2021, Pages: 56-58.
