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394Teylor qatori. Elementar funksiyalarni darajali qatorlarga
yoyish
Maxsud Tulqin o'g'li Usmonov [email protected] Mirzo Ulug'bek nomidagi O'zbekiston Milliy universiteti
Annotatsiya: Ushbu maqolada Oliy matematikaning qiziqarli mavzularidan biri bo'lgan Teylor qatori hamda Elementar funksiyalarni darajali qatorlarga yoyish haqida ma'lumotlar keltirildi va mavjud muammolar xal etildi. Funksiyaning Teylor qatori, Funksiyani Teylor qatoriga yoyish, Elementar funksiyalarni Teylor qatoriga yoyish. Bu hollarda qo'yilgan masalalarni yechishda quyida biz o'rganadigan qatorlar nazariyasi katta ahamiyatga ega.
Kalit so'zlar: Funksiyani Teylor qatoriga yoyish, Elementar funksiyalarni Teylor qatoriga yoyish.
Taylor line. Expansion of elementary functions into graded
series
Maxsud Tulqin oglu Usmonov [email protected] National University of Uzbekistan named after Mirzo Ulugbek
Abstract: In this article, one of the interesting topics of Higher Mathematics, Taylor's series and expansion of elementary functions into graded series, is presented, and existing problems are solved. Taylor series of a function, Taylor series expansion of a function, Taylor series expansion of elementary functions. The theory of series, which we will study below, is of great importance in solving the problems posed in these cases.
Keywords: Expanding a function into a Taylor series, Expanding elementary functions into a Taylor series.
10. Funksiyaning Teylor qatori. Aytaylik, f x funksiya x° e R nuqtaning biror Us(x0 ) = (x e R : x0 -S < x < x0 + S;S > 0}
atrofida isalgan tartibdagi hosilaga ega bo'lsin. Bu hol f (x) funksiyaning Teylor formulasini yozish imkonini beradi:
f (xo )(v v \ , f" (xo )(v v V , , f (n )(xo „ V
f(x) = f (x ) + ^(x - Xo )+ (x - Xo )2 +... + Äi (x - Xü )n + r, (x)
1! 2! n! ,
bunda r (x)-qoldiq had.
Modomiki, f (x) funksiya Us (xo ) da isalgan tartibdagi hosilaga ega ekan, unda f (xo )+^ (x - Xo ) +^ (x - Xo )2 +... + ¿^(x - Xo y +...
1! 2! n! (1)
darajali qatorni qarash mumkin bo'ladi.
(1) darajali qatorning koeffisientlari sonlar bo'lib, ular f (x) funksiya va uning
hosilalarining Xo nuqtadagi qiymatlari orqali ifodalangan.
(1) darajali qator f (x ) funksiyaning Teylor qatori deyiladi. x = 0
Xususan, x° Ü bo'lganda (1) darajali qator ushbu
f (o)+f^ x+x 2Ä) xn+...=]r fin^io) xn
~(n )(o) xn + =y /!n
1! 2! n! "' ¿1 n!
ko'rinishga keladi.
Faraz qilaylik, f (x) funksiya biror (- r'r) da (r > ü) isalgan tartibdagi hosilaga x = 0
ega bo'lib, uning x° Ü nuqtadagi Teylor qatori
f (0)+ f(0) x + IM X 2 + .+ Ä xn + ..
1! 2! n! (2)
bo'lsin. Bu qatorning qoldiq hadini rn (x) deylik:
f' (O) f "(o) 2 f (n)(0) n ( \
f(0)+^^x +J v 7x +... + --— x + r (x)
1! 2! n! .
1-teorema. (2) darajali qator (- r'r) da f (x) ga yaqinlashishi uchun ushbu
-(n
f (X )= f (Ü)+ ^ X + tf X 2 + .Ä" + r. (x)
1! 2! n!
Teylor formulasida, e (- r'r) uchun
lim rn (x )= 0
bo'lishi zarur va etarli.
Zarurligi. Aytaylik, (2) darajali qator (- r'r) da yaqinlashuvchi, yi\indisi
f (x )
bo'lsin. Ta'rifga binoan
lim s. (x) = f (x) (x e (- r>r))
bo'ladi, bunda
Sn (x ) = f (0)+ № x + f® x 2 +... «'
nV ' w 1! 2! n! .
y ( ) lim Sn (x )= f (x) Ravshanki, Vx r'r) da n—x bo'lishidan
lim[f (x)-Sn (x)] = lim rn (x)= 0
n—x n—x
bo'lishi kelib chiqadi.
,, Vx e(- r,r) Ha ,liimFn(x)= 0
Etarliligi. Aytaylik, Vx V r'r) da n—x bo'lsin. U holda
f (x )= f (0) + № x + ^ x2 + .Ä" +
lim[, f (x)-Sn (x)] = lim r„ (x) = 0
n—^x n—^x
bo'lib, undan
lim Sn (x )= f (x )
n—x
bo'lishi kelib chiqadi. Demak,
1! 2! _n!
bo'ladi.
Odatda, bu munosabat o'rinli bo'lsa, f(x) funksiya Teylor qatoriga yoyilgan deyiladi.
2°. Funksiyani Teylor qatoriga yoyish. Faraz qilaylik, f (x) funksiya biror (- r'r) da isalgan tartibdagi hosila-larga ega bo'lsin. 2-teorema. Agar 3M > 0' Vx e r' r) Vn ~ 0 da
f(" >(x M
bo'lsa, f (x) funksiya (- r'r) da Teylor qatoriga yoyiladi:
f(x)= £ fin)(0)= f (0) + fM x + f^ x^ + f y + ...
y> ¿0 n! Jy ' 1! 2! n! (3)
Ma'lumki, f (x) funksiyaning Lagranj ko'rinishidagi qoldiq hadli Teylor formulasi quyidagicha bo'ladi:
in)
f (x )=f (0)+x+^ x2+..ff0- xn+r„ (x) 1! 2! n!
bunda,
n (x ) = irixr x"+1 . (0 <*< 1)
(n +1)!
Teoremaning shartidan foydalanib topamiz:
\rn (x
f(n ]{0x )
(n +1)!
x ' n+1 x
n+1
< M ■
(n +1) ! '
(x e (- r, r))
Ravshanki,
И+1
lim
n^^ln +
(n +1) !
= 0
Demak,
Vx e (- r, r )
da
lim rn (x )= 0
bo'lib, undan qaralayotgan
f (x ) funksiyaning Teylor qatoriga yoyilishi kelib
chiqadi.
30. Elementarfunksiyalarni Teylor qatoriga yoyish.
a) Ko'rsatkichli va giperbolik funksiyalarni Teylor qatorlarini topamiz. Aytaylik,
f (x ) = ex
bo'lsin. Ravshanki, f(o) =1' f(n)(o) = 1 (n e N) bo'lib, Vx e (- a,a)
(a> 0)
da
0 < f (x)< ea, 0 < f(n)(x)< f (x ) = ex
bo'ladi. Binobarin, 2-teoremaga ko'ra J (x) e funksiya ( a,a) da Teylor qatoriga yoyiladi va (3) formulada foydalanib topamiz:
œ xn
2
-y* » ^ ^x ^ ^x ^x
e = ä— = 1 + — + — +... + — +... , ч ä n! 1! 2! n! (0!= 1)
. (4)
a > 0 ixtiyoriy musbat son. Demak, (4) darajali qatorning yaqinlashish radiusi r = +œ bo'ladi.
(4) munosabatda x ni - x ga almashtirib topamiz:
e-x = ä ML = 1 - x + ^ -... + (_ 1)n ■ +...
ä n! 1! 2! n!
Ma'lumki giperbolik sinus hamda giperbolik kosinus funksiyalari quyidagicha
, ex - e_x , ex + e_x shx =-, chx =
2
2
ta'riflanar edi. Yuqoridagi
v л x x x
e = 1 + - + — +... + — + .
1! 2!
n!
e "x = 1 - x + ^ -... +(- 1)n_ + , 1! 2! v 7
\n x
~n!
formulalardan foydalanib topamiz:
Y Y3 y2«+1 œ 2n+1
shx= — +---+... + 7-^ +... = > T-r-
1! 3! (2n +1)! n=o (2n +1)!
y2 y4 2n œ 2n
„| x x x » ^ x
chx = 1 +---i---+... + 7—r- + ... = > t—r-
2! 4! (2n)! П>о (2n)!.
С /lY /ому
Bu cnx funksiyalarining Teylor qatorlari bo'lib, ular ifodalangan darajali qatorlarning yaqinlashish radiuslari r = bo'ladi.
b) Trigonometrik funksiyalarning Teylor qatorlarini topamiz. Aytaylik,
f(x)= sinx bo'lsin. Ravshanki, Vx - R' Vn - N da
\f(x)< 1, f(n)(x)< 1
bo'lib, f(°X f'(0) = 1 f(2n)(0) = 0 f (2П+1)(0) =(-1)П (n - N) bo'ladi. Demak, 2-teoremaga ko'ra
f (x) = nx funksiya Teylor qatoriga yoyiladi va (3) formulaga
binoan
(- 1)П
„2n+1 1 „3 , 1 „5
sin x — > / \ x — x x + x ... n>0 (2n +1)! 3! 5!
bo'ladi. Aytaylik,
f (x) = cos x bo'lsin. Bu funksiya uchun Vx G R' Vn G N da
(5)
\f(x)< 1, \f(n)(x)
< 1
bo'lib,
f (0)— 1, Г(0) — 0, f(2n)(0)= (- l)n, f (2n+l)(0)= 0 (n - N)
bo'ladi. Unda 2-teoremaga ko'ra f (x) = cos x funksiya Teylor qatoriga yoyiladi va (3) formulaga binoan
^ (— l) 2n 1 l 2 l 4
cosx = —v- x = l--x +— x —...
n=0 (2n)! 2! 4! (6)
bo'ladi.
(5) va (6) darajali qatorlarning yaqinlashish radiusi r = bo'ladi. v) Logarifmik funksiyaning Teylor qatorini topamiz. Aytaylik,
f (x )= ln(l + x)
bo'lsin. Ma'lumki,
г-)(x)=(— f{n;,l)! (« - n)
(l + x)
bo'lib,
f(n >(<>)_(-1)"-1
n! n
bo'ladi. Bu funksiyaning Teylor formulasi
r2 r3 r4 , xn
ln(l + x)- x - f- + ir - X4 +... + (- ir - + rn (x)
2 3 4 n (7)
ko'rinishga ega.
f (x)-ln(l + x) funksiyani Teylor qatoriga yoyishda 1-teoremadan foydalanmiz. Buning uchun (7) formulada rn (x) ning 0 ga intilishini ko'rsatish etarli bo'ladi. Aytaylik, x G [0'1] bo'lsin. Bu holda Lagranj ko'rinishida yozilgan
rn (x):
(-1)
nxn+1
qoldiq had uchun
bo'ladi va
(n + l)(l + 0x)n+1 (0 <0< l)
|rn (x
n +1 lim r (x)- 0
n^ro
tenglik bajariladi.
Aytaylik, x G [-a'0] bo'lsin, bunda 0 < a <1. Bu holda Koshi ko'rinishida yozilgan
rn (x
(x) (- 1)n (1 -01 )n • ( ) (1 + 0 x )n+1
n xn+1
(0 < 0 < 1)
qoldiq had uchun
bo'lib, bo'ladi.
Demak, Vx G (-1'1] Unda 1-teoremaga ko'ra
rn (x
„n+1
(x)*f-
1 -a
lim r (x)- 0
n^ro
lim r (x) - 0
n^ro
» (-1V-1 r2 r3 , rn
ln(l + x= x - ^ +-... + (- l)n-1 + ...
n=i n 2 3 n (g)
bo'ladi.
(8) darajali qatorning yaqinlashish radiusi r = 1 ga teng.
Agar yuqoridagi ln(l + x) ning yoyilmasida x ni - x ga almashtirilsa, unda
i /,. \ » ^ x x x x ln(1 - x )= — = - x-----...---...
n=1 n 2 3 n
formula kelib chiqadi.
g) Darajali funksiyaning Teylor qatorini topamiz. Aytaylik,
f (x ) = (1 + x )a (ae R)
bo'lsin. Ma'lumki,
f(n)(x) = a(a - 1)(a - 2)...(a - n +1)(1 + x)a-n (n e N)
bo'lib,
f(n }(0) = a(a - 1)(a - 2)...(a - n +1) bo'ladi. Bu funksiyaning Teylor formulasi ushbu
(1 + x )a = 1 + a x + a(a- 1) x2 + ... + a(a- 1).(a- n + 1) xn + r (x) 1! 2! n!
ko'rinishga ega.
Endi n ^^ da rn (x)^ 0 bo'lishini ko'rsatamiz.
Ma'lumki, Teylor formulasidagi qoldiq hadning Koshi ko'rinishi quyidagicha
(x)=(a- 1)(a- 2)...[(a- 1)-(n - 1)] x„a . x(1 + gx)a-1
n!
'j-g^ n v 1 + Ox y
(0 <g<1) bo'lar edi.
Aytaylik, x e 1»1) bo'lsin. Bu holda:
lim-(a - 1)(a - 2)...[(a -1) - (n - 1)]xn = 0
1) n! bo'ladi,
chunki, limit ishorasi osidagi ifoda yaqinlashuvchi ushbu
, ^ a(a - 1)...(a - n +1) n 1 + ^ —-—-- x
n=1 n!
qatorning umumiy hadi;
|a-x|(1 -|x|)a 1 <a-x(1 + Ox)a-1 <|a-x|(1 + |x|)a \
2) ;
1 -0
1 + 0 x
<
1 -0
1 + 0 x
< 1
3)
bo'ladi. Bu munosabatlardan foydalanib, (-1,1) da
lim rn (x)- 0
n—^ro
bo'lishini topamiz. 1-teoremaga ko'ra
c(a-1) 2 a(a- 1)...(a- n +1)
/1 a 1 a a(a -1) 2
(1 + x) -1 + — x + ^—¿x + ... + ■
1!
2!
n!
xn +...
(9)
bo'ladi.
r -1
Bu darajali qatorning yaqinlashish radiusi »'N bo'lganda 1 ga teng: .
(9) munosabatda a - -1 deb olinsa, unda ushbu
1
- Z(- 1)"x" - 1 - x + x2 - x3 + x4 -... + (-1)
1 + x n-0
formula hosil bo'ladi. Bu formulada x ni - x ga almashtirib topamiz:
1
1 - x
- Z(- 1)nxn - 1 + x + x2 +... + xn +.
n-0
1-misol. Ushbu
f (x)- ln
1 + x
1 - x
funksiya Teylor qatoriga yoyilsin. Ma'lumki,
ln— - ln(1 + x) - ln(1 - x)
1 - x
bo'ladi. Biz yuqorida
2 3 x x
M
ln(1 + x)-x - — + — -... +(- 1)n — + .
2 3
\n-1 x
n
x x x
ln(1 - x)- -x-----...--...
2 3 n
bo'lishini ko'rgan edik. Bu munosabatlardan foydalanib topamiz:
y2 r3 , y"
ln(1 +
)- ln(1 x) — x +—-...+(- 1)n-1—+...
2 3
xx
x
2 3
n
n
2 x3 2 x5
2x
2n-1
- 2x +-+-+... +
3 5 2n -1
+
Demak,
n
lni+i = 2
1 - X
C 3 5
X X X
X +---1---+ ... +--
3 5 2n -1
v
2n-1 A
+...
J
(10)
(10) darajali qatorning yaqinlashish radiusi r = 1 bo'lib, yaqinlashish to'plamsi
(-1'1) bo'ladi.
2-misol. Ushbu
x sint
f (x )=J-dt
funksiya Teylor qatoriga yoyilsin. Ma'lumki,
t3 t5
\n-1
t
2n-1
sint = t - — + — -... + (- 1)n . , 3! 5! 7 (2n -1)!
+...
Unda
Sin t
t2 t4
1-1 t
2n-2
-= 1 - - + - - .. +(-1)" 7-V
t 3! 5! (2n -1)!
bo'ladi. Bu darajali qatorni hadlab integrallab topamiz:
+...
x + x f J 2 + 4 + 2
XSintdt =i|1 - - + - - ... + (- 1)n-17t-.
J t il 3! 5! ' (2n -1)!
0
2n-2
v
X X
x---+
3!-3 5!-5
"... + (- 1)n-1
X
2n-1
(2n - 1)!-(2n -1)
+...
+...
dt
Keyingi darajali qatorning yaqinlashish radiusi r = bo'ladi. 3-misol. Ushbu
f (x ) =
2 x -1
x + x - 6
funksiya Teylor qatoriga yoyilsin va bu qatorning yaqinlashish radiusi topilsin. Avvalo f(x)
funksiyani quyidagicha yozib olamiz:
f (x ) =
2x -1
1
+
1
1
1
x2 + x - 6 x + 2 x - 3
/
2
1
\
1 + x
v 2 J
/
3
1
\
1 — x v 3 J
Ma'lumki,
1 + x
= E(- 1)n • xn
n=0
1 TO
= Z xn
I - x n=0
Bu formulalardan foydalanib topamiz:
0
1
Demak,
1 ^ 1/ ^ r 1 Y \)n n
f 1 A
2
1 +1 x v 2 ,
1
= Z 1 (- 1)n -1 x = 2
n=0
2
— x v 2 ,
n=0
2
n+1
x
(r = 2)
'i 1 '
1 — x
v 3 ,
ro i
= 221
n=0 3
— x
v 3 y
= Z
n=o3
n+1
x
ro 1 X>
„n X""* 1 „n
2x -1 ro (_i)n
2x 1 =yi-ü-xn _V-^-xn = Y
x + x _ O n=0 2 n=0 3 n=0
(r = 3) (_ 1)" 1
n+1 n+1
v 2 3 y
x
bo'ladi.
Bu darajali qatorning yaqinlashish radiusi r = 2 bo'ladi.
1
Foydalanilgan adabiyotlar
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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,М.Т. (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,М.Т. (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 OSHIRISHISTIQBOLLARI
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,МТ, 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 еchimlar». 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.
