O'zbekiston respublikasi oliy va o'rta maxsus talim vazirligi samarqand davlat universiteti haydarov Akram matematik fizika va analizning zamonaviy usullari va nokorrekt masalalari

u(x,y) = e^y sinx bo'lsa, /(z) funksiyani Gursa formulasi orqali toping.

u(x,y) = shx sin у bo'lsa, /(z) analitik funksiyani Gursa formulasi yordamida toping.

u(x,y) = chx cos у bo'lsa, /(z) analitik funksiyani Gursa formulasi orqali toping.

Agar u(x,y) = shx cosy bo'lsa, /(z) analitik funksiyani Gursa formulasi orqali toping.

Agar u(x,y) = chx sin у bo'lsa, /(z) analitik funksiyani Gursa formulasi orqali toping.

Markazi (0,0) nuqtada bo'lgan quyi yarim aylananing garmonik o'lchovini toping.

Markazi (0,0) nuqtada bo'lgan yuqori yarim aylananing garmonik o'lchovini toping.

Markazi (0,0) nuqtada bo'lgan ikkinchi chorak aylanasining garmonik o'lchovini toping.

Markazi (0,0) nuqtada bo'lgan uchinchi chorak aylanasining garmonik o'lchovini toping.

Markazi (0,0) nuqtada bo'lgan to'rtinchi chorak aylanasining garmonik o'lchovini toping.

u'" +u' = f{x), x £ R s(x) £ C(R),f(x) = 0 agar |x| > const

bo'lsa, tenglamaning xususiy yechimini toping.
d.² d

2. 
   A = —- — 2 1-1 operatoming fundamental yechimini toping.
   

dx dx
d² d

2. 
   A = —- — 2 1-2 operatoming fundamental yechimini toping.
   

dx dx

2. 
   3u"(x) — u'(x) = 8(x),x £ R tenglamaning yechimini toping.
   
3. 
   и"(х) — 3и'(х) + 2и(х) = S(x) х Е R tenglama yechimini toping.
   
4. 
   Koeffitsientlari l₂ fazo metrikasida taqribiy berilgan Fure qatorining jamlash masalasining korrekt emasligini isbotlang.
   

dxt

2. 
   Au(x,y) = 0 u(x, 0) = f(x), —(x,0) = (pipe) masala yechimi
   

korrekt emasligini isbotlang.

2. 
   Sohaning qismida berilgan analitik funksiyani butun sohaga davom ettirish masalasini korrekt emasligini isbotlang.
   
3. 
   Gravimetriya teskari masalasi yechimining korrekt emasligini isbotlang.
   
4. 
   Issiqlik tarqalishi tenglamasi uchun teskari vaqtli Koshi masalasini yechishning korrekt emasligini isbotlang.
   
5. 
   
   
   u_xx Wyy 0
   

masalaning korrekt qo'yilmaganligini isbotlang.
36) u_tt = u_xx tenglamaning xarakteristik to'rtburchakda berilgan Dirixle masalasini korrekt qo'yilmaganligini isbotlang.
Ma'ruzalarni mustahkamlash uchun test savollari

1. 
   F(s) = /₀°°e ^stf(t)dt Laplas almashtirishda originalni ко'rsating
   

= X(-D 19
|| 1| ~ ^ 31
\l, 0<(р<7Г 68
0, 123
1, 123
2,—<в<2п A 123

4. 
   Agar fit) F(s), git) C(s) bo'lsa, Dyuamel integralini ko'rsating
   
   1. 
      sF(s)G(s) -> fiO)git) + (/' * = flr(0)/(t) + Q7' * /)
      
   2. 
      sFis)Gis) -> /(O)flf(t)
      
   3. 
      sFis)Gis) = if'*g)it)
      
   4. 
      sFis)Gis)^ig*f)it)
      
5. 
   - tasvirning originalini toping
   
   1. 
      1
      
   2. 
      t
      

1

3. 
   -
   

t

4. 
   t + 1
   
4. 
   x(ⁿ) + a, x^-¹) + ••• + a_nx = 1, x(0) = x'(0) + ••• + x⁽ⁿ-^1}(0) = 0 masala echimi x_t(t) bo'lsa, x^ + a_t + —I- a_nx = fit), x(0) = x'(0) = ••• = x(^n-1)(0) = 0 masala echimi qanday aniqlanadi
   
   1. 
      x(t) = _Xl(t)f(0) + /₀^Сх{(т) ■ f(t - т)dx = т) ■ fit + r)dr
      
   2. 
      x(t) = x(0)/'(t) + /₀^Сх1(т)/а - r)dt
      
   3. 
      x(t) = x^t)/'(t) + /₀^Сх;(т)Г (t - T)dT
      
   4. 
      *(t) = /₀VW*i(t-T)dT
      
5. 
   f{t) = cost funksiyaning tasvirini toping
   

s
a. ——

b.
s²+l s
s²-l s
s²+ 4 s
^C' s² + 2

4. 
   /(t) = sint ning tasvirini toping
   

l

1. 
   ——
   

s²+1 s
s²-l

2. 
   -f-
   

s²+4 s
C. ——
s² + 2

4. 
   /(t) = sinlt originalning tasvirini toping
   

2

1. 
   ——
   

s²+4 1
s² +1

2. 
   -f-
   

s²+l s
C. ——
s²+4
11. D sohada regulyar va analitik bo'lgan /(z) funksiya D da uzluksiz bo'lsa, u holda/(z) funksiyaning zeD dagi analitik davomini ko'rsating
ь. с 1
12.Agar Z) markazi koordinata boshida bo'lib, radiusi birga teng bo'lsa, u holda yuqori yarim aylananing garmonik o'lchovini toping

1. 
   a)(z) = — Г —-—-dw
   

^{v y} 27T^j0 l—2rcos(w—
2

2. 
   a)(z) = - f⁴ —-—-dbv
   

7T^JU l-2rcos(4»-^)+r^z
C. 0)(z) = Г -dw
^{v y J}0 l-2rcos(v-
2
d. o)(z) = Г^ln —-—
^{v y J}0 l-2rcos(w-(o)+r²
13.Birlik aylana birinchi choragining garmonik o'lchovini toping

a.	l -arq IT	[z-i [z-1	e	.n-, -T]
b.	1 -arq IT	[ z+i [z+1	e	.n-i
c.	1 -arq IT	'z+i .z+1	e	■ 7Tn I— 4
d.	1 -arg ТС	z-i' z-1.

14.Birlik doirada Г₁ f(z) < C, zeV₁ да /(z) < s bo'lganda, analitik davom ettirish masalasi turg'unligini baholang

1. 
   |/(z)| < C¹-^) ■ _£"(^z)
   
2. 
   |/(z)| < C¹-^
   
3. 
   \f{z)\<£^
   
4. 
   \f(z)\^1+a}^ ■