Kompleks o’zgaruvchili funktsiyaning integrali va uning xossalari.
Kompleks o’zgaruvchili funktsiyaning integrali deb quyidagi limitga aytiladi:
Kabi belgilanadi . Demak
(1)
Teorema
Shunday f(z) uchun F(z) boshlang`ich funksiya mavjudki, unda Nyuton-Leybnits formulasi o’rinli
Ya’ni, bu integral L chiziqning berilishiga emas, balki boshlang’ich va quyi nuqtalarning joylashuviga bog’liq.
Agar L- yopiq bo’lsa u holda Koshi teoremasi o’rinli va
Agar nuqta L egri chiziqning ichida bo’lsa, Koshi integral formulasi o’rinli
va
(egri chiziq bo’ylab soat strelkasiga teskari yo’nalishda o’tilgan).
Auditoriya topshirig’i
1–Misol. Quyidagi integralni hisoblang
Bu yerda L chiziq
0 nuqtadan nuqtagacha bo’lgan to’g’ri chiziq.
0 nuqtadan nuqtagacha bo’lgan parabola yoyi
Yechish:
a)L- to’g’ri chiziq kesmasi va bo’lganligini e’tiborga olsak, u holda
![](data:image/png;base64,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)
x
0
b)L uchu quyidagiga egamiz:
2–Misol.
xisoblansin.
bunda, ![](data:image/png;base64,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)
3–Misol.
xisoblansin.
![](data:image/png;base64,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)
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