Xoshimov Muhammad Qosimo‘g‘li Axborottexnologiyalari vamenejment universiteti magistranti

Аннотация

Mazkur dissertatsiyada kompleks o‘zgaruvchili funksiyalar nazariyasining muhim bo‘limi bo‘lgan qoldiqlar nazariyasi va uning matematik tahlil, sonli hisoblash va fizik modellardagi ba’zi tatbiqlari yoritilgan. Ishda qoldiq tushunchasining analitik va geometrik mazmuni, Loran qatori yordamida qoldiqni topish usullari, oddiy va yuqori tartibli qutblar uchun qoldiqlarni aniqlash formulalari hamda Koshi qoldiqlar teoremasining umumiy ko‘rinishi keltirilgan. Amaliy misollar orqali Koshi teoremasining qo‘llanilishi, ayniqsa, xosmas integrallarni hisoblashdagi afzalliklari ko‘rsatib berilgan. Shuningdek, yuqori yarim tekislikdagi qutblarni aniqlash va ularning qoldiqlari yordamida haqiqiy integralni kompleks tekislikdahisoblash usuliasoslab berilgan. Ishda mashhurmanbalar – I.I. Privalov hamda Yu.V. Sidorov, M.V. Fudoryuk va M.I. Shabuninlarning darsliklariga tayanilgan. Tadqiqot natijalari kompleks funksiyalar nazariyasining chuqurroq o‘zlashtirilishiga xizmat qiladi va matematik fizika, muhandislikmatematikasi hamda sonlianalizda keng tatbiq etilishimumkin. Kalit so‘zlar: Qoldiq, Loran qatori, Koshi teoremasi, qutb, analitik funksiya, ajralgan maxsus nuqta, xosmas integral, kompleks tekislik,yuqori yarim tekislik,kompleks funksiyalarnazariyasi. Annotation. This master's thesis considers an important section of the theory of functions of a complex variable - the theory of residues and some of its applications in mathematical analysis, numerical calculations and physical models. The work reveals the analytical and geometric essence of the concept of residue, presents methods for calculating residues using the expansion of a function in a Laurent series, formulas for finding residues at simple and multiple poles, as well as a general idea of ​ ​ the Cauchy theorem on residues. Through specific examples, the application of Cauchy's theorem is shown, especially its effectiveness in calculating improper integrals. Particular attention is paid to the definition of poles located in the upper half-plane and the calculation of the corresponding residues for solving integrals on the real axis using the methods of complex analysis. The dissertation uses authoritative sources - the works of I. I. Privalov, Yu. V. Sidorov, M. V. Fedoryuk and M. I. Shabunin. The results of the studycontribute to the in-depth study of the theory of functions of a complex variable and can be useful in mathematical physics, applied mathematics and numerical analysis. Keywords: Residue, Laurent series, Cauchy theorem, pole, analytic function, isolated singular point, improper integral, complex plane, upper half-plane, theory offunctions ofa complex variable. Ta’rif1. Agar f (z) funksiya 0< z -a < R halqada analitik bo‘lsa, uning ajralgan maxsus a nuqtaga nisbatan qoldig‘i deb, ushbu 1 f (z)dz 2pi G www.wordlyjournals.com AXBOROTTEXNALOGIYALARIVAMENEJMENTUNIVERSITETI INTERNATIONAL JOURNAL OFSCIENCE ScientificBulletin ISSN:306-4583 VOLUME 3,ISSUE1,2025 integralning qiymatiga aytiladi vau 1 res f (z) = f (z)dz (1) r=a 2pi G ko‘rinishda yoziladi[1],[2]. Berilgan f (z) funksiyani 0< z -a < R halqadaLoran qatoriga yoyaylik: C C C f (z) = C (z -a)n =L+ -n +L+ -2 + -1 +C + n (z -a)n (z -a)2 z -a 0 n=- +C (z -a)+C (z -a)2+L+C (z -a)n+L (2) 1 2 n Ma’lumki,agar k -1 bo‘lsa, (z -a)kdz =0 (3) G Agar k = -1 bo‘lsa, dz (z - a)-1dz = = 2pi (4) z - a G G bo‘ladi[7]. Shu sababli (2) ning ikkala tomonidan G bo‘ylab integral olinsa, barcha integrallar nolga aylanib, faqat bitta dz f (z)dz =C =C 2pi -1 z - a -1 G G had qoladi. Bundan 1 C = f (z)dz = res f (z) (5) -1 2pi z=a G 1 Shunday qilib, f (z) funksiyaning a nuqtadagi qoldig‘i uning Loran qatoridagi had oldidagi z -a koeffisientdan iborat ekan. Demak, f (z) ning biror z = a nuqtadagi qoldig‘ini topish uchun C ni -1 topish kifoya. Agar f (z) ning ajralgan maxsus z = a nuqtasi chetlantiriladigan nuqta bo‘lsa, (2) Loran qatori bosh qismga ega bo‘lmaydi, ya’ni C =0 , k =1,2,3,K bo‘ladi[4]. Shu sababli z = a -k chetlantiriladigan nuqtada qoldiqnolga teng, ya’ni www.wordlyjournals.com AXBOROTTEXNALOGIYALARIVAMENEJMENTUNIVERSITETI INTERNATIONAL JOURNAL OFSCIENCE ScientificBulletin ISSN:306-4583 VOLUME 3,ISSUE1,2025 1 res f (z) = f (z)dz =C =0. (6) z=a 2pi -1 G 2.Funksiyaning qutbga nisbatan qoldig‘i. z = a nuqtaqutbbo‘lsa, uholdaqoldiqquyidagicha izlanadi: a) agar z = a nuqta f (z) funksiyaning oddiy, ya’ni birinchi tartibliqutb bo‘lsa, uholda res f (z) =lim [ f (z)(z -a) ] (7) z=a z®a b)agar z = a nuqta f (z) funksiyaning n-tartibli qutbibo‘lsa, uholda 1 dn-1 res f (z) = lim f (z)(z -a)n (8) z=a (n-1)!z®a dzn-1 Ba’zan f (z) funksiya j(z) f (z) = (9) y(z) kasr shaklida bo‘lsa, bundan j(z) va y(z) analitik funksiyalar bo‘lib, y(a)=0 , y(a) 0 , j(a) 0 bo‘lsin. Demak, z = a oddiy qutbdan iborat bo‘lsa, uholda qoldiq j(z) res f (z) = (10) z=a y(z) bo‘ladi. 3.Qoldiqlarhaqida Koshi teoremasi. Agar f (z) funksiya G yopiq chiziq bilan o‘ralgan G yopiq sohaning ajralgan maxsus a ,a ,K,a nuqtalaridan boshqa barcha nuqtalarida ana