MUODILAI INTEGRALI

MUODILAI INTEGRALI, muodilaest, ki funksiyai nomalumi on tahti alomati integral voqe ast. Muodilai integralii jinsi du­yumi Fredgolm az soddatarin Muodilai integralist, ki dar on 1r(j)—funksiyai nomalum, /(j)— azoi ozod, K (j, t)—yadroi Muodilai integrali, X — parametr (ham qimathoi haqiqi va ham qimathoi kompleksi qa­bul karda metavonad) meboshad. Dar mavridi /(j) “0 budan, Muodilai integrali yakjinsa nomida meshavad. On qimati X, ki baroi on Muodilai integralii yakjinsa tanho halli sifri dorad, qimati durust va on qimati X, ki baroi on Muodilai integralii yakjinsa halli gayrisifri dorad, adadi kharakteristiki nomi­da meshavad. Agar tarafi rosti Muodilai integrali /(j) va yadroi on K (j, t) sharthoi malum, masalan, sharthoro qone gardonand, teoremahoi zerini Fredgolm joizand: 1) Majmui adadkhoi kharakteristikii Muodilai integrali az majmui hisobi ziyod nabuda, faqat nuqtai beokhir nuqtai hududii on shuda metavonad.

2) Agar qimati X durust boshad, Muodilai integrali va khalli muodilai payvastai on baroi har guna g(j) halli yagona do­rad. Muodilai integralii yakjinsai ba on muvofiq faqat halli sifri dorad.

Agar qimati X kharakteristiki boshad, Muodilai integralii yakjinsa va Muodilai integralii ba on hamrohshuda khalhoi gayrisifri dorand. Miqdori halhoi khatti novobastai Muodilai integralii yakjinsa va Muodilai integralii ba on hamrodshuda ba hamdigar barobarand.

Baroi halshavanda budani Muodilai integrali zarur va kofist, ki tarafi ros­ti on /(j) Qa hamai halhoi Muodilai integralii yakjinsai hamrohshuda ortogonal boshad.

Aksar khosiyat va natijahoi nazariyai Muodilai integralii baroi sistemai muodilahoi integrali niz joizand. Agar dar muodilai (•) funksiyai K (j, 1) bo funksiyai matrisali, funksiyahoi / (j) va * (j) bo vektor-funk- siyaho ivaz karda shavand, sistemai Muodilai integralii dar shakli (*) hosil meshavad. Muodilai integralii namudi niz muoina karda meshavad, ki onro muodilai jinsi yakumi Fredgolm menomand. Agar hangomi t > kh budan K (j, *)=0 boshad, muodilai muodilai integralii Volter meno­mand. Agar funksiyai nomalum ba ifodai tahtiintegrali dar shakli gayrikhatti dokhil shavad, Muodilai integraliro gayrikhatti menomand. Nisbat ba funksiyahoi bisyortagyiryobanda niz muodilahoi integrali mavjudand, masalan, muodilai ki dar on D sohai fazoi duchenai Evklid meboshad, az hamin qabil ast. Bisyor masalahoi fizika va fizikai matematiki bo yorii Muodilai integrali hal karda meshavand. Dar aksar mavridho ba Muodilai integrali badal kardani mas­alahoi kanori baso tadqiqbob ast va ahamiyati amali dorad.

D. Murtazoev