Bu yerda deb belgilash kiritdik. Endi ihtiyoriy va larni chetki (2) shartdan topamiz. da (10) dan
Bu tenglikdan,
Koeffisientlarni aniqlab, (10) ga qo’yamiz. Trigonometrik almashtirishlr bajarib, ushbuni hosil qilamiz:
Kvadrat qavs ichidagi ifodani soddalashtiramiz:
Hosil bo’lgan ifodani (13) ga qo’yamiz:
Bu formula Puasson integrali deyiladi va Dirihle masalasini doira uchun yechimini ifodalaydi.
Misol : Radiusi ga teng bo’lgan yupqa bir jinsli plastinkaning yuqori yarim qismining temperaturasi ni saqlaydi, quyi yarim qismida temperatura ga teng bo’lsa, issiqlikning stasionar tarqalish taqsimotini toping.
Yechish:
va . Issiqlikning tarqalishi
Integral bilan aniqlanadi. nuqta yuqori yarim aylanada joylashgan bo’lsin, ya`ni . U holda dan gacha o’zgaradi va bu uzunligi ga teng interval nuqtalarni o’z ichiga olmaydi. Shuning uchun almashtirish bajaraylik.
U holda
Yoki
Ifodaning o’ng qismi manfiy, demak, da tengsizlikni qanoatlantiradi. Bu hol uchun yechim:
yoki
ga teng.
Agar nuqta quyi yarim aylanada joylashgan bo’lsa: , u hoda intervalda o’zgaradi. Bu interval nuqtani esa bu intervalda yotmaydi. Shuning uchun bu yerda
U holda ning bu qiymatlari uchun:
Yuqoridagidek almashtirish bajarib
Ni topamiz. Bu yerda o’ng tomon musbat bo’lganligi uchun dan ![](data:image/png;base64,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)
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