Bessel potensiali
Matematikada Bessel potentsiali Riesz potentsialiga o'xshash potentsialdir (Fridrix Vilgelm Bessel nomi bilan atalgan), lekin cheksizlikda yaxshiroq parchalanish xususiyatlariga ega.
Agar s musbat haqiqiy qismga ega kompleks son bo'lsa, s tartibli Bessel potensiali operatori:
bu yerda Δ - Laplas operatori va kasr quvvati Furye transformlari yordamida aniqlanadi.
Yukava potensiali Bessel potentsiallarining 3 o'lchovli fazoda xususiy holatlaridir
Furye fazosida ko'rinishi[tahrir | manbasini tahrirlash]
Bessel potentsiali Furye o'zgarishlariga ko'paytirish orqali ta'sir qiladi: har bir uchun:
Integral ko'rinishlari[tahrir | manbasini tahrirlash]
bo'lganda, Bessel potensiali da quyidagicha ifodalanishi mumkin:
bu yerda Bessel yadrosi uchun integral formula bo'yicha[1] ifodalanadi
Bu yerda Gamma funksiyasini bildiradi. Bessel yadrosi[2] orqali ham ifodalanishi mumkin:
Ushbu oxirgi ifodani o'zgartirilgan Bessel funksiyasi[3] nuqtai nazaridan qisqaroq yozish mumkin, shu sababli potensial o'z nomini oladi:
Asimptotiklar[tahrir | manbasini tahrirlash]
Kelib chiqishida birida ,[4]
Xususan, bo'lganda Bessel potensiali Riesz potensiali kabi asimptotik tarzda o'zini tutadi.
Cheksizlikda, ,[5]
Shuningdek qarang:[tahrir | manbasini tahrirlash]
Manbalar[tahrir | manbasini tahrirlash]
- ↑ Stein, Elias. Singular integrals and differentiability properties of functions. Princeton University Press, 1970. ISBN 0-691-08079-8.
- ↑ N. Aronszajn; K. T. Smith (1961). „Theory of Bessel potentials I“. Ann. Inst. Fourier. 11-jild. 385–475, (4,2).
- ↑ N. Aronszajn; K. T. Smith (1961). „Theory of Bessel potentials I“. Ann. Inst. Fourier. 11-jild. 385–475.
- ↑ N. Aronszajn; K. T. Smith (1961). „Theory of Bessel potentials I“. Ann. Inst. Fourier. 11-jild. 385–475, (4,3).
- ↑ N. Aronszajn; K. T. Smith (1961). „Theory of Bessel potentials I“. Ann. Inst. Fourier. 11-jild. 385–475-bet.
- Grafakos, Loukas (2009), Modern Fourier analysis, Graduate Texts in Mathematics, 250-jild (2nd-nashr), Berlin, New York: Springer-Verlag, doi:10.1007/978-0-387-09434-2, ISBN 978-0-387-09433-5, MR 2463316
- Stein, Elias (1970), Singular integrals and differentiability properties of functions, Princeton, NJ: Princeton University Press, ISBN 0-691-08079-8