On the zeros of the MacDonald functions

• Autorzy: Hamana Yuri , Matsumoto Hiroyuki , Shirai Tomoyuki

Identyfikatory

DOI

10.7494/OpMath.2019.39.3.361

• Języki publikacji: EN

Abstrakty

EN

We are concerned with the zeros of the Macdonald functions or the modified Bessel functions of the second kind with real index. By using the explicit expressions for the algebraic equations satisfied by the zeros, we describe the behavior of the zeros when the index moves. Results by numerical computations are also presented.

Słowa kluczowe

EN

zeros; Macdonald functions; Bessel functions

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2019
• Tom: Vol. 39, no. 3
• Strony: 361--382
• Opis fizyczny: Bibliogr. 12 poz.

Twórcy

autor

Hamana Yuri

• Kumamoto University Department of Mathematics Kurokami 2-39-1, Kumamoto 860-8555, Japan

autor

Matsumoto Hiroyuki

• Aoyama Gakuin University Department of Physics and Mathematics Fuchinobe 5-10-1, Sagamihara 252-5258, Japan

autor

Shirai Tomoyuki

• Kyushu University Institute of Mathematics for Industry Motooka 744, Fukuoka 819-0395, Japan

Bibliografia

• [1] Y. Hamana, The expected volume and surface area of the Wiener sausage in odd dimensions, Osaka J. Math. 49 (2012), 853-868.
• [2] Y. Hamana, H. Matsumoto, The probability densities of the first hitting times of Bessel processes, J. Math-for-Industry 4 (2012), 91-95.
• [3] Y. Hamana, H. Matsumoto, The probability distributions of the first hitting times of Bessel processes, Trans. Amer. Math. Soc. 365 (2013), 5237-5257.
• [4] Y. Hamana, H. Matsumoto, Hitting times of Bessel processes, volume of the Wiener sausages and zeros of Macdonald functions, J. Math. Soc. Japan 68 (2016), 1615-1653.
• [5] L. Hormander, An Introduction to Complex Analysis in Several Variables, 3rd ed., North-Holland, 1990.
• [6] M.K. Kerimov, S.L. Skorokhodov, Calculation of the complex zeros of the modified Bessel function of the second kind and its derivatives, U.S.S.R. Comput. Math, and Math. Phys. 24 (1984), 115-123; Russian original, Zh. Vychisl. Mat. i Mat. Fiz. 24 (1984), 1150-1163.
• [7] N.N. Lebedev, Special Functions and Their Applications, Dover, 1972.
• [8] W. Magnus, F. Oberhettinger, R.P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, 3rd ed., Springer, 1966.
• [9] M. Marden, Geometry of Polynomials, Amer. Math. Soc, 1966.
• [10] R. Parnes, Complex zeros of the modified Bessel function Kn(z), Math. Comp. 26 (1972), 949-953.
• [11] G.N. Watson, A Treatise on the Theory of Bessel Functions, Reprinted of 2nd ed., Cambridge Univ. Press, 1995.
• [12] M.V. Zavolzhenskii, A.Kh. Terskov, The zeros of the cylinder functions Kn(z), U.S.S.R. Comput. Math, and Math. Phys., 17 (1978), 192-195; Russian original, Zh. Vychisl. Mat. i Mat. Fiz. 17 (1977), 759-762.