Asymptotics of Modified Bessel Functions of High Order
نویسندگان
چکیده
In this work, we present two sets of full asymptotic expansions for the modified Bessel functions Iν(z) and Kν(z) and a full asymptotic expansion for Iν(z)Kν(z) as ν → ∞ and z is fixed with | arg z| < π. In particular, we show that Iν(z) ∼ (12z) ν Γ(ν + 1) ∞ ∑ m=0 bm(z) νm as ν → ∞, | arg ν| ≤ π − δ, and Kν(z) ∼ 1 2 Γ(ν) (12z) ν ∞ ∑ m=0 (−1) bm(z) νm as ν → ∞, | arg ν| ≤ 12π − δ, where, for each m, bm(z) is a polynomial of degree m in z 2, whose coefficients alternate in sign. Actually, b0(z) = 1; bm(z) = m ∑ k=1 (−1) S(m,k) k! ( 4z ), m = 0, 1, . . . , where S(m,k) are the Stirling numbers of the second kind. We also compare the asymptotic expansions of this work with those existing in the literature. AMS Subject Classification: 33C10, 34E05, 35C20, 41A60
منابع مشابه
Asymptotics of the modified bessel and the incomplete gamma matrix functions
h thii paper, an asymptotic expression of the incomplete gamma matrix function and integral expressions of Bessel matrix functions are given. Results are applied to study the asymptotic behavior of the modified Bessel function. @ 2003 Elsevier Science Ltd. All rights reserved. Keywords-Bessel function, Incomplete gamma function, Asymptotics.
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