Browsing by Subject "Cauchy type integral"
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Item Open Access Analytic and asymptotic properties of non-symmetric Linnik's probability densities(1999) Erdoǧan, M.B.The function φθα(t) =1/1 + e-iθsgnt|t|α, α ε (0, 2), θ ε (-π, π], is a characteristic function of a probability distribution iff |θ| ≤ min(πα/2, π - πα/2). This distribution is absolutely continuous; for θ = 0 it is symmetric. The latter case was introduced by Linnik in 1953 [13] and several applications were found later. The case θ ≠ 0 was introduced by Klebanov, Maniya, and Melamed in 1984 [9], while some special cases were considered previously by Laha [12] and Pillai [18]. In 1994, Kotz, Ostrovskii and Hayfavi [10] carried out a detailed investigation of analytic and asymptotic properties of the density of the distribution for the symmetric case θ = 0. We generalize their results to the non-symmetric case θ ≠ 0. As in the symmetric case, the arithmetical nature of the parameter a plays an important role, but several new phenomena appear.Item Open Access Analytic and asymptotic properties of non-symmetric Linnik's probability densities(1995) Erdoğan, M. BurakWe prove that the function 1 , a 6 (0 ,2 ), ^ e R, 1 + is a characteristic function of a probability distribution if and only if ( a , 0 e P D = {{a,e) : a € (0,2), \d\ < m in (f^ , x - ^ ) (mod 27t)}. This distribution is absolutely continuous, its density is denoted by p^(x). For 0 = 0 (mod 2tt), it is symmetric and was introduced by Linnik (1953). Under another restrictions on 0 it was introduced by Laha (1960), Pillai (1990), Pakes (1992). In the work, it is proved that p^{±x) is completely monotonic on (0, oo) and is unimodal on R for any (a,0) € PD. Monotonicity properties of p^(x) with respect to 9 are studied. Expansions of p^(x) both into asymptotic series as X —»· ±oo and into conditionally convergent series in terms of log |x|, \x\^ (^ = 0 ,1 ,2 ,...) are obtained. The last series are absolutely convergent for almost all but not for all values of (a, 0) € PD. The corresponding subsets of P D are described in terms of Liouville numbers.Item Open Access Riemann boundary value problem(2000) Derya, Ayşe MutluWe study the Riemann boundary value problem with finite and infinite index. The basis of the solution of the problem with the finite index is the Plemelj-Sokhotski formula. The solution of the problem with infinite index depends not only on the Plemelj-Sokhotski formula but also on some results of the entire functions theory.