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Extra resources for 3-manifold Groups and Nonpositive Curvature

Example text

33) This function is called a cylindrical function of order ????. The Bessel functions satisfy the relations ) ???? ???? ???? ( −???? (???? ???????? (????)) = ???? ???? ????????−1 (????), ???? ???????? (????) = −???? −???? ????????+1 (????). 34) The same relations are valid for Neumann functions. In addition, the following Lommel–Hankel formula holds, ???????? (????)????????+1 (????) − ????????+1 (????)???????? (????) = −2/(????????). We now prove the next result that will be used later. 35) 22 Chapter 1. 5. Let ???? ∕= 0. Then there exists a positive number ???? = ????(????) such that ????(????/2)+???? (????)????(????/2)−1 (????) ∕= ????(????/2)+???? (????)????(????/2)−1 (????) for all integer ???? > ????.

Proof. 85) holds in ℝ ∖{0}. If ???? = 0 ????,???? ????,???? then by the deﬁnitions of Φ0,???? and Ψ0,???? we infer that ???????? = ???????? = 0 for all ???? = 0, . . , ????. Assume that ???? ∈ ℂ∖(−∞, 0]. 84)). This implies ???????? = ???????? = 0. Repeating the arguments we arrive at the required assertion. □ For ???? ∈ ℂ ∖ (−∞, 0], the following equality is valid: Φ????,???? ????,???? (????) ∫ ????????+???? = (2????)????/2 (????) ????????−1 ????????????(????,????) (????, ????)???? ???????? (????)????????(????). 87) where the constant ????(????, ????) > 0 depends only on ????, ????.

We now establish some useful statements concerning the distribution of zeros of Bessel functions. 7. 47) and lim inf (????????+1 − ???????? ) < +∞. Then either ???? = ???? = 1/2, ???? ∈ ℚ, or ???? = ????, ???? = 1. ????→∞ Proof. It follows from the hypothesis that there exists an increasing sequence {???????? }∞ ????=1 ∈ ℕ, such that ???????????? +1 −???????????? < ????, ???? = 1, 2, . . , where ???? > ( 0 is independent ) of ????. For brevity, we set ???????????? = ???????? , ???? ???????? −????????/2−????/4 = ???????? , ???????? = ???? ???????????? +(2????−1)/4 , where ???????? is a number of the positive root ???????? of function ???????? .