On the Square of the First Zero of the Bessel Function J_v(z)

Abstract

Abstract. Let j ν,1 be the smallest (first) positive zero of the Bessel function J ν (z), ν > −1, which becomes zero when ν approaches −1. Then j 2 ν,1 can be continued analytically to −2 < ν < −1, where it takes on negative values. We show that j 2 ν,1 is a convex function of ν in the interval −2 < ν ≤ 0, as an addition to an old result [Á. Elbert and A. Laforgia, SIAM J. Math. Anal. 15(1984), [206][207][208][209][210][211][212], stating this convexity for ν > 0. Also the monotonicity properties of the functionsare determined. Our approach is based on the series expansion of Bessel function J ν (z) and it turned out to be effective, especially when −2 < ν < −1.

Anomalous Transient Heat Conduction in Fractal Metamaterials

“…Clearly, increasing d s will decrease the characteristic time, which means larger dimension can promote the heat diffusion. Note ξ 1 (d s ), as the first zero of the Bessel function J ds/2−1 , is a function of the fractal dimension d s and has the theorem [25,26]:…”