Inequalities for the Zeros of the Airy Functions

Abstract

By using a theorem of Sturm type the authors show that the approximations obtained by truncating the asymptotic series for the real zeros of the Airy functions Ai (x) or Bi (x) are in fact lower and upper bounds. A lower bound for the zeros of the derivatives of these functions is also derived.

Spectral properties of the massless relativistic harmonic oscillator

“…Making use of the asymptotic expansions and estimates for the zeroes of the Airy function and its derivative (see formulae (1.4), (1.7) and (5.1) in [21], we also refer to [11,20,24]) yields Corollary 3.6. We have Using the above asymptotic formulae we can derive a result on the asymptotic behaviour of the trace of the semigroup at zero.…”

“…Making use of the asymptotic expansions and estimates for the zeroes of the Airy function and its derivative [14,6] yields Corollary 3.6. We have…”

On the Reversion of an Asymptotic Expansion and the Zeros of the Airy Functions

“…Only if strict and realistic bounds are known for the remainder terms, as opposed to neglected terms, can we be quite certain that the specified precision is attained. At present the only bounds that appear to be available in the literature are those of Pittaluga and Sacripante [10]. For the expansions of a s and b s they showed that the Jth error term (that is, the error on stopping the expansion at j = J − 1) is bounded by the first neglected term and has the same sign as this term when J = 1, 2, 3, 4, 5, and also that the sixth error term has the opposite sign to the fifth term.…”

Proofs of two conjectures on the real zeros of the cylinder and Airy functions

“…The most important result in this direction is due to Pittaluga and Sacripante [25]. For the particular cases of a k and b k , they showed that the N th error term (that is, the error on stopping the expansion (1.5) at n = N − 1) does not exceed the first neglected term in absolute value and has the same sign as this term when N = 1, 2, 3, 4, 5, and also that the sixth error term has the opposite sign to the fifth term.…”