Bounds on differences of adjacent zeros of Bessel functions and iterative relations between consecutive zeros

Bounds for the distance |cν,s− cν±1,s′ | between adjacent zeros of cylinder functions are given; s and s′ are such that @cν,s′′ ∈ ]cν,s, cν±1,s′ [; cν,k stands for the kth positive zero of the cylinder (Bessel) function Cν(x) = cosαJν(x)− sinαYν(x), α ∈ [0, π[, ν ∈ R. These bounds, together with the application of modified (global) Newton methods based on the monotonic functions fν(x) = x2ν−1Cν(x)/Cν−1(x) and gν(x) = −x−(2ν+1)Cν(x)/Cν+1(x), give rise to forward (cν,k → cν,k+1) and backward (cν,k+1 → cν,k) iterative relations between consecutive zeros of cylinder functions. The problem of finding all the positive real zeros of Bessel functions Cν(x) for any real α and ν inside an interval [x1, x2], x1 > 0, is solved in a simple way.