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We prove the enveloping property of the known divergent asymptotic expansionsof the large real zeros of the cylinder and Airy functions, and therebyanswering in the affirmative two conjectures posed by Elbert and Laforgia andby Fabijonas and Olver, respectively. The essence of the proof is theconstruction of analytic functions that return the zeros when evaluated alongcertain discrete sets of real numbers. By manipulating contour integrals ofthese functions, we derive the asymptotic expansions of the large zerostruncated after a finite number of terms plus remainders that can be estimatedefficiently. The conjectures are then deduced as corollaries of theseestimates. An analogous result for the associated phase function is alsodiscussed.

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