It is shown that the function , frequently arising in problems of water wave run-up on a beach, has no zeroes in the upper half plane. The method is as usual, the use of the principle of the argument for a semicircular arc in the upper half plane (and along the real axis). Estimations and asymptotics of Bessel functions together with the numerical evaluation of an improper integral give all necessary remedies. - But a conjecture arises which should be attractive for experts in Bessel function theory: equals the integral of along the whole x-axis (from - to to the value /2? This means, is the integral of
along the whole x-axis equal to /2? Here is an interesting fact: if we replace the denominator by its asymptotic expression
we have an integral simply to be calculated and found to be equal to the conjectured value /2.