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Zbl 0652.33003
Synolakis, Costas Emmanuel
On the roots of $f(z)=J\sb 0(z)-iJ\sb 1(z)$.
(English)
[J] Q. Appl. Math. 46, No.1, 105-107 (1988). ISSN 0033-569X; ISSN 1552-4485/e

It is shown that the function $f(z)=J\sb 0(z)-iJ\sb 1(z)$, 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 $J\sb 1(x)/(xf(x))$ along the whole x-axis (from -$\infty$ to $+\infty)$ to the value $\pi$ /2? This means, is the integral of $$J\sb 0(x)J\sb 1(x)/(x(J\sp 2\sb 0(x)+J\sp 2\sb 1(x)))$$ along the whole x-axis equal to $\pi$ /2? Here is an interesting fact: if we replace the denominator by its asymptotic expression $$x(J\sp 2\sb 0(x)+J\sp 2\sb 1(x))\sim 2/\pi \quad for\quad x\to \pm \infty,$$ we have an integral simply to be calculated and found to be equal to the conjectured value $\pi$ /2.
[E.Lanckau]
MSC 2000:
*33C10 Cylinder functions, etc.
76B15 Wave motions (fluid mechanics)

Keywords: water waves; Bessel functions

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