zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
On the roots of f(z)=J 0 (z)-iJ 1 (z). (English) Zbl 0652.33003

It is shown that the function f(z)=J 0 (z)-iJ 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 1 (x)/(xf(x)) along the whole x-axis (from - to +) to the value π /2? This means, is the integral of

J 0 (x)J 1 (x)/(x(J 0 2 (x)+J 1 2 (x)))

along the whole x-axis equal to π /2? Here is an interesting fact: if we replace the denominator by its asymptotic expression

x(J 0 2 (x)+J 1 2 (x))2/πforx±,

we have an integral simply to be calculated and found to be equal to the conjectured value π /2.

Reviewer: E.Lanckau
33C10Bessel and Airy functions, cylinder functions, 0 F 1
76B15Water waves, gravity waves; dispersion and scattering, nonlinear interaction