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 exceptional zeros of cross-products of derivatives of spherical Bessel functions. (English) Zbl 0588.33005

The author obtains an asymptotic expansion for the lowest exceptional root of the equation

j ' ν (x)y ' ν (ρx)-j ' ν (ρx)y ' ν (x)

where ' =d/dx and j ν and y ν denote the spherical Bessel functions of the first and secind kind, respectively. The result is valid for ρ 1, and it can be thought of as complementing that given by McMahon which is useful for larger zeros.

Reviewer: A.Laforgia
33C10Bessel and Airy functions, cylinder functions, 0 F 1
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
[1]M. Abramowitz and I. Stegun (Eds),Handbook of mathematical functions, Dover, New York 1972.
[2]G. Arfken,Mathematical methods for physicists. 2nd edn, Academic Press, New York 1970.
[3]H. F. Bauer,Tables of zeros of cross product Bessel functions J p ?(?)Y p ?(k?)?J p ?(k?)Y p ?(?) =0. Math. Comp.18, 128-135 (1964).
[4]J. F. Bridge and S. W. Angrist,An extended table of roots of J n ?(x)Y n ?(?x)?J n ?(?x)Y n ?(x)=0. Math. Comp.16, 198-204 (1962).
[5]H. Buchholz,Besondere Reihenentwicklungen f?r eine h?ufig vorkommende zweireihige Determinante mit Zylinderfunktionen und ihre Nullstellen. Z. angew. Math. Mech.29, 356-367 (1949).
[6]J. A. Cochran,Remarks on the zeros of cross-product Bessel functions. J. Soc. Indust. Appl. Math.12, 580-587 (1964). · Zbl 0132.05601 · doi:10.1137/0112049
[7]H. B. Dwight,Table of roots for natural frequencies in coaxial type cavities. J. Math. & Phys.27, 84-89 (1948).
[8]H. B. Dwight,Mathematical tables of elementary and some higher mathematical functions. 3rd edn, Dover, New York 1961.
[9]H. P. W. Gottlieb,Eigenvalues of the Laplacian with Neumann boundary conditions. J. Austral. Math. Soc. Ser. B26, 293-309 (1985). · Zbl 0567.35067 · doi:10.1017/S0334270000004525
[10]A. Gray, G. B. Mathews and T. M. MacRobert,A treatise on Bessel functions and their applications to physics. 2nd edn, Dover, New York 1966.
[11]F. B. Hildebrand,Advanced calculus for applications. Prentice-Hall, Englewood Cliffs 1962.
[12]D. Kirkham,Graphs and formulas for zeros of cross product Bessel functions. J. Math. & Phys.36, 371-377 (1957).
[13]J. McMahon,On the roots of the Bessel and certain related functions. Ann. of Math.9, 23-40 (1894-95). · doi:10.2307/1967501
[14]R. Truell,Concerning the roots of J n ?N n ?(kx)?J n ?(kx)N n ?(x)=0. J. appl. Phys.14 350-352 (1943). · Zbl 0063.07856 · doi:10.1063/1.1714997
[15]R. A. Waldron,Theory of the helical waveguide of rectangular cross-section. J. Brit. I. R. E.17 577-592 (1957).