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)
The zeros of the third derivative of Bessel functions of order less than one. (English) Zbl 0843.33001
Main results: “If $\lambda_1$, $\lambda_2$ are zeros of $J'''_\nu(x)$, $0\leq \nu< 1$, $\lambda_1< \lambda_2< j_{\nu 1}$, then $\lambda_1$ is a steadily decreasing function of $\nu$ as $\nu$ increases to 1 and $\lambda_2$ is a steadily increasing function of $\nu$ as $\nu$ increases to 1. Further, there exists a (unique) value of $\nu= \nu_0$ such that $J'''_\nu(x)$ has two zeros when $\nu_0< \nu< 1$ and none when $0< \nu< \nu_0$. When $\nu= \nu_0$, $J'''_\nu(x)$ has a douoble zero in $0< x< j_{\nu 1}$;” “Let $\lambda_1 (\nu)< \lambda_2 (\nu)$ denote the zeros of $J'''_\nu(x)$ in $0< x< j_{\nu 1}$, $\nu_0< \nu< 1$. Then $\lambda_1 (\nu) \downarrow 0$, $\lambda_2 (\nu) \uparrow \sqrt {3}= j'''_{11}$ as $\nu \uparrow 1$” (Theorems 4.1 and 4.2 in the paper, respectively). Some inequalities for the zeros studied are also established. Here $\nu_0= 0.755578\dots$; $j_{\nu 1}$ is, as usual, the first positive zero of the Bessel function $J_\nu (x)$ and $j'''_{\nu k}$ $(k= 1, 2, \dots)$ stands for the positive zeros of the third derivative $J'''_\nu(x)$ and $J_\nu(x)$. The above results complete other ones found in another paper by the author and {\it P. Szegö} [Methods Appl. Anal. 2, 103-111 (1995; Zbl 0833.33003)].

33C10Bessel and Airy functions, cylinder functions, ${}_0F_1$