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Zbl 0902.30022
Kamimoto, Joe
On an integral of Hardy and Littlewood. (English)
[J] Kyushu J. Math. 52, No.1, 249-263 (1998). ISSN 1340-6116

The entire function $\varphi (x)=\int_{-\infty}^{\infty}\exp (-w^{2m}+ixw) dw$, $m\in \bbfN$ is studied. It appears in many areas: in Waring's problem, as a solution of a special form of Turrittin's differential equation, as a generalization of the Airy function, in questions about analytic hypoellipticity of the tangential Cauchy-Riemann operator, in the representation of the Bergman and Szeg\H{o} kernel of weakly pseudoconvex domains in $\bbfC^2$ and in a connection between Brownian motion and a generalized heat equation. First the asymptotic behavior of $\varphi$ at infinity is considered, then the asymptotic expansion of $\varphi$ is computed. It is also shown that $\varphi$ can be approximated by the Bessel function. In the final part of the paper the properties of the zeroes of $\varphi$ are investigated. It is added as a note that meanwhile the conjecture that all zeroes of $\varphi$ are simple has been verified.
[F.Haslinger (Wien)]

MSC 2000:: *30D10 Representations of entire functions by series and integrals
41A60 Asymptotic problems in approximation
30E15 Asymptotic representations in the complex domain
34E20 Asymptotic singular perturbations, methods (ODE)
32W05 $\overline\partial$ and $\overline\partial$-Neumann operators
30C40 Kernel functions and appl. (one complex variable)

Keywords: Bergman kernel; Airy function; Cauchy-Riemann operator; Szeg\H{o} kernel; Brownian motion; Bessel function

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