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On an integral of Hardy and Littlewood. (English) Zbl 0902.30022
The entire function φ(x)= - exp(-w 2m +ixw)dw, m 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 Szegő kernel of weakly pseudoconvex domains in 2 and in a connection between Brownian motion and a generalized heat equation. First the asymptotic behavior of φ at infinity is considered, then the asymptotic expansion of φ is computed. It is also shown that φ can be approximated by the Bessel function. In the final part of the paper the properties of the zeroes of φ are investigated. It is added as a note that meanwhile the conjecture that all zeroes of φ are simple has been verified.
MSC:
30D10Representations of entire functions by series and integrals
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
30E15Asymptotic representations in the complex domain
34E20Asymptotic singular perturbations, turning point theory, WKB methods (ODE)
32W05 ¯ and ¯-Neumann operators
30C40Kernel functions and applications (one complex variable)