Global asymptotics of Hermite polynomials via Riemann-Hilbert approach
R. Wong L. Zhang
Discrete & Continuous Dynamical Systems - B 2007, 7(3): 661-682 doi: 10.3934/dcdsb.2007.7.661
In this paper, we study the asymptotic behavior of the Hermite polynomials $H_{n}((2n+1)^{1/2}z)$ as $n\rightarrow \infty$. A globally uniform asymptotic expansion is obtained for $z$ in an unbounded region containing the right half-plane Re $z \geq 0$. A corresponding expansion can also be given for $z$ in the left half-plane by using the symmetry property of the Hermite polynomials. Our approach is based on the steepest-descent method for Riemann-Hilbert problems introduced by Deift and Zhou.
keywords: Riemann-Hilbert problems airy functions. Global asymptotics Hermite polynomials
Integral and series representations of the dirac delta function
Y. T. Li R. Wong
Communications on Pure & Applied Analysis 2008, 7(2): 229-247 doi: 10.3934/cpaa.2008.7.229
Mathematical justifications are given for several integral and series representations of the Dirac delta function which appear in the physics literature. These include integrals of products of Airy functions, and of Coulomb wave functions; they also include series of products of Laguerre polynomials and of spherical harmonics. The methods used are essentially based on the asymptotic behavior of these special functions.
keywords: Airy function Coulomb wave function spherical harmonics. Dirac delta function Laguerre polynomials Liouville-Green (WKB) approximation

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