Parabolic Cylinder Functions:Examples of Error Bounds For Asymptotic Expansions

Rigorous Asymptotics for the Lamé and Mathieu Functions and their Respective Eigenvalues with a Large Parameter

By application of the theory for second-order linear differential equations with two turning points developed in [Olver F.W.J., Philos. Trans. Roy. Soc. London Ser. A 278 (1975), 137–174], uniform asymptotic approximations are obtained in the first part of this paper for the Lamé and Mathieu functions with a large real parameter. These approximations are expressed in terms of parabolic cylinder...

Numerical and Asymptotic Aspects of Parabolic Cylinder Functions

Several uniform asymptotics expansions of the Weber parabolic cylinder functions are considered, one group in terms of elementary functions, another group in terms of Airy functions. Starting point for the discussion are asymptotic expansions given earlier by F.W.J. Olver. Some of his results are modified to improve the asymptotic properties and to enlarge the intervals for using the expansions...

Asymptotic expansions for degenerate parabolic equations

Article history: Received 14 May 2014 Accepted after revision 26 September 2014 Available online 11 October 2014 Presented by Olivier Pironneau We prove asymptotic convergence results for some analytical expansions of solutions to degenerate PDEs with applications to financial mathematics. In particular, we combine short-time and global-in-space error estimates, previously obtained in the unifo...

Symbolic Evaluation of Coefficients in Airy-type Asymptotic Expansions

Computer algebra algorithms are developed for evaluating the coefficients in Airy-type asymptotic expansions that are obtained from integrals with a large parameter. The coefficients are defined from recursive schemes obtained from integration by parts. An application is given for the Weber parabolic cylinder function. 1991 Mathematics Subject Classification: 41A60, 33C10, 33C15, 33F05, 65D20.

Absence of zeros and asymptotic error estimates for Airy and parabolic cylinder functions