Mathieu Functions of General Order: Connection Formulae, Base Functions and Asymptotic Formulae: IV. The Liouville-Green Method Applied to the Mathieu Equation

• Published in: Philosophical transactions of the Royal Society of London. Series A: Mathematical, physical, and engineering sciences Vol. 301; no. 1459; p. 115
• Main Author: W. Barrett
• Format: Journal Article
• Language: English
• Published: The Royal Society 06.05.1981
• ISSN: 1364-503X; 1471-2962
• DOI: 10.1098/rsta.1981.0101

The methods described in part III and the formulae derived in part II are applied to the construction of a comprehensive set of asymptotic formulae relating to the Mathieu equation y$^{\prime \prime}$ + ($\lambda $ +2h$^{2}$ cos 2z) y = 0 with real parameters. These comprise formulae both (a) for the auxiliary parameters and (b), in terms of exponential and circular functions, for the fundamental solution, a function of a complex variable, and the various pairs of real-variable base-functions, all introduced in part II. With the aid of these, together with connection formulae also obtained in part II, approximations can readily be obtained for Mathieu functions of various types, including in particular periodic functions. Formulae for solutions are applicable on the half-strip {z:0 $\leq $ Re z $\leq {\textstyle\frac{1}{2}}\pi $, Im z $\leq $ 0} with the transition point of the differential equation which lies on its frontier removed, or in the case of real-variable solutions of the ordinary or modified equation, on the interval [0, ${\textstyle\frac{1}{2}}\pi $] or [0, $\infty $] respectively, with the same qualification as for the half-strip when this is relevant. The formulae cover the full range of the parameters subject to $\lambda \leq $ +- 2h$^{2}$. The O-terms providing error estimates are uniformly valid on any subdomain of the independent variable and parameters on which they remain bounded.