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Asymptotic approximations for functions defined by series, with some applications to the theory of guided waves. (English) Zbl 0832.41020
The author is interested in obtaining asymptotic approximations for functions defined by series of the form $$(*)$$ $$\sum_{n=1}^\infty c_n u(\mu_n x)$$, where $$c_n$$ are known constants, $$\mu_n$$ is an increasing sequence with $$n=1, 2,\dots$$ and the function $$u(y)$$ is defined for all $$y>0$$. Two cases are considered in this paper. First, series $$(*)$$ for which the function $$u(y)$$ is sampled at points $$y= \mu_n x$$ whose variation with $$n$$ and $$x$$ is separated (separable series). Second, series $$(*)$$ where $$u(y)$$ is sampled at nonseparable points, $$y= \lambda_n (x)$$, where $$\lambda_n (x)$$ is a known function of $$x$$ for each $$n$$ (nonseparable series). Both separable and nonseparable series arise in various physical situations such as electromagnetic waveguide and linear water-waves problem.
The author deals with several examples, where, making use of Mellin transforms, asymptotic approximations of certain functions defined by each one of both types of series are obtained. In the case of nonseparable ones, the Mellin transforms techniques are applied using suitable separable approximations to $$\lambda_n (x)$$; specially, certain water-wave problem solved by C. M. Linton and D. V. Evans [Q. J. Mechanic Appl. Math. 44, No. 3, 487-506 (1991; Zbl 0746.76012)] concerning the situation that, physically, corresponds to the case in which the water is infinitely deep, is examined in detail by the author.

##### MSC:
 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.) 44A15 Special integral transforms (Legendre, Hilbert, etc.)
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