An approximation theoretic alternative to asymptotic expansions for specialfunctions

Meinardus, Günter

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URN: urn:nbn:de:bsz:180-madoc-19513
Document Type: Working paper
Year of publication: 1985
Publication language: English
Institution: School of Business Informatics and Mathematics > Sonstige - Fakultät für Mathematik und Informatik
MADOC publication series: Veröffentlichungen der Fakultät für Mathematik und Informatik > Institut für Mathematik > Mannheimer Manuskripte
Subject: 510 Mathematics
Classification: MSC: 41A10 33C05 41A20 33B20 33B15 ,
Subject headings (SWD): Lanczos-Verfahren , Carathéodory-Klasse , Normalverteilung , Unvollständige Gammafunktion
Keywords (English): Carathéodory-Féjer method , Gaussian probability
Abstract: Some classes of functions, which are solutions of ordinary linear homogeneous differential equations of second order with an irregular singularity at infinity possess asymptotic expansions with respect to a real positive variable at infinity. In the case of non-oscillatory behavior of such functions these asymptotic expansions can be replaced by near-best relative approximations by polynomials of the reciprocal variable and by approximations with rational functions, using the socalled Carathéodory-Féjèr method. The investigations include Kummer functions resp. Whittaker functions (confluent hypergeometric functions) with this behaviour. A large class of special functions can be considered as Kummer functions resp. Whittaker functions. Two examples concerning the incomplete Gamma function and the transformed Gaussian probability function are given in some detail.
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