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Error Estimates for Luke's Approximation Formulas for Bessel and Hankel Functions
Author(s) -
Krumhaar Hans
Publication year - 1965
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.19650450410
Subject(s) - bessel function , hankel transform , mathematics , approximation error , order (exchange) , hankel matrix , struve function , bessel polynomials , bessel process , function (biology) , cylindrical harmonics , mathematical analysis , pure mathematics , orthogonal polynomials , jacobi polynomials , classical orthogonal polynomials , gegenbauer polynomials , finance , evolutionary biology , economics , biology , macdonald polynomials
Error estimates are presented for approximation formulas for Bessel functions J n (z) and for the Hankel functions H 0 (1) (z) and H 1 (2) (z). These approximation formulas are obtained by means of the trapezoidal rule according to Luke. The error estimates are valid in the case of Bessel funtions for arbitrary real or complex argument z, they hold in the case of the Hankel functions for arguments z with a positive imaginary part. Use is made of Luke's results, some of them are repeated in order to make this paper self‐contained. In order to derive the error estimates for the Hankel functions some inequalities are engaged for the Gamma function with certain complex arguments and for Bessel functions of certain complex orders.