López, José L.; Pérez Sinusía, Ester New series expansions for the confluent hypergeometric function \(M(a,b,z)\). (English) Zbl 1334.33019 Appl. Math. Comput. 235, 26-31 (2014). Summary: Three new series expansions of the confluent hypergeometric function \(M(a,b,z)\) in terms of elementary functions are given. The starting point is an integral representation of \(M(a,b,z)\). Then, different multi-point Taylor expansions of an appropriate function in the integrand are used. The new expansions are used to evaluate \(M(a,b,z)\). They provide accurate evaluations of the confluent hypergeometric function, in particular improving the results in the region of small or moderate \(\mid z\mid\) where the power series definition is recommended for the evaluation of \(M(a,b,z)\). Cited in 3 Documents MSC: 33C15 Confluent hypergeometric functions, Whittaker functions, \({}_1F_1\) Keywords:confluent hypergeometric function; series expansion; Taylor expansion; multi-point Taylor expansion Software:DLMF PDFBibTeX XMLCite \textit{J. L. López} and \textit{E. Pérez Sinusía}, Appl. Math. Comput. 235, 26--31 (2014; Zbl 1334.33019) Full Text: DOI Digital Library of Mathematical Functions: §13.11 Series ‣ Kummer Functions ‣ Chapter 13 Confluent Hypergeometric Functions §13.11 Series ‣ Kummer Functions ‣ Chapter 13 Confluent Hypergeometric Functions References: [1] Abad, J.; Sesma, J., Computation of the regular confluent hypergeometric function, Math. J., 5, 74-76 (1995) [2] Abad, J.; Sesma, J., A new expansion of the confluent hypergeometric function in terms of modified Bessel functions, J. Comput. Appl. Math., 78, 97-101 (1997) · Zbl 0931.33002 [3] Buchholz, H., The Confluent Hypergeometric Function (1969), Springer-Verlag: Springer-Verlag Heidelberg · Zbl 0169.08501 [4] Gautschi, W., Gauss quadrature approximations to hypergeometric and confluent hypergeometric functions, J. Comput. Appl. Math., 139, 173-187 (2002) · Zbl 0998.65032 [5] López, J. L.; Temme, N. M., Two-point Taylor expansions of analytic functions, Stud. Appl. Math., 109, 297-311 (2002) · Zbl 1141.30300 [6] López, J. L.; Temme, N. M., Multi-point Taylor expansions of analytic functions, Trans. Am. Math. Soc., 356, 4323-4342 (2004) · Zbl 1047.30002 [7] López, J. L.; Temme, N. M., New series expansions of the Gauss hypergeometric function, Adv. Comput. Math., 39, 349-365 (2013) · Zbl 1276.33006 [9] Luke, Y. L., Mathematical Functions and their Approximations (1975), Academic Press · Zbl 0318.33001 [10] Muller, K. E., Computing the confluent hypergeometric function, \(M(a, b, x)\), Numer. Math., 90, 179-196 (2001) · Zbl 0995.65029 [11] Olde Daalhuis, A. B., Confluent Hypergeometric Functions, NIST Handbook of Mathematical Functions (2010), NIST and Cambridge University Press: NIST and Cambridge University Press 2010, (Chapter 13) <http://dlmf.nist.gov/> · Zbl 1190.33002 [13] Relph, A. P., Algorithm 192, confluent hypergeometric, Commun. Assoc. Comput. Mach., 6, 388 (1963) [14] Slater, L. J., Confluent Hypergeometric Functions (1960), Cambridge University Press · Zbl 0086.27502 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.