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High-precision computation of the confluent hypergeometric functions via Franklin-Friedman expansion. (English) Zbl 1393.33008

Summary: We present a method of high-precision computation of the confluent hypergeometric functions using an effective computational approach of what we termed Franklin-Friedman expansions. These expansions are convergent under mild conditions of the involved amplitude function and for some interesting cases the coefficients can be rapidly computed, thus providing a viable alternative to the conventional dichotomy between series expansion and asymptotic expansion. The present method has been extensively tested in different regimes of the parameters and compared with recently investigated convergent and uniform asymptotic expansions.

MSC:

33C15 Confluent hypergeometric functions, Whittaker functions, \({}_1F_1\)
33F05 Numerical approximation and evaluation of special functions
41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series)
65D20 Computation of special functions and constants, construction of tables
68W30 Symbolic computation and algebraic computation

Software:

Mathematica; DLMF; mpmath; Arb
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Full Text: DOI

References:

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