RUI: A Numerical Evaluation of the Confluent HypergeometricFunction with Large, Complex Arguments

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ABSTRACT This investigation utilizes a new approach for evaluating the confluent hypergeometric function for a wide range of arguments both real and complex. The approach consist of the direct summation of Kummer's series and extension of the computing machine precision to control overflow. This direct approach 8seems promising for the accurate evaluation of the entire subset of special case functions: Bessel functions, Coulomb functions, Laguerre polynomials, etc. The goals of the research are to: determine the absolute accuracy of the method; make the function evaluator as robust as possible; ensure the portability of the evaluator; and vectorize the function evaluator algorithm to achieve computation times competitive with existing evaluators. Because a wide range of complex arguments are now possible, this function evaluator will be an accurate, versatile, and portable package applicable to a wide spectrum of problems in science and engineering. In addition, the approach used to control overflow in the complex arithematic may be applicable to a broader class of problems requiring the summation of a power series.