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A stable recurrence for the incomplete gamma function with imaginary second argument. (English) Zbl 1102.65028

Summary: Even though the two term recurrence relation satisfied by the incomplete gamma function is asymptotically stable in at least one direction, for an imaginary second argument there can be a considerable loss of correct digits before stability sets in. We present an approach to compute the recurrence relation to full precision, also for small values of the arguments, when the first argument is negative and the second one is purely imaginary. A detailed analysis shows that this approach works well for all values considered.

MSC:

65D20 Computation of special functions and constants, construction of tables
65Q05 Numerical methods for functional equations (MSC2000)
33B20 Incomplete beta and gamma functions (error functions, probability integral, Fresnel integrals)
33F05 Numerical approximation and evaluation of special functions
65E05 General theory of numerical methods in complex analysis (potential theory, etc.)
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