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On the evaluation of Bessel functions. (English) Zbl 0802.33004

The author presents an algorithm for the evaluation of Bessel functions J ν (x), Y ν (x) and H ν (j) (x) (j=1,2) of arbitrary positive orders and arguments. This algorithm consists of two parts: One of them combines the evaluation of the function H ν (1) (x) via Taylor expansions and via numerical computation of the Sommerfield integral along contours of steepest descents (the Debye contours); the other one computes H ν (1) (x) by means of the Debye asymptotic expansions.

The algorithm can be easily implemented for the evaluation of J ν (x), Y ν (x) and H ν (2) (x) making use of the well- known connection formulas between the three kinds of Bessel functions.

MSC:
33C10Bessel and Airy functions, cylinder functions, 0 F 1
65D20Computation of special functions, construction of tables