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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 $J\sb \nu(x)$, $Y\sb \nu(x)$ and $H\sb \nu\sp{(j)}(x)$ $(j=1,2)$ of arbitrary positive orders and arguments. This algorithm consists of two parts: One of them combines the evaluation of the function $H\sb \nu\sp{(1)}(x)$ via Taylor expansions and via numerical computation of the Sommerfield integral along contours of steepest descents (the Debye contours); the other one computes $H\sb \nu\sp{(1)} (x)$ by means of the Debye asymptotic expansions. The algorithm can be easily implemented for the evaluation of $J\sb \nu(x)$, $Y\sb \nu(x)$ and $H\sb \nu\sp{(2)} (x)$ making use of the well- known connection formulas between the three kinds of Bessel functions.

33C10Bessel and Airy functions, cylinder functions, ${}_0F_1$
65D20Computation of special functions, construction of tables
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