On the evaluation of Bessel functions (Q1317210)

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