The numerical computation of Bessel functions of the first and second kind for integer orders and complex arguments

A rigorous algorithm for computing integer order Bessel functions of the first and second kind with complex arguments is discussed. The algorithm makes use of backward recurrence for computing Bessel functions of the first kind where applicable, and of Hankel´s asymptotic expansion for large arguments. The argument is limited only by the ability of the computer to handle large numbers. Orders up to a few thousand have been computed to 13-digit accuracy