Hankel functions incorrect for large imaginary parts and standard precision

[paulmasson]

For example:

>>> hankel1(1,1+30j)
mpc(real='0.0', imag='0.0')
>>> hankel2(1,1-30j)
mpc(real='0.0', imag='0.0')

whereas Mathematica gives

In[1]:= HankelH1[1, 1. + 30 I]
Out[1]= -7.25495*10^-15 - 1.17346*10^-14 I
In[2]:= HankelH2[1, 1. - 30 I]
Out[2]= -7.25495*10^-15 + 1.17346*10^-14 I

Correct values can be found by increasing the precision, but this should be automatic as clearly Mathematica is doing.

The problem arises because the standard definitions of the functions involve two terms with many digits in common, and there need to be sufficient digits available when they are combined.

[fredrik-johansson]

Yeah, a simple fix is to use sum_accurately in the implementations of both Hankel functions. A better fix is to implement the appropriate asymptotic expansions of these functions.