3F2 returns NaNs

[christofbradly]

I have found several cases where _3F2 incorrectly returns NaN. For example:

julia> _₃F₂(-9.0, 16.5, 12.5, 29.0, -32.5, 1.0)
NaN

It may be similar to #73 . But I do not see any simple pattern in my cases. The suggestion in that thread to perturb the argument gives the correct results (compared to e.g. Mathematica).

All cases I have found NaNs:
(all evaluated at z=1.0):

-9.0, 16.5, 12.5, 29.0, -32.5, 1.0
-9.0, 24.5, 16.5, 41.0, -43.5, 1.0
-13.0, 4.5, 3.5, 8.0, -17.5, 1.0
-13.0, 5.5, 4.5, 10.0, -20.5, 1.0
-13.0, 6.5, 5.5, 12.0, -23.5, 1.0
-13.0, 7.5, 6.5, 14.0, -26.5, 1.0
-13.0, 8.5, 7.5, 16.0, -29.5, 1.0
-13.0, 9.5, 8.5, 18.0, -32.5, 1.0
-13.0, 10.5, 9.5, 20.0, -35.5, 1.0
-13.0, 11.5, 10.5, 22.0, -38.5, 1.0
-13.0, 12.5, 11.5, 24.0, -41.5, 1.0
-13.0, 13.5, 12.5, 26.0, -44.5, 1.0
-13.0, 14.5, 13.5, 28.0, -47.5, 1.0
-13.0, 15.5, 14.5, 30.0, -50.5, 1.0
-13.0, 16.5, 15.5, 32.0, -53.5, 1.0
-17.0, 7.5, 3.5, 11.0, -26.5, 1.0
-17.0, 6.5, 4.5, 11.0, -28.5, 1.0
-17.0, 12.5, 6.5, 19.0, -41.5, 1.0
-17.0, 11.5, 7.5, 19.0, -43.5, 1.0
-21.0, 3.5, 2.5, 6.0, -23.5, 1.0
-21.0, 4.5, 3.5, 8.0, -28.5, 1.0
-21.0, 5.5, 4.5, 10.0, -33.5, 1.0
-21.0, 6.5, 5.5, 12.0, -38.5, 1.0
-21.0, 10.5, 3.5, 14.0, -35.5, 1.0
-21.0, 9.5, 4.5, 14.0, -39.5, 1.0
-21.0, 7.5, 6.5, 14.0, -43.5, 1.0
-21.0, 8.5, 7.5, 16.0, -48.5, 1.0
-21.0, 9.5, 8.5, 18.0, -53.5, 1.0
-21.0, 16.5, 3.5, 20.0, -38.5, 1.0
-21.0, 10.5, 9.5, 20.0, -58.5, 1.0
-25.0, 6.5, 2.5, 9.0, -32.5, 1.0
-25.0, 10.5, 2.5, 13.0, -35.5, 1.0
-25.0, 9.5, 3.5, 13.0, -41.5, 1.0
-25.0, 13.5, 3.5, 17.0, -44.5, 1.0
-25.0, 12.5, 4.5, 17.0, -50.5, 1.0
-25.0, 10.5, 6.5, 17.0, -58.5, 1.0
-25.0, 9.5, 7.5, 17.0, -60.5, 1.0
-25.0, 16.5, 4.5, 21.0, -53.5, 1.0
-29.0, 3.5, 2.5, 6.0, -32.5, 1.0
-29.0, 4.5, 3.5, 8.0, -39.5, 1.0
-29.0, 5.5, 4.5, 10.0, -46.5, 1.0
-29.0, 6.5, 5.5, 12.0, -53.5, 1.0
-29.0, 7.5, 6.5, 14.0, -60.5, 1.0
-29.0, 16.5, 3.5, 20.0, -53.5, 1.0
-33.0, 4.5, 2.5, 7.0, -39.5, 1.0
-33.0, 3.5, 3.5, 7.0, -41.5, 1.0
-37.0, 3.5, 2.5, 6.0, -41.5, 1.0
-37.0, 5.5, 2.5, 8.0, -46.5, 1.0
-37.0, 4.5, 3.5, 8.0, -50.5, 1.0
-37.0, 5.5, 4.5, 10.0, -59.5, 1.0
-37.0, 11.5, 2.5, 14.0, -53.5, 1.0
-41.0, 6.5, 2.5, 9.0, -53.5, 1.0
-41.0, 5.5, 3.5, 9.0, -59.5, 1.0
-41.0, 4.5, 4.5, 9.0, -61.5, 1.0
-41.0, 10.5, 2.5, 13.0, -58.5, 1.0
-45.0, 3.5, 2.5, 6.0, -50.5, 1.0
-45.0, 4.5, 3.5, 8.0, -61.5, 1.0
-45.0, 7.5, 2.5, 10.0, -60.5, 1.0
-53.0, 3.5, 2.5, 6.0, -59.5, 1.0

[christofbradly]

I have also noticed this was not an issue in v0.3.20.

[MikaelSlevinsky]

Thank you for the bug report! To be brief, I do know how to fix it: it appears to be a pole/pole situation. I will try to get a fix pushed, but don't hold your breath! The epsilon perturbation is a good patch for the moment because the rational approximations are stable in the neighbourhood of a pole/pole (like the barycentric formula).

In case you're wondering how the pole/pole get simplified technically, it just means a careful analysis of the case where μhi is not 1/0. Here's how pole hopping takes place for the 0F0 case. eee6021