Hi,
I have some code that I prototyped in Mathematica and am now writing
in C using GSL, that makes use of Mathieu functions. I have different
results between the two of them, and I cannot figure out whether this
is a bug in GSL, Mathematica or simply some misunderstanding from my
part.
I am using `MathieuC` function in latest Mathematica
(http://reference.wolfram.com/language/ref/MathieuC.html) which should
be the same function as `gsl_sf_mathieu_ce`
(https://www.gnu.org/software/gsl/manual/html_node/Angular-Mathieu-Functions.html#Angular-Mathieu-Functions)
except that the former one takes a single `a` argument being the
characteristic value whereas the GSL 2.3 implementation takes the
order `n` and the `q` parameter directly.
So, I guess,
```
N[MathieuC[MathieuCharacteristicA[0, -1], -1, 2*Pi/180]]
1.41071
```
should be equivalent to
```
gsl_sf_mathieu_ce(0, -1.0, 2.0 * M_PI / 180.0)
```
which gives a totally different value: 0.99751942347886335.
I tried to debug with different values, and the discrepancies between
Mathematica and GSL seems to appear only when the `q` parameter (-1.0
here) is negative. If I take 1.0 instead, I get values in agreement. I
tried to find yet another implementation to debug it, and found Scipy
(https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.mathieu_cem.html#scipy.special.mathieu_cem)
which relies on Fortran SPECFUN library apparently, and is in
agreement with GSL.
I am missing something? Thanks!