> What are the values returned by MathieuCharacteristicA[0, -1]

You can check this with Wolfram|Alpha (which is powered by Mathematica): http://www.wolframalpha.com/input/?i=MathieuCharacteristicA%5B0,+-1%5D

The result is

-0.45513860410741354823263318752888586691651984219666547103...

from phyks =at= phyks =dot= me

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.