[Bug-gsl] [bug #50343] Different value for mathieu_ce in Mathematica and GSL

[Patrick Alken]

URL:
  <http://savannah.gnu.org/bugs/?50343>

                 Summary: Different value for mathieu_ce in Mathematica and
GSL
                 Project: GNU Scientific Library
            Submitted by: psa
            Submitted on: Fri 17 Feb 2017 10:16:59 PM UTC
                Category: Runtime error
                Severity: 3 - Normal
        Operating System: 
                  Status: None
             Assigned to: None
             Open/Closed: Open
                 Release: 
         Discussion Lock: Any

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Details:

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. 




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