Re: [Help-gsl] Different value for mathieu_ce in Mathematica and GSL

[maxgacode]

Il 17/02/2017 23:17, Patrick Alken ha scritto:

> > > 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.


Looking at Abramovitz and Stegun I found the following power serie for 
Ce(0,q,z) ( for small |q| ).

Ce(0,q,z) = ( 1/sqrt(2) ) * [ 1 - q * cos(2 z)/2 + q^2 * ((cos(4 z)/32) 
- 1/16) +........


for  q= -1 , z = 2 pi / 180

Ce(0,q,z) =~ 1.04 + ....

That is not proving anything but my guess is that GSL implementation 
agrees with Abramovitz and Stegun.

Moreover Scilab (using the Mathieu Toolbox from R.Coisson & G. Vernizzi, 
Parma University, 2001-2002.)

-->mathieu_ang_ce(0,-1, 2 * %pi / 180 ,1)
 ans  =

    0.9975194

again in agreement with GSL, Specfun and Abramovitz.

The Wolfram site says

"For nonzero q, the Mathieu functions are only periodic in z for certain 
values of a. Such characteristic values are given by the Wolfram 
Language functions MathieuCharacteristicA[r, q] and 
MathieuCharacteristicB[r, q] with r an integer or rational number. These 
values are often denoted a_r and b_r. In general, both a_r and b_r are 
multivalued functions with very complicated branch cut structures. 
Unfortunately,

there is no general agreement on how to define the branch cuts.

As a result, the Wolfram Language's implementation simply picks a 
convenient sheet. "


What are the values returned by

MathieuCharacteristicA[0, -1]

Hope this helps

Max