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

[Patrick Alken]

Follow-up Comment #1, bug #50343 (project gsl):

from maxgacode =at= gmail =dot= com

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] 

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