Sat 18 Feb 2017 02:53:30 PM UTC, comment #1:
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, 20012002.)
>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]
