[Help-gsl] Need help with negative arguments to gsl_sf_ellint_Kcomp, complete elliptic integral of the first kind

[Jerry]

Hi list,

I'm having trouble understanding how to use gsl_sf_ellint_Kcomp, the complete 
elliptic integral of the first kind, K(k).

First, gsl_sf_ellint_Kcomp has a domain limited to -1 < k < +1 and raises an 
error for arguments outside that range.

Second, gsl_sf_ellint_Kcomp defines a function with even symmetry around k = 0.
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The corresponding Octave, Mathematica, and I presume Matlab functions are all 
non-symmetric and they all decay towards zero as the argument tends to from +1 
to -infinity. (Mathematica also returns a real result for arguments > 1.) 
Octave and Mathematica both reference Abramowitz & Stegun without 
qualification, whereas the GSL reference says that "Note that Abramowitz & 
Stegun define this function in terms of the parameter m = k^2."

For arguments between 0 and 1, taking the square root before passing to 
gsl_sf_ellint_Kcomp returns a result consistent with the other references 
herein (Octave, Mathematica). However, I don't know how to get results for 
arguments less than zero that are also consistent with those references. 
Abramowitz & Stegun provides a bunch of ways of handling various kinds of 
arguments but I can't find one that is suitable.

For what it's worth, this function arises in the probability density function 
of the sum of two random sine variables. For unit-amplitude sine RVs, the 
argument to gsl_sf_ellint_Kcomp is always between 0 and 1 so to proceed with 
programming that problem I don't really need to have the question herein 
answered, but I would like to know how to handle it in the future should it 
arise.

Thanks for any help.

Jerry