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. - 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