— Function: double gsl_sf_expm1 (double x)
— Function: int gsl_sf_expm1_e (double x, gsl_sf_result * result)

These routines compute the quantity \exp(x)-1 using an algorithm that is accurate for small x.

— Function: double gsl_sf_exprel (double x)
— Function: int gsl_sf_exprel_e (double x, gsl_sf_result * result)

These routines compute the quantity (\exp(x)-1)/x using an algorithm that is accurate for small x. For small x the algorithm is based on the expansion (\exp(x)-1)/x = 1 + x/2 + x^2/(2*3) + x^3/(2*3*4) + \dots.

— Function: double gsl_sf_exprel_2 (double x)
— Function: int gsl_sf_exprel_2_e (double x, gsl_sf_result * result)

These routines compute the quantity 2(\exp(x)-1-x)/x^2 using an algorithm that is accurate for small x. For small x the algorithm is based on the expansion 2(\exp(x)-1-x)/x^2 = 1 + x/3 + x^2/(3*4) + x^3/(3*4*5) + \dots.

— Function: double gsl_sf_exprel_n (int n, double x)
— Function: int gsl_sf_exprel_n_e (int n, double x, gsl_sf_result * result)

These routines compute the N-relative exponential, which is the n-th generalization of the functions gsl_sf_exprel and gsl_sf_exprel2. The N-relative exponential is given by,

          exprel_N(x) = N!/x^N (\exp(x) - \sum_{k=0}^{N-1} x^k/k!)
                      = 1 + x/(N+1) + x^2/((N+1)(N+2)) + ...
                      = 1F1 (1,1+N,x)