— Function: int gsl_integration_qagi (gsl_function * f, double epsabs, double epsrel, size_t limit, gsl_integration_workspace * workspace, double * result, double * abserr)

This function computes the integral of the function f over the infinite interval (-\infty,+\infty). The integral is mapped onto the semi-open interval (0,1] using the transformation x = (1-t)/t,

          \int_{-\infty}^{+\infty} dx f(x) =
               \int_0^1 dt (f((1-t)/t) + f((-1+t)/t))/t^2.

It is then integrated using the QAGS algorithm. The normal 21-point Gauss-Kronrod rule of QAGS is replaced by a 15-point rule, because the transformation can generate an integrable singularity at the origin. In this case a lower-order rule is more efficient.

— Function: int gsl_integration_qagiu (gsl_function * f, double a, double epsabs, double epsrel, size_t limit, gsl_integration_workspace * workspace, double * result, double * abserr)

This function computes the integral of the function f over the semi-infinite interval (a,+\infty). The integral is mapped onto the semi-open interval (0,1] using the transformation x = a + (1-t)/t,

          \int_{a}^{+\infty} dx f(x) =
               \int_0^1 dt f(a + (1-t)/t)/t^2

and then integrated using the QAGS algorithm.

— Function: int gsl_integration_qagil (gsl_function * f, double b, double epsabs, double epsrel, size_t limit, gsl_integration_workspace * workspace, double * result, double * abserr)

This function computes the integral of the function f over the semi-infinite interval (-\infty,b). The integral is mapped onto the semi-open interval (0,1] using the transformation x = b - (1-t)/t,

          \int_{-\infty}^{b} dx f(x) =
               \int_0^1 dt f(b - (1-t)/t)/t^2

and then integrated using the QAGS algorithm.