Information on the exponential integrals can be found in Abramowitz & Stegun, Chapter 5.  These functions are declared in the header file gsl_sf_expint.h.

Exponential Integral
--------------------

 -- Function: double gsl_sf_expint_E1 (double X)
 -- Function: int gsl_sf_expint_E1_e (double X, gsl_sf_result * RESULT)
     These routines compute the exponential integral E_1(x),

          E_1(x) := \Re \int_1^\infty dt \exp(-xt)/t.

 -- Function: double gsl_sf_expint_E2 (double X)
 -- Function: int gsl_sf_expint_E2_e (double X, gsl_sf_result * RESULT)
     These routines compute the second-order exponential integral E_2(x),

          E_2(x) := \Re \int_1^\infty dt \exp(-xt)/t^2.

 -- Function: double gsl_sf_expint_En (int N, double X)
 -- Function: int gsl_sf_expint_En_e (int N, double X, gsl_sf_result * RESULT)
     These routines compute the exponential integral E_n(x) of order n,

          E_n(x) := \Re \int_1^\infty dt \exp(-xt)/t^n.

Ei(x)
-----

 -- Function: double gsl_sf_expint_Ei (double X)
 -- Function: int gsl_sf_expint_Ei_e (double X, gsl_sf_result * RESULT)
     These routines compute the exponential integral Ei(x),

          Ei(x) := - PV(\int_{-x}^\infty dt \exp(-t)/t)

     where PV denotes the principal value of the integral.

Hyperbolic Integrals
--------------------

 -- Function: double gsl_sf_Shi (double X)
 -- Function: int gsl_sf_Shi_e (double X, gsl_sf_result * RESULT)
     These routines compute the integral Shi(x) = \int_0^x dt \sinh(t)/t.

 -- Function: double gsl_sf_Chi (double X)
 -- Function: int gsl_sf_Chi_e (double X, gsl_sf_result * RESULT)
     These routines compute the integral  Chi(x) := \Re[ \gamma_E + \log(x) + \int_0^x dt (\cosh[t]-1)/t] , where \gamma_E is the Euler constant (available as the macro M_EULER).

Ei_3(x)
-------

 -- Function: double gsl_sf_expint_3 (double X)
 -- Function: int gsl_sf_expint_3_e (double X, gsl_sf_result * RESULT)
     These routines compute the third-order exponential integral Ei_3(x) = \int_0^xdt \exp(-t^3) for $x \ge 0$ x >= 0.

Trigonometric Integrals
-----------------------

 -- Function: double gsl_sf_Si (const double X)
 -- Function: int gsl_sf_Si_e (double X, gsl_sf_result * RESULT)
     These routines compute the Sine integral Si(x) = \int_0^x dt \sin(t)/t.

 -- Function: double gsl_sf_Ci (const double X)
 -- Function: int gsl_sf_Ci_e (double X, gsl_sf_result * RESULT)
     These routines compute the Cosine integral Ci(x) = -\int_x^\infty dt \cos(t)/t for x > 0.

Arctangent Integral
-------------------

 -- Function: double gsl_sf_atanint (double X)
 -- Function: int gsl_sf_atanint_e (double X, gsl_sf_result * RESULT)
     These routines compute the Arctangent integral, which is defined as AtanInt(x) = \int_0^x dt \arctan(t)/t.