| @cindex exponential integrals |
| @cindex integrals, exponential |
| |
| Information on the exponential integrals can be found in Abramowitz & |
| Stegun, Chapter 5. These functions are declared in the header file |
| @file{gsl_sf_expint.h}. |
| |
| @menu |
| * Exponential Integral:: |
| * Ei(x):: |
| * Hyperbolic Integrals:: |
| * Ei_3(x):: |
| * Trigonometric Integrals:: |
| * Arctangent Integral:: |
| @end menu |
| |
| @node Exponential Integral |
| @subsection Exponential Integral |
| @cindex E1(x), E2(x), Ei(x) |
| |
| @deftypefun double gsl_sf_expint_E1 (double @var{x}) |
| @deftypefunx int gsl_sf_expint_E1_e (double @var{x}, gsl_sf_result * @var{result}) |
| These routines compute the exponential integral @math{E_1(x)}, |
| @tex |
| \beforedisplay |
| $$ |
| E_1(x) := \Re \int_1^\infty dt \exp(-xt)/t. |
| $$ |
| \afterdisplay |
| @end tex |
| @ifinfo |
| |
| @example |
| E_1(x) := \Re \int_1^\infty dt \exp(-xt)/t. |
| @end example |
| |
| @end ifinfo |
| @noindent |
| @comment Domain: x != 0.0 |
| @comment Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW, GSL_EUNDRFLW |
| @end deftypefun |
| |
| @deftypefun double gsl_sf_expint_E2 (double @var{x}) |
| @deftypefunx int gsl_sf_expint_E2_e (double @var{x}, gsl_sf_result * @var{result}) |
| These routines compute the second-order exponential integral @math{E_2(x)}, |
| @tex |
| \beforedisplay |
| $$ |
| E_2(x) := \Re \int_1^\infty dt \exp(-xt)/t^2. |
| $$ |
| \afterdisplay |
| @end tex |
| @ifinfo |
| |
| @example |
| E_2(x) := \Re \int_1^\infty dt \exp(-xt)/t^2. |
| @end example |
| |
| @end ifinfo |
| @noindent |
| @comment Domain: x != 0.0 |
| @comment Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW, GSL_EUNDRFLW |
| @end deftypefun |
| |
| @node Ei(x) |
| @subsection Ei(x) |
| |
| @deftypefun double gsl_sf_expint_Ei (double @var{x}) |
| @deftypefunx int gsl_sf_expint_Ei_e (double @var{x}, gsl_sf_result * @var{result}) |
| These routines compute the exponential integral @math{Ei(x)}, |
| @tex |
| \beforedisplay |
| $$ |
| Ei(x) := - PV\left(\int_{-x}^\infty dt \exp(-t)/t\right) |
| $$ |
| \afterdisplay |
| @end tex |
| @ifinfo |
| |
| @example |
| Ei(x) := - PV(\int_@{-x@}^\infty dt \exp(-t)/t) |
| @end example |
| |
| @end ifinfo |
| @noindent |
| where @math{PV} denotes the principal value of the integral. |
| @comment Domain: x != 0.0 |
| @comment Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW, GSL_EUNDRFLW |
| @end deftypefun |
| |
| |
| @node Hyperbolic Integrals |
| @subsection Hyperbolic Integrals |
| @cindex hyperbolic integrals |
| @cindex Shi(x) |
| @cindex Chi(x) |
| |
| @deftypefun double gsl_sf_Shi (double @var{x}) |
| @deftypefunx int gsl_sf_Shi_e (double @var{x}, gsl_sf_result * @var{result}) |
| These routines compute the integral @math{Shi(x) = \int_0^x dt \sinh(t)/t}. |
| @comment Exceptional Return Values: GSL_EOVRFLW, GSL_EUNDRFLW |
| @end deftypefun |
| |
| |
| @deftypefun double gsl_sf_Chi (double @var{x}) |
| @deftypefunx int gsl_sf_Chi_e (double @var{x}, gsl_sf_result * @var{result}) |
| These routines compute the integral @math{ Chi(x) := \Re[ \gamma_E + \log(x) + \int_0^x dt (\cosh[t]-1)/t] }, where @math{\gamma_E} is the Euler constant (available as the macro @code{M_EULER}). |
| @comment Domain: x != 0.0 |
| @comment Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW, GSL_EUNDRFLW |
| @end deftypefun |
| |
| |
| @node Ei_3(x) |
| @subsection Ei_3(x) |
| |
| @deftypefun double gsl_sf_expint_3 (double @var{x}) |
| @deftypefunx int gsl_sf_expint_3_e (double @var{x}, gsl_sf_result * @var{result}) |
| These routines compute the third-order exponential integral |
| @math{Ei_3(x) = \int_0^xdt \exp(-t^3)} for @c{$x \ge 0$} |
| @math{x >= 0}. |
| @comment Exceptional Return Values: GSL_EDOM |
| @end deftypefun |
| |
| @node Trigonometric Integrals |
| @subsection Trigonometric Integrals |
| @cindex trigonometric integrals |
| @cindex Si(x) |
| @cindex Ci(x) |
| @deftypefun double gsl_sf_Si (const double @var{x}) |
| @deftypefunx int gsl_sf_Si_e (double @var{x}, gsl_sf_result * @var{result}) |
| These routines compute the Sine integral |
| @math{Si(x) = \int_0^x dt \sin(t)/t}. |
| @comment Exceptional Return Values: none |
| @end deftypefun |
| |
| |
| @deftypefun double gsl_sf_Ci (const double @var{x}) |
| @deftypefunx int gsl_sf_Ci_e (double @var{x}, gsl_sf_result * @var{result}) |
| These routines compute the Cosine integral @math{Ci(x) = -\int_x^\infty dt |
| \cos(t)/t} for @math{x > 0}. |
| @comment Domain: x > 0.0 |
| @comment Exceptional Return Values: GSL_EDOM |
| @end deftypefun |
| |
| |
| @node Arctangent Integral |
| @subsection Arctangent Integral |
| @cindex arctangent integral |
| @deftypefun double gsl_sf_atanint (double @var{x}) |
| @deftypefunx int gsl_sf_atanint_e (double @var{x}, gsl_sf_result * @var{result}) |
| These routines compute the Arctangent integral, which is defined as @math{AtanInt(x) = \int_0^x dt \arctan(t)/t}. |
| @comment Domain: |
| @comment Exceptional Return Values: |
| @end deftypefun |
| |