# Exponential Integrals¶

## Exponential Integral¶

gsl_sf_expint_E1(x)

This routine computes the exponential integral $$\operatorname{E_1}(x)$$,

$\operatorname{E_1}(x) := \operatorname{Re} \int_1^\infty \exp(-xt)/t dt.$
gsl_sf_expint_E2(x)

This routine computes the second-order exponential integral $$\operatorname{E_2}(x)$$,

$\operatorname{E_2(x)} := \operatorname{Re} \int_1^\infty \exp(-xt)/t^2 dt.$
gsl_sf_expint_En(n, x)

This routine computes the exponential integral $$\operatorname{E_n}(x)$$ of order $$n$$,

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

## Ei(x)¶

gsl_sf_expint_Ei(x)

These routines compute the exponential integral $$\operatorname{Ei}(x)$$,

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

where $$PV$$ denotes the principal value of the integral.

## Hyperbolic Integrals¶

gsl_sf_Shi(x)

This routine computes the integral

$\operatorname{Shi}(x) = \int_0^x \sinh(t)/t dt.$
gsl_sf_Chi(x)

This routine computes the integral

$\operatorname{Chi}(x) := \operatorname{Re}[ \gamma_E + \log(x) + \int_0^x (\cosh(t)-1)/t dt],$

where $$\gamma_E$$ is the Euler constant.

## Ei_3(x)¶

gsl_sf_expint_3(x)

This routine computes the third-order exponential integral

$\operatorname{Ei_3}(x) = \int_0^x \exp(-t^3) dt \text{ for } x \geq 0.$

## Trigonometric Integrals¶

gsl_sf_Si(x)

This routine computes the Sine integral

$\operatorname{Si}(x) = \int_0^x \sin(t)/t dt.$
gsl_sf_Ci(x)

This routine computes the Cosine integral

$\operatorname{Ci}(x) = -\int_x^\infty \cos(t)/t dt \text{ for } x > 0.$

## Arctangent Integral¶

gsl_sf_atanint(x)

This routine computes the Arctangent integral, which is defined as

$\operatorname{AtanInt}(x) = \int_0^x \arctan(t)/t dt.$