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

\[\begin{split}\operatorname{Ci}(x) = -\int_x^\infty \cos(t)/t dt \text{ for } x > 0.\end{split}\]

Arctangent Integral

gsl_sf_atanint(x)

This routine computes the Arctangent integral, which is defined as

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