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