15.5 Exponential Integrals

The Exponential Integral and related functions are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, A&S Chapter 5.

Function: expintegral_e1 (z) ¶

The Exponential Integral E1(z) defined as

\[E_1(z) = \int_z^\infty {e^{-t} \over t} dt \]

with \(\left| \arg z \right| < \pi\) . (A&S eqn 5.1.1) and (DLMF 6.2E2)

Categories: Exponential Integrals · Special functions ·

Function: expintegral_ei (x) ¶

The Exponential Integral Ei(x) defined as

\[Ei(x) = - -\kern-10.5pt\int_{-x}^\infty {e^{-t} \over t} dt = -\kern-10.5pt\int_{-\infty}^x {e^{t} \over t} dt \]

with \(x\) real and \(x > 0\). (A&S eqn 5.1.2) and (DLMF 6.2E5)

Categories: Exponential Integrals · Special functions ·

Function: expintegral_li (x) ¶

The Exponential Integral li(x) defined as

\[li(x) = -\kern-10.5pt\int_0^x {dt \over \ln t} \]

with \(x\) real and \(x > 1\). (A&S eqn 5.1.3) and (DLMF 6.2E8)

Categories: Exponential Integrals · Special functions ·

Function: expintegral_e (n,z) ¶

The Exponential Integral En(z) (A&S eqn 5.1.4) defined as

\[E_n(z) = \int_1^\infty {e^{-zt} \over t^n} dt \]

with \({\rm Re}(z) > 1\) and \(n\) a non-negative integer.

Categories: Exponential Integrals · Special functions ·

Function: expintegral_si (z) ¶

The Exponential Integral Si(z) (A&S eqn 5.2.1) defined as

\[{\rm Si}(z) = \int_0^z {\sin t \over t} dt \]

Categories: Exponential Integrals · Special functions ·

Function: expintegral_ci (z) ¶

The Exponential Integral Ci(z) (A&S eqn 5.2.2) defined as

\[{\rm Ci}(z) = \gamma + \log z + \int_0^z {{\cos t - 1} \over t} dt \]

with \(|\arg z| < \pi\) .

Categories: Exponential Integrals · Special functions ·

Function: expintegral_shi (z) ¶

The Exponential Integral Shi(z) (A&S eqn 5.2.3) defined as

\[{\rm Shi}(z) = \int_0^z {\sinh t \over t} dt \]

Categories: Exponential Integrals · Special functions ·

Function: expintegral_chi (z) ¶

The Exponential Integral Chi(z) (A&S eqn 5.2.4) defined as

\[{\rm Chi}(z) = \gamma + \log z + \int_0^z {{\cosh t - 1} \over t} dt \]

with \(|\arg z| < \pi\) .

Categories: Exponential Integrals · Special functions ·

Option variable: expintrep ¶

Default value: false

Change the representation of one of the exponential integrals, expintegral_e(m, z), expintegral_e1, or expintegral_ei to an equivalent form if possible.

Possible values for expintrep are false, gamma_incomplete, expintegral_e1, expintegral_ei, expintegral_li, expintegral_trig, or expintegral_hyp.

false means that the representation is not changed. Other values indicate the representation is to be changed to use the function specified where expintegral_trig means expintegral_si, expintegral_ci, and expintegral_hyp means expintegral_shi or expintegral_chi.

Categories: Exponential Integrals ·

Option variable: expintexpand ¶

Default value: false

Expand expintegral_e(n,z) for half integral values in terms of erfc or erf and for positive integers in terms of expintegral_ei.

Categories: Exponential Integrals ·