Values of the exponential integral

edit · history · discussion · files · long url · special functions special values

Numbers

normalisation

$x$

value

$\operatorname{Ei}(x)$

1.895117816355936755466520934331634269017060581732707591646228431882513834533804153548900710126138957

$\operatorname{Ei}(x)$

1/3:

-0.1580921089711557103135773062301122859227892527545194175342259120266969176086114508383033262467941875

$\operatorname{Ei}(x)$

1/2:

0.4542199048631735799205238126628023652814055543526420451628177866881528969266622561361796585714373140

$\operatorname{Ei}(x)$

4.954234356001890163379505130227035275518053562420042054527095254570707913298407190883096971852850163

$\operatorname{Ei}(x)$

9.933832570625416558008336019216765262990653022987582119693034065242496228332697443187187398902087851

$\operatorname{Ei}(x)$

19.63087447005622002264572027972383885313207975115315216815404620859056206761114761767084729515828299

$\operatorname{Ei}(x)$

40.18527535580317745509142179379586709541908739919593043418287525164264481392822057865779970191749916

$\operatorname{Ei}(x)$

85.98976214243920480358340030799069003929678516476342843823240236713215826636768251657012220573383910

$\operatorname{Ei}(x)$

440.3798995348382689974245966593933924469328441823334772514628449299946652759842114114326427664576163

$\operatorname{Ei}(x)$

$\log 2$:

1.045163780117492784844588889194613136522615578151201575832909144075013205210359530172717405626383356

$\operatorname{Ei}(x)$

$\log 3$:

2.163588594667191972876922367347721366542116212453868873985823001863092853305919390199866424856772226

$\operatorname{Ei}(x)$

$\log 5$:

3.634588310032651864392342706860811271370411550229815556665838870218151370805808813685376286938805576

$\operatorname{Ei}(x)$

$\log 7$:

4.757051765997606069292390575450131824861568757165315950696408010698927727109200529964652542349845415

$E_1(x)$

0.2193839343955202736771637754601216490310472934069082075779786130735686985591415447222102510351372500

$E_1(x)$

1/3:

0.8288877453485866361202069811634756826450010649093967332272735913382854921305272444761315598181019478

$E_1(x)$

1/2:

0.5597735947761608117467959393150852352268468903163535152482932191073398988302742688603417524733251125

$E_1(x)$

0.04890051070806111956723983522804952231449218496302311632732287371169292871415219127926896100745164177

$E_1(x)$

0.01304838109419703741250074582864502294847763408022360117699903114085916543429526306181307900231552789

$E_1(x)$

0.003779352409848906478874860132466414851716547042489580360734020313584606172362404699077971896569385492

$E_1(x)$

0.001148295591275325797330561969819722076266095470697866764845494043583439236859430913119905624993329877

$E_1(x)$

0.0003600824521626586592953941157717971888273892837201121793388199201625965852366091758674599327787750408

$E_1(x)$

0.00003766562284392490177255799595075272089048784609372541214111741941813534667453112306971822882358190202

Definition

For $x>0$ the exponential integral is the principal value $\operatorname{Ei}(x)=\operatorname{PV}\!\int_{-\infty}^x e^t\,\mathrm{d}t/t$, and $E_1(x)=\int_x^{\infty} e^{-t}\,\mathrm{d}t/t$ [4]. The two are one function in two normalisations, $E_1(x)=-\operatorname{Ei}(-x)$, and this table holds both at selected real $x>0$.

Parameters

normalisation

— normalisation

$x$

— argument ($x>0$)

Formulas

(1)

For $x>0$, $\operatorname{Ei}(x)=-\operatorname{PV}\!\int_{-x}^{\infty} e^{-t}\,\mathrm{d}t/t=\operatorname{PV}\!\int_{-\infty}^x e^t\,\mathrm{d}t/t$ and $\operatorname{Ei}(-x)=-\int_x^{\infty} e^{-t}\,\mathrm{d}t/t$ [4].

(2)

For $x>0$, $\operatorname{Ei}(x)=\gamma+\log x+\sum_{k=1}^{\infty}x^k/(k\,k!)$, where $\gamma$ is the Euler-Mascheroni constant [4].

(3)

For $x>0$, $E_1(x)=\int_x^{\infty} e^{-t}\,\mathrm{d}t/t=-\operatorname{Ei}(-x)$ [4].

(4)

For $x>0$, $\operatorname{Ei}(x)=\operatorname{Shi}(x)+\operatorname{Chi}(x)$ [4].

Comments

(5)

For positive real $x$, $E_1$ is the case $n=1$ of the generalized exponential integral $E_n(x)=\int_1^{\infty} e^{-xt}t^{-n}\,\mathrm{d}t$ [6].

(6)

The reciprocal subunit inputs and small positive integer endpoints are deliberately sparse reference examples for the exponential kernel in [4], not an exhaustive rational mesh or individually documented applications. The logarithms of the first four primes are selected through $\operatorname{Ei}(\log p)=\operatorname{li}(p)$, connecting the two integral normalisations. The unit reference retains $eE_1(1)=\int_0^\infty e^{-t}/(1+t)\,dt$, the Euler-Gompertz constant. The finite endpoints are reference-budget choices.

Programs

(P1)

Sage

import numberdb.sage as numberdb
from sage.rings.rational_field import QQ
from sage.rings.real_arb import RealBallField

RBF = RealBallField(numberdb.bits(100, losing=64))
x = RBF(QQ(1))
x.Ei()               # Ei(1)
-RBF(-QQ(1)).Ei()    # E_1(1) = -Ei(-1)

(P2)

Python

import mpmath

mpmath.mp.dps = 100
mpmath.ei(1)
mpmath.e1(1)

Links

[1]

Wikipedia: Exponential integral

[2]

Wikipedia: Logarithmic integral function

[3]

Wikipedia: Trigonometric integral

[4]

DLMF 6.2: Definitions and Basic Properties

[5]

DLMF 6.4: Analytic Continuation

[6]

DLMF 8.19: Generalized Exponential Integral

[7]

OEIS A091725: decimal expansion of $\operatorname{Ei}(1)$

[8]

OEIS A099285: decimal expansion of $E_1(1)$

Similar tables

Values of the logarithmic integral $\operatorname{li}(x)$ — $\operatorname{li}(x)=\operatorname{Ei}(\log x)$, so that table is this one after a change of variable

Values of the hyperbolic sine integral $\operatorname{Shi}(x)$ — $\operatorname{Ei}(x)=\operatorname{Shi}(x)+\operatorname{Chi}(x)$ (4)

Values of the hyperbolic cosine integral $\operatorname{Chi}(x)$ — the other half of $\operatorname{Ei}=\operatorname{Shi}+\operatorname{Chi}$

Wilbraham-Gibbs constant — stores $\operatorname{Si}(\pi)$ and its normalisations

Stieltjes constants — contains $\gamma_0$, the Euler-Mascheroni constant appearing in $\operatorname{Ei}$, $\operatorname{Ci}$ and $\operatorname{Chi}$

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it holds both normalisations at $x=1/3,1/2,1,2,3,4,5,6,8$, and $\operatorname{Ei}(\log p)$ for $p=2,3,5,7$; these bounded exploratory integral references preserve the unit anchors and connect to logarithmic-integral values)

How they were obtained:

All retained numeric strings, exact values and row annotations are unchanged. The standalone generate.py, balanced_values.py and selection.json reproduce the entire current selection. Each finite Arb enclosure at 733 bits must fit inside the interval claimed by the written last place; a second run uses 912 bits. New nonexact entries have 100 significant digits.

Every new entry has an independent mpmath quadrature, closed-form or inverse-function check at 150 and 200 decimal digits, with a nonzero known-integral control. Arguments are exact rationals or propagated balls from exact expressions; root arguments use strict sign-certified rational brackets. Lambert and saddle roots also pass endpoint sign tests on the written last-place interval. The measurements are in coverage-checks.json. Older attachments are retained for provenance; their previous enumerations are historical, and the active enumeration is selection.json.