Values of the Lambert $W$ function

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Numbers

$k$

$x$

$W_k(x)$

0.5671432904097838729999686622103555497538157871865125081351310792230457930866845666932194469617522946

comment: This is the omega constant.

$e$:

$-e^{-1}$:

-1

$-3e^{-3}$:

-0.1785606278779211065968086697055148046541182559268859077201423060587856954944826087543134346521603556

$-5e^{-5}$:

-0.03488576825572369630124086867710605594441501320274902718555385521953601204254702776172954933999039170

1/2:

0.3517337112491958260249093009299510651714642155171118040466438461099606107203387108968323038321915693

0.8526055020137254913464724146953174668984533001514035087721073946525150656742630448965773783502494847

1.049908894964039959988697070552897904589466943706341452932871583316649050444442957885678666822434674

1.202167873197042939212074165495153447501512521829625981739203590700634132981772677227826104976568377

1.326724665242200223635099297758079660128793554638047479789290393025342679920536226774469916608426789

1.432404775898300311234078007212058694786434608804302025655948496343399593259831116857638422299445651

1.524345204984144369122476060675779385408719418799609942927160045982360820247712017874947206364556560

1.605811996320177596033746190198001992299603560529966120743330224549154103132734177725394347125840549

-1/3:

-0.6190612867359451121523269940209222333014717772629693524598360744929373522550887346110469261882588407

-1/4:

-0.3574029561813889030688111040559047533165905550760120436276204485896714025961457962896168513444411851

-1/5:

-0.2591711018190737450566519502154067057135883397008937032218391275127128217957580326598838960236799743

-1/6:

-0.2044814493399155336177577545103568904881569994079249050071973880762179473713908151298328671141408253

-1/7:

-0.1691926038562251029660671049831814435964394192529518648529863026037438278908130553175907447244744749

-1/8:

-0.1444213531375097291689673973929506964945240188378136452921781603535255447194649839438887511551181045

$-2 e^{-2}$:

-0.4063757399599599076769581241248397582109975751811406350004954883039150151838120497672500723381559286

$-4 e^{-4}$:

-0.07930960512711365643910864738646377947437262879201546959882499420949435163329424256634339810516341095

$-6 e^{-6}$:

-0.01509877359740557476360693508389744385615494896221645326746163303613865451910837099612016904594982078

$-7 e^{-7}$:

-0.006424313271845135863437353893289741731049823619995592650398476692350343175210188083335263051309738152

$-8 e^{-8}$:

-0.002690932406490997127050829632963503405486452465341603069568204727042846197026901930002599529544357542

$-9 e^{-9}$:

-0.001111923924471259608643527479682127563067093251106781615489422495008894491118989126159111410781614648

$-10 e^{-10}$:

-0.0004542055534648269009335640043022358905214241658879230694768319567056556983048244878515492922120518054

$-11 e^{-11}$:

-0.0001837524705611312071657186192425549984709494333021315594573699900529337164424119516528906514331608313

$-12 e^{-12}$:

-0.00007373598503498141382737575627422032375420765934024858345170549963681131322747126380326314905200011589

$-7/2 e^{-7/2}$:

-0.1190533345266324634703480294809152116003548988177328758963847209861624105568718469364355340087055385

comment: This exponent occurs in the square-root spectral-coordinate examples in [2], Table 2.

$-9/2 e^{-9/2}$:

-0.05269539502410005022807518817354492840423662987864424611039652671621441575103413231924204044292165681

comment: This exponent occurs in the square-root spectral-coordinate examples in [2], Table 2.

-1

$-e^{-1}$:

-1

$-e^{-1/2}/2$:

-1.756431208626169676982737616609216326916416831701323711125894727048300478541051903533664750947250849

-1

-1/3:

-1.512134551657842473896739678072038704603650385135359454259285473998977160511574827324265488152779835

-1

-1/4:

-2.153292364110349649169099150092981375536206485319477695884511507721362584650649378945727909538027439

-1

-1/5:

-2.542641357773526424293806156661848290161474907529431767116934699793352535483158781587362627961593449

-1

-1/6:

-2.833147892049342142611674642343132564014684277147565123233727179328355819234045938602374889059888660

-1

-1/7:

-3.066421345069269412410724506976364909123882310233083793276489033138648112911014247276124956339615294

-1

-1/8:

-3.261685684576488776905662364308739731721145393347809522040218079880635146768556728409460458942611242

-1

$-e^{-1/3}/3$:

-2.237147027773716818043473694161468460613698137895320779212723151615279208778585076269230554542403598

-1

$-e^{-1/4}/4$:

-2.586662982263053881182850266714011154679504739557109039685225120116767568104385781543593285556392922

-1

$-e^{-1/5}/5$:

-2.860399058463684990432673112693838761316192793557399779133621108141380344526679670369701126317582048

-1

$-e^{-1/6}/6$:

-3.084967142449719257970888386464166126636728540528251276611161874911544556431050402081615770545932062

-1

$-e^{-1/7}/7$:

-3.275122593397562963976279603726446189620357947110791848662996770640576271501084726443634642452761881

-1

$-e^{-1/8}/8$:

-3.439877361786054930901651570191749329550314788552505601434533572470055998782613615635980490671720637

-1

$-e^{-1/9}/9$:

-3.585130588077465616563182098514399770335244538133531270488022218468028335312049789365507729917542753

-1

$-e^{-1/10}/10$:

-3.714950427087530629681888723572661435176889879499709780542263575885132076037722882450886931283670500

-1

$-e^{-1/11}/11$:

-3.832258743265606790043834591534744870950376447346138788282036513799100792760871363195435273993105873

-1

$-e^{-1/12}/12$:

-3.939223641782293046285700070987597722905930280183991576987839220379300398980977307059847013533139044

Definition

The Lambert $W$ function is the multivalued inverse of $w\mapsto we^w$ [4] [1]. This table gives selected real values $W_k(x)$ on the principal branch $k=0$ and the lower real branch $k=-1$.

Parameters

$k$

— branch of $W$ ($0$ is the principal branch $W_0$ and $-1$ is the lower real branch $W_{-1}$)

$x$

— argument ($x\in\mathbb{R}$; for branch $0$, $x\geq-e^{-1}$; for branch $-1$, $-e^{-1}\leq x<0$)

Formulas

(1)

$W_k(x)e^{W_k(x)}=x$ [4].

(2)

If $y\geq-1$, then $W_0(ye^y)=y$. If $y\leq-1$, then $W_{-1}(ye^y)=y$ [4].

(3)

The omega constant is $\Omega=W_0(1)$ and satisfies $e^{-\Omega}=\Omega$ [6] [8].

(4)

Let $h$, $c$ and $k_B$ denote Planck's constant, the speed of light and Boltzmann's constant, and let $T>0$ be absolute temperature. For the frequency and wavelength spectral-energy densities, the respective peaks satisfy $h\nu_{\max}/(k_BT)=3+W_0(-3e^{-3})$ and $hc/(\lambda_{\max}k_BT)=5+W_0(-5e^{-5})$ [2].

(5)

The unique positive minimizer of $\lambda/(1-e^{-\lambda})^2$ satisfies $e^\lambda-1=2\lambda$, hence $\lambda=-W_{-1}(-e^{-1/2}/2)-1/2$ [3]. The principal branch gives $\lambda=0$, outside the positive-minimizer domain.

Comments

(6)

The branch convention is that of [4]. On $x\geq0$ the principal branch is the nonnegative real solution. On $-e^{-1}<x<0$, the principal branch satisfies $-1<W_0(x)<0$, while the lower real branch satisfies $W_{-1}(x)<-1$. The other branches are complex and are not stored here.

(7)

Wolfram Language calls the same function ProductLog [7].

(8)

The exact inverse-normalization values are $W_0(0)=0$ and $W_0(e)=1$. The two real branches meet at their square-root branch point $x=-e^{-1}$, with $W_0(-e^{-1})=W_{-1}(-e^{-1})=-1$ [1].

(9)

The spectral family uses the nonzero maximum of $x^n/(e^x-1)$, whose location is $n+W_0(-ne^{-n})$; the linear-frequency and linear-wavelength cases are $n=3,5$, and the half-integer cases occur in [2]. The integer prefix through $12$ is a bounded reference extension, not a claim that each exponent has a separate standard physical convention. For $k$-uniform hypergraphs set $u=k-1$; differentiating the minimisation in [3], equation (1), gives $e^\lambda-1=u\lambda$, hence $\lambda=-W_{-1}(-e^{-1/u}/u)-1/u$. This Lambert form is our derivation. The cutoff $k=13$ is practical. The positive and negative simple-input slices are exploratory references for small-coefficient inverse equations; no individual application is asserted for each. Only $0,e,-1/e$ are used as exact inverse anchors; identities giving arbitrary rational outputs are not added.

(10)

For blackbody spectral energy per unit linear frequency and per unit linear wavelength, respectively, the dimensionless peak positions involve $W_0(-3e^{-3})$ and $W_0(-5e^{-5})$ [2], Section 2 and Table 2. These are two specified spectral-density conventions, not a coordinate-independent peak or an arbitrary grid of exponents. The principal branch gives the positive peak; the lower branch gives the excluded zero solution.

(11)

The $2$-core of a random $3$-uniform hypergraph governs clustering in random $3$-XOR-SAT [3], Sections 1.1 and 1.2. Here degree $2$ is selected by deleting variables occurring in at most one equation, and edge size $3$ is the first nontrivial XOR-SAT case. Differentiating the published core-threshold minimization gives the Lambert argument $-e^{-1/2}/2$. The lower branch, not the principal branch, gives its positive minimizer. This Lambert expression is derived from the cited minimization, not quoted from the paper.

Programs

(P1)

Sage

import numberdb.sage as numberdb
from sage.rings.complex_arb import ComplexBallField

CBF = ComplexBallField(numberdb.bits(100, losing=160))
CBF(1).lambert_w(0).real()
argument = -(CBF(-1) / 2).exp() / 2
argument.lambert_w(-1).real()

(P2)

Python

import mpmath as mp

mp.mp.dps = 100
mp.lambertw(1, 0)
argument = -mp.exp(-mp.mpf(1) / 2) / 2
mp.lambertw(argument, -1)

References

[1]

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey and D. E. Knuth, On the Lambert W Function, Advances in Computational Mathematics 5 (1996), 329-359. (doi)

[2]

Seán M. Stewart, Wien peaks and the Lambert W function, Revista Brasileira de Ensino de Física 33 (2011), 3308. (doi)

[3]

Dimitris Achlioptas and Michael Molloy, The solution space geometry of random linear equations, Random Structures & Algorithms 46 (2015), 197-231. (arXiv) (doi)

Links

[4]

DLMF 4.13: Lambert W-Function

[5]

Wikipedia: Lambert W function

[6]

Wikipedia: Omega constant

[7]

MathWorld: Lambert W-Function

[8]

OEIS A030178: decimal expansion of the omega constant

Similar tables

Euler's constant $e$ — the real branch point is at $x=-e^{-1}$, where both real branches have value $-1$, and the principal branch also satisfies $W_0(e)=1$

Minimisers_in_the_core_threshold_formula_of_random_k-uniform_hypergraphs — its minimizer for edge size $3$ and minimum degree $2$ is $-W_{-1}(-e^{-1/2}/2)-1/2$

Values_of_in_the_Dickman-de_Bruijn_estimate — its positive root at $u=2$ satisfies $\xi(2)=-W_{-1}(-e^{-1/2}/2)-1/2$

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it retains all earlier anchors and includes $W_0(-ne^{-n})$ for $2\leq n\leq12$ and $n=7/2,9/2$; the hypergraph lower-branch arguments for $3\leq k\leq13$; the exploratory inverse inputs $1/2,1,2,\ldots,8$ on $W_0$ and $-1/m$ for $3\leq m\leq8$ on both real branches)

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.