## On the Lambert W Function (1996)

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Venue: | ADVANCES IN COMPUTATIONAL MATHEMATICS |

Citations: | 124 - 6 self |

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@INPROCEEDINGS{Corless96onthe,

author = {R. M. Corless and G. H. Gonnet and D. E. G. Hare and D. J. Jeffrey and D. E. Knuth},

title = {On the Lambert W Function},

booktitle = {ADVANCES IN COMPUTATIONAL MATHEMATICS},

year = {1996},

pages = {329--359},

publisher = {}

}

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### Citations

2017 | Choosability in graphs
- Erdős, Rubin, et al.
- 1979
(Show Context)
Citation Context ...nches of T and W . The epidemic or reachability problem is closely related to the size of the `giant component' in a random graph, a phenomenonsrst demonstrated in a famous paper by Erd}os and Renyi =-=[26]-=-. When a graph on n vertices has m = 12n edges chosen at random, for > 1, it almost surely has a connected component with approximatelysn vertices, wheresis given by (2.34). The study of the emerge... |

1087 |
Mathematical methods of classical mechanics
- Arnold
- 1989
(Show Context)
Citation Context ...at the coefficients of the asymptotic series for W were known in closed form, and gave us the reference to Comtet's work. J. Borwein pointed out a connection between (3.11) and the Legendre Transform =-=[4]-=-. The jet-fuel problem was communicated to us by Michael Kamprath, with commentary by Harald Hanche-Olsen. We thank G.-C. Rota and D. Loeb for discussions, and we thank Prof. D. E. Gerber for his help... |

257 |
Algorithms for Computer Algebra
- Geddes, Czapor, et al.
- 1992
(Show Context)
Citation Context ...inally, note that this technique allows the Risch algorithm to be applied to determine whether integrals containing W are elementary or not. For an introduction to the Risch algorithm see for example =-=[31]-=-. 4. Branches and Asymptotics We have seen that W has two branches on the real line. We have also seen, in the delay equations example, that complex values for W are required. Thus, to extend On the L... |

223 | Theory of Functions - Titchmarsh - 1968 |

185 |
Differential-Difference Equations
- Bellman, Cooke
- 1963
(Show Context)
Citation Context ...s0, and hence in terms of W . Solution of linear constant-coefficient delay equations Perhaps the most signicant use of theW function is in the solution of linear constantcoefficient delay equations =-=[8,69]-=-. Many of the complex-variable properties of this function (and generalizations of it) were proved by workers in thisseld, motivated by the appearance of W in the solution of simple delay equations su... |

111 |
Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen
- Pólya
- 1937
(Show Context)
Citation Context ... labelled points. The exponential generating function T (x) = ∑ tnx n=n! satises the functional equation T (x) = x+ xT (x) + xT (x)2=2! + = xeT (x), so T (x)eT (x) = x and T (x) = W (x). In =-=[57]-=-, Polya used this approach and Lagrange inversion to deduce that tn = n n1, a formula that had previously been proved in other ways [10,13,67]. If we put U(x) = T (x) 12T (x)2 ; (2:1) then one can ... |

100 |
Introduction to Non-Linear Differential and Integral Equations
- Davis
- 1962
(Show Context)
Citation Context ...snd an implicit analytic phase plane solution of the Volterra equations dx dt = ax(1 y); dy dt = cy(1 x) ; (2:27) essentially in terms of the W function (see equations (11) and (12) on page 104 of =-=[22]-=-). These equations, with a = 2 and c = 1, appear as problem B1 of the DETEST test suite for numerical methods for integration of ordinary differential equations [40]. The analytic solution is a closed... |

91 |
Maple V language reference manual
- Char, Geddes, et al.
- 1991
(Show Context)
Citation Context ...lgebra system Maple has had an arbitrary precision implementation of this same realvalued branch of W for many years, and since Release 2 has had an arbitrary precision implementation of all branches =-=[14]-=-. The purposes of this paper are to collect existing results on this function for convenient reference, and to present some new results on the asymptotics, complex analysis, numerical analysis, and sy... |

91 | Problems and Theorems in Analysis - Pólya, Szegö - 1970 |

84 | A note on the height of binary search trees
- Devroye
- 1986
(Show Context)
Citation Context ...s, when n = m and < 1, is approximately T (m log2 )= log 1 for large m. Another, quite different, application to information retrieval concerns the expected height of random binary search trees =-=[23,60]-=-. Let binary search trees with n nodes be constructed by standard insertions from a random permutation of 1; : : : ; n; let hn be a random variable giving the height of such trees. Devroye proved in [... |

83 |
Expected length of the longest probe sequence in hash code searching
- Gonnet
- 1981
(Show Context)
Citation Context ...known as hashing with uniform probing: each of n items is mapped into a random permutation (p1; : : : pm) of f1; : : : ;mg and stored in thesrst cell pj that is currently unoccupied. Gonnet proved in =-=[34]-=- that the expected maximum number of probes (the maximum j) over all n items, when n = m and < 1, is approximately T (m log2 )= log 1 for large m. Another, quite different, application to inform... |

65 |
Singular Perturbation Methods for Ordinary Dierential Equations
- O’MALLEY
- 1991
(Show Context)
Citation Context ... ; (2:10) A Ae �if ;1 A<0. A ;2 W 2 ;1 Once wisknown,thethrustspecicfuelconsumptionfollowsfrom c=;log w. Solution of a model combustion problem Theproblem dy dt = y2 (1 ; y)� y(0)=" >0 (2:11) isusedin=-=[54,59]-=-toexploreperturbationmethods.Weshowherethatanexplicitanalytic solutionispossible,intermsof W,andthusalltheperturbationresultsin[54]canbe simplytestedbycomparisonwiththeexactsolution.Themodelproblemiss... |

65 | Finite operator calculus - Rota, Kahaner, et al. - 1973 |

63 |
The birth of the giant component. Random Structures and Algorithms
- Janson, Knuth, et al.
- 1993
(Show Context)
Citation Context ...s on the right-hand side: log x = v + 21 2! v2 + 32 3! v3 + 43 4! v4 + 54 5! v5 + etc: (1:4) This series, which can be seen to converge for jvj < 1=e, denes a function T (v) called the tree function =-=[41]-=-. It equals W (v), where W (z) is dened to be the function satisfying W (z)eW (z) = z : (1:5) This paper discusses both W and T , concentrating on W . The two functions are used in many application... |

56 | TITCHMARSH, “The theory of functions - C - 1939 |

54 | A History of Greek Mathematics - Heath - 1921 |

50 |
Gaussian limiting distributions for the number of components in combinatorial structures
- Flajolet, Soria
- 1990
(Show Context)
Citation Context ...e polynomial of order n (see [47]) and is generated by 1 (1 T (z))y = ∑ n0 tn(y) zn n! : (2:3) One application of these functions is to derive the limiting distribution of cycles in random mappings =-=[29]-=-. Chaotic maps of the unit interval usingsoating-point arithmetic can be studied in this way; an elementary discussion that looks only at the expected length of the longest cycle can be found in [18].... |

46 |
The quasi steady-state assumption: a case study in perturbation
- Segel, Slemrod
- 1989
(Show Context)
Citation Context ...ansion itself is done in terms of the W function. In [54] the Michaelis{Menten model of enzyme kinetics is solved with a perturbation technique. A similar model, with a better scaling, is examined in =-=[61]-=-. The outer solution is taken to be of the form s() = s0() + "s1() + : : : and c() = c0() + "c1() + : : :, and the leading order terms s0 and c0 are found to satisfy c0 = ( + 1)s0 s0 + 1 (2:14... |

45 |
Connectivity of Random Nets
- Solomonoff, Rapoport
- 1951
(Show Context)
Citation Context ... the infected person, the total number of infected people will be approximatelysn for large n, wheres= 1 es: (2:33) This formula, derived heuristically forsxed integer by Solomonoff and Rapoport =-=[65]-=-, then proved rigorously by Landau [50], holds also when is an expected value (notsxed for all individuals, and not necessarily an integer). Since (2.33) can be written e = (1s)e( 1) ; we hav... |

43 |
Comparing numerical methods for ordinary differential equations
- Hull, Enright, et al.
- 1972
(Show Context)
Citation Context ...tions (11) and (12) on page 104 of [22]). These equations, with a = 2 and c = 1, appear as problem B1 of the DETEST test suite for numerical methods for integration of ordinary differential equations =-=[40]-=-. The analytic solution is a closed loop in the phase plane. If the upper branch is y+ and the lower y, then y+ = W1 ( Cxc=aecx=a ) ; y = W0 ( Cxc=aecx=a ) ; (2:28) where C is an arbitrary co... |

39 |
Branch cuts for complex elementary functions. In The State of the Art m Numerical Analyszs
- KAHAN
- 1987
(Show Context)
Citation Context ...rly dashed lines in Figure 2, with closure at B. (CCC) around the branch point, which is a mnemonic principle that gives some uniformity to choices for the branch cuts of all the elementary functions =-=[44]-=-. Here, this convention distinguishes between two possibilities, namely the choice of attaching the image of the boundary in the p-plane to the top or to the bottom of the branch cut in the z-plane. T... |

38 | A recurrence related to trees
- Knuth, Pittel
- 1989
(Show Context)
Citation Context ... (x) [41, 72]. The number of mappings from f1; 2; : : : ; ng into itself having exactly k component cycles is the coefficient of yk in tn(y), where tn(y) is called the tree polynomial of order n (see =-=[47]-=-) and is generated by 1 (1 T (z))y = ∑ n0 tn(y) zn n! : (2:3) One application of these functions is to derive the limiting distribution of cycles in random mappings [29]. Chaotic maps of the unit in... |

37 | Photometria, sive de Mensura et Gradibus Luminis, Colorum et Umbrae - LAMBERT - 1760 |

34 |
On the orthogonality of eigenvectors computed by divide-and-conquer techniques
- Sorensen, Tang
- 1991
(Show Context)
Citation Context ...d only be computed to (n 1)dk1 digits, since thesrst dk1 digits will not be affected by this correction term, and the sum will then be correct to ndk1 digits. This analysis is similar to that in =-=[66]-=-, and we remark that the residual in the Newton-like method must be carefully computed to ensure the correction term is accurate. To estimate the cost of such a scheme, suppose convergence begins with... |

33 |
A non-linear difference-differential equation
- Wright
- 1955
(Show Context)
Citation Context ... it is the logarithm of a special case (s= = 1) of Lambert's series (1.2). Fortuitously, the letter W has additional signicance because of the pioneering work on many aspects of W by E. M. Wright =-=[69,70,71,72]-=-. In [14], the function is also called the Omega function. If x is real, then for 1=e x < 0 there are two possible real values of W (x) (see Figure 1). We denote the branch satisfying 1 W (x) by ... |

31 |
The height of binary search trees
- Robson
- 1979
(Show Context)
Citation Context ...s, when n = m and < 1, is approximately T (m log2 )= log 1 for large m. Another, quite different, application to information retrieval concerns the expected height of random binary search trees =-=[23,60]-=-. Let binary search trees with n nodes be constructed by standard insertions from a random permutation of 1; : : : ; n; let hn be a random variable giving the height of such trees. Devroye proved in [... |

27 |
Roots of the transcendental equation associated with a certain differential difference equation
- Hayes
- 1950
(Show Context)
Citation Context ...27], Euler made brief mention of the complex roots of x = ax when a is real, but thesrst person to explain how all values Wk(x) could be calculated for real x was apparently Lemeray [51,52]. Then in =-=[36]-=- Hayes showed how tosnd all the values Wk(x) when x is complex, and how to bound their real part. Wright made further studies, reported in [70], and then wrote a comprehensive paper [71] containing a ... |

26 |
Theory of Functions of a Complex Variable
- Carathéodory
- 1958
(Show Context)
Citation Context ...ction. In fact, it is probably the simplest function that exhibits both algebraic and logarithmic singularities. It also provides a simple example of the application of the Lagrange inversion theorem =-=[12]-=-. The asymptotic analysis of the W function might protably be used in a later course. 3. Calculus The principal branch of W is analytic at 0. This follows from the Lagrange inversion theorem (see e.g... |

18 |
private communication
- Coppersmith
(Show Context)
Citation Context ...n be reverted to give W (z) = 1∑ ℓ=0 ℓp ℓ = 1 + p 13p2 + 1172p3 + : (4:22) This series converges for jpj < p2. It can be computed to any desired order from the following recurrence relations =-=[17]-=-: k = k 1 k + 1 (k2 2 + k2 4 ) k 2 k1 k + 1 ; (4:23) k = k1∑ j=2 jk+1j ; 0 = 2; 1 = 1 ; (4:24) where 0 = 1 and 1 = 1. This relation, which follows from 2pW = ( p2 2 1 ) d(1 ... |

18 |
A theorem on trees", Quarterly
- Cayley
(Show Context)
Citation Context ...o be the function satisfying W (z)eW (z) = z : (1:5) This paper discusses both W and T , concentrating on W . The two functions are used in many applications: for example, in the enumeration of trees =-=[10, 13, 25, 41, 67]-=-; in the calculation of water-wave heights [64]; and in problems considered by Polya and Szego [56, Problem III.209, p. 146]. Wright used the complex branches ofW , and roots of more general exponen... |

17 | Handbook of Algorithms and Data Structures Addison-Wesley - Gonnet - 1984 |

16 | de Bruijn, Asymptotic Methods - G - 1961 |

16 |
Ueber eine der Interpolation entsprechende Darstellung der Eliminations-Resultante
- Borchardt
(Show Context)
Citation Context ...o be the function satisfying W (z)eW (z) = z : (1:5) This paper discusses both W and T , concentrating on W . The two functions are used in many applications: for example, in the enumeration of trees =-=[10, 13, 25, 41, 67]-=-; in the calculation of water-wave heights [64]; and in problems considered by Polya and Szego [56, Problem III.209, p. 146]. Wright used the complex branches ofW , and roots of more general exponen... |

13 |
Exponentials reiterated
- Knoebel
- 1981
(Show Context)
Citation Context ...on converges, it converges to h(z) = T (log z) log z = W ( log z) log z ; (2:5) as can be seen on solving h(z) = zh(z) for h(z) by taking logarithms. This immediately answers the question posed in =-=[46]-=- about the analytic continuation of h(z). Euler observed in [27] that the equation g = zz g sometimes has real roots g that are not roots of h = zh. A complete analysis of such questions, considering ... |

13 |
1783) De serie Lambertina Plurimisque eius insignibus proprietatibus. Acta Acad
- Euler
- 1921
(Show Context)
Citation Context ... W . 1. Introduction In 1758, Lambert solved the trinomial equation x = q+xm by giving a series development for x in powers of q. Later, he extended the series to give powers of x as well [48,49]. In =-=[28]-=-, Euler transformed Lambert's equation into the more symmetrical form x xs= (s)vx+s(1:1) by substituting xsfor x and setting m = sand q = ( s)v. Euler's version of Lambert's series solution ... |

12 |
A class of exact solutions for Richards’ equation
- Barry, Parlange, et al.
- 1993
(Show Context)
Citation Context ... signicant for small enough R. Similarity solution for the Richards equation Recent work uses both real branches ofW to give a new exact solution for the Richards equation for water movement in soil =-=[7]-=-. By a similarity transformation, the Richards equation for the moisture tensions, d ds@s@t = @ @z [ K( ) @s@z K( ) ] ; (2:24) is reduced, in a special case, to the ordinary differential equation ... |

11 |
Algorithm 443: Solution of the transcendental equation wew = x
- Fritsch, Shafer, et al.
- 1973
(Show Context)
Citation Context ...Polya and Szego [56, Problem III.209, p. 146]. Wright used the complex branches ofW , and roots of more general exponential polynomials, to solve linear constantcoefficient delay equations [69]. In =-=[30]-=-, Fritsch, Shafer and Crowley presented an algorithm for thesxed-precision computation of one branch of W (x) for x > 0. The computer algebra system Maple has had an arbitrary precision implementation... |

9 |
Observationes variae in mathesin puram
- Lambert
- 1758
(Show Context)
Citation Context ...s containing W . 1. Introduction In 1758, Lambert solved the trinomial equation x = q+xm by giving a series development for x in powers of q. Later, he extended the series to give powers of x as well =-=[48,49]-=-. In [28], Euler transformed Lambert's equation into the more symmetrical form x xs= (s)vx+s(1:1) by substituting xsfor x and setting m = sand q = ( s)v. Euler's version of Lambert's series ... |

8 |
Random Structures and Algorithms
- Luczak
- 1990
(Show Context)
Citation Context ...ft-handsideanda niceseriesontheright-handside: log x=v+ 21 2! v2 + 32 3! v3 + 43 4! v4 + 54 5! v5 +etc: (1:4) Thisseries,whichcanbeseentoconvergefor jvj <1=e,denesafunction T(v)calledthe tree function=-=[41]-=-.Itequals ;W(;v),where W(z)isdenedtobethefunctionsatisfying W(z)e W (z) = z: (1:5) Thispaperdiscussesboth Wand T,concentratingon W. Thetwofunctionsareusedinmanyapplications:forexample,intheenumeration... |

8 |
On the convergence of Halley's method
- Alefeld
- 1981
(Show Context)
Citation Context ... k and complex z. Taking full advantage of the features of iterative rootnders outlined above, we compared the efficiency of three methods, namely, (1) Newton's method, (2) Halley's method (see e.g. =-=[1]-=-), which is a third-order method, and (3) the fourth-order method described in [30] (as published, this last method evaluates only the principal branch of W at positive real arguments, but it easily e... |

8 |
De formulis exponentialibus replicatis, Leonhardi Euleri
- Euler
- 1777
(Show Context)
Citation Context ...z) log z ; (2:5) as can be seen on solving h(z) = zh(z) for h(z) by taking logarithms. This immediately answers the question posed in [46] about the analytic continuation of h(z). Euler observed in =-=[27]-=- that the equation g = zz g sometimes has real roots g that are not roots of h = zh. A complete analysis of such questions, considering also the complex roots, involves the T function, as shown by Hay... |

8 |
Sur quelques problèmes posés par Ramanujan
- Karamata
- 1960
(Show Context)
Citation Context ...lier when we said that the expansion at the branch point 1=e was complicated by the choice of locations for the branch cuts. The relation between W1 and W0 near the branch point was investigated in =-=[45]-=- by Karamata, who studied the coefficients cn in the power series = + 23 2 + 49 3 + 44135 4 + = ∑ n1 cn n ; (4:26) being the solution to (1 + )e = (1 )e ; = +O(2) : (4:27) Th... |

7 |
Introduction to nonlinear dierential and integral equations
- Davis
- 1962
(Show Context)
Citation Context ...giverisetophysicallymeaningfulsolutions.Ifweuse W0the solutioncorrespondsto capillary rise,whileifinsteadweuseW;1thesolutioncanbe interpretedas in ltration. Volterra equations for population growth In=-=[22,pp102{109]-=-,wendanimplicitanalyticphaseplanesolutionoftheVolterra equations dx dy = ax(1 ; y)�= ;cy(1 ; x) � (2:27) dt dt essentiallyintermsofthe Wfunction(seeequations(11)and(12)onpage104of[22]). Theseequations... |

7 |
What good are numerical simulations of chaotic dynamical systems
- Corless
- 1994
(Show Context)
Citation Context ... [29]. Chaotic maps of the unit interval usingsoating-point arithmetic can be studied in this way; an elementary discussion that looks only at the expected length of the longest cycle can be found in =-=[18]-=-. Iterated exponentiation The problem of iterated exponentiation is the evaluation of h(z) = zz zz ; (2:4) whenever it makes sense. Euler was thesrst to prove that this iteration converges for rea... |

7 |
On the change of systems of independent variables
- Sylvester
(Show Context)
Citation Context ...o be the function satisfying W (z)eW (z) = z : (1:5) This paper discusses both W and T , concentrating on W . The two functions are used in many applications: for example, in the enumeration of trees =-=[10, 13, 25, 41, 67]-=-; in the calculation of water-wave heights [64]; and in problems considered by Polya and Szego [56, Problem III.209, p. 146]. Wright used the complex branches ofW , and roots of more general exponen... |

6 | Well ... it isn’t quite that simple - Corless, Jeffrey - 1992 |

6 |
The linear difference-differential equation with constant coefficients
- Wright
- 1949
(Show Context)
Citation Context ...dered by Polya and Szego [56, Problem III.209, p. 146]. Wright used the complex branches ofW , and roots of more general exponential polynomials, to solve linear constantcoefficient delay equations =-=[69]-=-. In [30], Fritsch, Shafer and Crowley presented an algorithm for thesxed-precision computation of one branch of W (x) for x > 0. The computer algebra system Maple has had an arbitrary precision imple... |

5 |
It Isn't Quite That Simple
- Corless, Jeffrey, et al.
- 1992
(Show Context)
Citation Context ...l points. The equation above is a typical example, for Maple V Release 3 will return W (x)=(x(1 + W (x))) when asked to differentiate W , and hence is able to compute W ′0(0) = 1 only as a limit. See =-=[20]-=- for further discussion On the Lambert W function 340 of the handling of special cases (the so-called specialization problem) by computer algebra systems. Taking further derivatives, we can see by ind... |

5 | Squaring the Circle - Hobson - 1953 |

5 | Solution of the Equation ze z =a - Wright - 1959 |