×

zbMATH — the first resource for mathematics

On the Lambert \(w\) function. (English) Zbl 0863.65008
The tree function \(T\) defined by series \(T(v)=v+{2\over 2!}v^2+{3^2\over 3!}v^3+{4^3\over 4!}v^4+\dots\) converges for \(|v|< {1\over e}\). It equals \(-w(-v)\), where \(w(z)\) is defined to be the function satisfying \(w(z)e^{w(z)}=z\). This paper discusses both \(w\) and \(T\), concentrating on \(w\). The authors present a new discussion of the complex branches for \(w\), an asymptotic expansion valid for all branches, an efficient numerical procedure for evaluating the function to arbitrary precision, and a method for the symbolic integration of expressions containing \(w\).
Reviewer: R.S.Dahiya (Ames)

MSC:
65E05 General theory of numerical methods in complex analysis (potential theory, etc.)
30E10 Approximation in the complex plane
65D20 Computation of special functions and constants, construction of tables
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] G. Alefeld, On the convergence of Halley’s method, Amer. Math. Monthly 88 (1981) 530–536. · Zbl 0486.65035
[2] American National Standards Institute/Institute of Electrical and Electronic Engineers:IEEE Standard for Binary Floating-Point Arithmetic, ANSI/IEEE Std 754–1985, New York (1985).
[3] J.D. Anderson,Introduction to Flight, 3rd ed. (McGraw-Hill, New York, 1989).
[4] V.I. Arnold,Mathematical Methods for Classical Mechanics (Springer-Verlag, 1978). · Zbl 0386.70001
[5] I.N. Baker and P.J. Rippon, Convergence of infinite exponentials, Ann. Acad. Sci. Fenn. Ser. AI Math. 8 (1983) 179–186. · Zbl 0503.30022
[6] I.N. Baker and P.J. Rippon, A note on complex iteration, Amer. Math. Monthly 92 (1985) 501–504. · Zbl 0605.30027
[7] D.A. Barry, J.-Y. Parlange, G.C. Sander and M. Sivaplan, A class of exact solutions for Richards’ equation, J. Hydrology 142 (1993) 29–46.
[8] R.E. Bellman and K.L. Cooke,Differential-Difference Equations (Academic Press, 1963). · Zbl 0105.06402
[9] W.H. Beyer (ed.),CRC Standard Mathematical Tables, 28th ed. (1987).
[10] C.W. Borchardt, Ueber eine der Interpolation entsprechende Darstellung der Eliminations-Resultante, J. reine angewandte Math. 57 (1860) 111–121. · ERAM 057.1508cj
[11] N.G. de Bruijn,Asymptotic Methods in Analysis (North-Holland, 1961). · Zbl 0109.03502
[12] C. Carathéodory,Theory of Functions of a Complex Variable (Chelsea, 1954).
[13] A. Cayley, A theorem on trees, Quarterly Journal of Mathematics, Oxford Series 23 (1889) 376–378. · JFM 21.0687.01
[14] B.W. Char, K.O. Geddes, G.H. Gonnet, B.L. Leong, M.B. Monagan and S.M. Watt,The Maple V Language Reference Manual (Springer-Verlag, 1991). · Zbl 0758.68038
[15] L. Comtet,Advanced Combinatorics (Reidel, 1974).
[16] S.D. Conte and C. de Boor,Elementary Numerical Analysis, 3rd ed. (McGraw-Hill, 1980). · Zbl 0496.65001
[17] D. Coppersmith, private communication.
[18] R.M. Corless, What good are numerical simulations of chaotic dynamical systems?, Computers Math. Applic. 28 (1994) 107–121. · Zbl 0813.65101
[19] R.M. Corless,Essential Maple (Springer-Verlag, 1994).
[20] R.M. Corless and D.J. Jeffrey, Well, It Isn’t Quite That Simple, SIGSAM Bulletin 26 (3) (1992) 2–6.
[21] R.M. Corless, G.H. Gonnet, D.E.G. Hare, and D.J. Jeffrey, Lambert’sW function in Maple, The Maple Technical Newsletter 9 (1993) 12–22.
[22] H.T. Davis,Introduction to Nonlinear Differential and Integral Equations (Dover, 1962). · Zbl 0106.28904
[23] L. Devroye, A note on the height of binary search trees, J. ACM 33 (1986) 489–498. · Zbl 0741.05062
[24] O. Dziobek, Eine Formel der Substitutionstheorie, Sitzungsberichte der Berliner Mathematischen Gesellschaft 17 (1917) 64–67. · JFM 46.0106.03
[25] G. Eisenstein, Entwicklung von {\(\alpha\)}{\(\alpha\)}, J. reine angewandte Math. 28 (1844) 49–52. · ERAM 028.0815cj
[26] P. Erdös and A. Rényi, On the evolution of random graphs, Magyar Tud. Akad. Mat. Kut. Int. Közl. 5 (1960) 17–61. Reprinted in P. Erdös,The Art of Counting (1973) pp. 574–618, and inSelected Papers of Alfréd Rényi (1976), pp. 482–525. · Zbl 0103.16301
[27] L. Euler, De formulis exponentialibus replicatis, Leonhardi Euleri Opera Omnia, Ser. 1, Opera Mathematica 15 (1927) [original date 1777] 268–297.
[28] L. Euler, De serie Lambertina plurimisque eius insignibus proprietatibus, Leonhardi Euleri Opera Omnia, Ser. 1, Opera Mathematica 6 (1921) [orig. date 1779] 350–369.
[29] P. Flajolet and M. Soria, Gaussian Limiting Distributions for the Number of Components in Combinatorial Structures, J. Combinatorial Theory, Series A 53 (1990) 165–182. · Zbl 0691.60035
[30] F.N. Fritsch, R.E. Shafer, and W.P. Crowley, Algorithm 443: Solution of the transcendental equationwe w =x, Comm. ACM 16 (1973) 123–124.
[31] K.O. Geddes, S.R. Czapor and G. Labahn,Algorithms for Computer Algebra (Kluwer Academic Publishers, 1992). · Zbl 0805.68072
[32] Ch.C. Gillispie (ed.),Dictionary of Scientific Biography (Scribners, N.Y., 1973).
[33] G.H. Gonnet,Handbook of Algorithms and Data Structures (Addison-Wesley, 1984). · Zbl 0665.68001
[34] G.H. Gonnet, Expected length of the longest probe sequence in hash code searching, J. ACM 28 (1981) 289–304. · Zbl 0456.68067
[35] R.L. Graham, D.E. Knuth and O. Patashnik,Concrete Mathematics (Addison-Wesley, 1994).
[36] N.D. Hayes, Roots of the transcendental equation associated with a certain difference-differential equation, J. Lond. Math. Soc. 25 (1950) 226–232. · Zbl 0038.24102
[37] N.D. Hayes, The roots of the equationx=(c exp) n x and the cycles of the substitution (x|ce x ), Q. J. Math. 2 (1952) 81–90. · Zbl 0046.30601
[38] T.L. Heath,A Manual of Greek Mathematics (Dover, 1963). · Zbl 0113.00105
[39] E.W. Hobson,Squaring the Circle (Chelsea, 1953). · Zbl 0052.16301
[40] T.E. Hull, W.H. Enright, B.M. Fellen and A.E. Sedgwick, Comparing numerical methods for ordinary differential equations, SIAM J. Numer. Anal. 9 (1972) 603–637. · Zbl 0221.65115
[41] S. Janson, D.E. Knuth, T. Łuczak and B. Pittel, The birth of the giant component,Random Structures and Algorithms 4 (1993) 233–358. · Zbl 0795.05127
[42] D.J. Jeffrey, R.M. Corless, D.E.G. Hare and D.E. Knuth, Sur l’inversion dey a e y e y au moyen de nombres de Stirling associés, C. R. Acad. Sc. Paris, Série I, 320, 1449–1452. · Zbl 0846.05002
[43] D.J. Jeffrey, R.M. Corless and D.E.G. Hare, Unwinding the branches of the LambertW function, The Mathematical Scientist, 21, 1–7. · Zbl 0852.33001
[44] W. Kahan, Branch cuts for complex elementary functions, InThe State of the Art in Numerical Analysis: Proc. Joint IMA/SIAM Conf. on the State of the Art in Numerical Analysis, University of Birmingham, April 14–18, 1986, eds. M.J.D. Powell and A. Iserles (Oxford University Press, 1986).
[45] J. Karamata, Sur quelques problèmes posés par Ramanujan, J. Indian Math. Soc. 24 (1960) 343–365. · Zbl 0217.32101
[46] R.A. Knoebel, Exponentials reiterated, Amer. Math. Monthly 88 (1981) 235–252. · Zbl 0493.26007
[47] D.E. Knuth and B. Pittel, A recurrence related to trees, Proc. Amer. Math. Soc. 105 (1989) 335–349. · Zbl 0672.41024
[48] J.H. Lambert, Observationes variae in mathesin puram, Acta Helvetica, physico-mathematico-anatomico-botanico-medica 3, Basel (1758) 128–168.
[49] J.H. Lambert, Observations Analytiques, inNouveaux mémoires de l’Académie royale des sciences et belles-lettres, Berlin (1772) vol. 1, for 1770.
[50] H.G. Landau, On some problems of random nets, Bull. Mathematical Biophysics 14 (1952) 203–212.
[51] E.M. Lémeray, Sur les raciens de l’equationx=a x , Nouvelles Annales de Mathématiques (3) 15 (1896) 548–556.
[52] E.M. Lémeray, Sur les racines de l’equationx=a x . Racines imaginaires, Nouvelles Annales de Mathématiques (3) 16 (1897) 54–61.
[53] E.M. Lémeray, Racines de quelques équations transcendantes. Intégration d’une équation aux différences mèlées. Racines imaginaires, Nouvelles Annales de Mathématiques (3) 16 (1897) 540–546.
[54] R.E. O’Malley, Jr.,Singular Perturbation Methods for Ordinary Differential Equations (Springer-Verlag Applied Mathematical Sciences 89, 1991).
[55] F.D. Parker, Integrals of inverse functions, Amer. Math. Monthly 62 (1955) 439–440.
[56] G. Pólya and G. Szegö,Problems and Theorems in Analysis (Springer-Verlag, 1972). · Zbl 0236.00003
[57] G. Pólya, Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen, Acta Mathematica 68 (1937) 145–254. English translation by Dorothee Aeppli in G. Pólya and R.C. Read,Combinatorial Enumeration of Groups, Graphs, and Chemical Compounds (Springer-Verlag, 1987). · Zbl 0017.23202
[58] K.B. Ranger, A complex variable integration technique for the two-dimensional Navier-Stokes equations, Q. Applied Math. 49 (1991) 555–562. · Zbl 0731.76017
[59] E.L. Reiss, A new asymptotic method for jump phenomena, SIAM J. Appl. Math. 39 (1980) 440–455. · Zbl 0444.34054
[60] J.M. Robson, The height of binary search trees, Aust. Comput. J. 11 (1979) 151–153.
[61] L.A. Segel and M. Slemrod, The quasi-steady-state assumption: a case study in perturbation, SIAM Review 31 (1989) 446–477. · Zbl 0679.34066
[62] T.C. Scott, J.F. Babb, A. Dalgarno and J.D. Morgan III, Resolution of a paradox in the calculation of exchange forces forH 2 + , Chem. Phys. Lett. 203 (1993) 175–183.
[63] T.C. Scott, J.F. Babb, A. Dalgarno and J.D. Morgan III, The calculation of exchange forces: general results and specific models, J. Chem. Phys. 99 (1993) 2841–2854.
[64] O. Skovgaard, I.G. Jonsson, and J.A. Bertelsen, Computation of wave heights due to refraction and friction, J. Waterways Harbours and Coastal Engineering Division, February, 1975, pp. 15–32.
[65] R. Solomonoff and A. Rapoport, Connectivity of random nets, Bull. Math. Biophysics 13 (1951) 107–117.
[66] D.C. Sorensen and Ping Tak Peter Tang, On the orthogonality of eigenvectors computed by divide-and-conquer techniques, SIAM J. Numer. Anal. 28 no. 6 (December 1991) 1752–1775. · Zbl 0743.65039
[67] J.J. Sylvester, On the change of systems of independent variables, Q. J. Pure and Applied Math. 1 (1857) 42–56.
[68] E.C. Titchmarsh,Theory of Functions, 2nd ed. (Oxford, 1939). · Zbl 0022.14602
[69] E.M. Wright, The linear difference-differential equation with constant coefficients, Proc. Royal Soc. Edinburgh, A 62 (1949) 387–393. · Zbl 0033.12002
[70] E.M. Wright, A non-linear difference-differential equation, J. für reine und angewandte Mathematik 194 (1955) 66–87. · Zbl 0064.34203
[71] E.M. Wright, Solution of the equationze z =a, Proc. Roy. Soc. Edinburgh A 65 (1959) 193–203. · Zbl 0093.27204
[72] E.M. Wright, The number of connected sparsely edged graphs, J. Graph Theory 1 (1977) 317–330. · Zbl 0363.05040
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.