On the LambertW function

The LambertW function is defined to be the multivalued inverse of the functionw →wew. It has many applications in pure and applied mathematics, some of which are briefly described here. We present a new discussion of the complex branches ofW, 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 containingW.

[1]  E. M. Wright.  Solution of the equation $ze^z = a$ , 1959 .

[2]  K. Knopp.  Theory of Functions , 1958 .

[3]  Keith O. Geddes,et al.  Algorithms for computer algebra , 1992 .

[4]  S. N. Afriat.  BOUNDS FOR THE CHARACTERISTIC VALUES OF MATRIX FUNCTIONS , 1951 .

[5]  E.-M. Lémeray.  Sur les racines de l’équation $x=a^x$. Racines imaginaires , 1896 .

[6]  D. A. Barry,et al.  A class of exact solutions for Richards' equation , 1993 .

[7]  Philippe Flajolet,et al.  Gaussian limiting distributions for the number of components in combinatorial structures , 1990, J. Comb. Theory, Ser. A.

[8]  H. Davis.  Introduction to Nonlinear Differential and Integral Equations , 1964 .

[9]  J. R. E. O’Malley.  Singular perturbation methods for ordinary differential equations , 1991 .

[10]  C. W. Borchardt.  Ueber eine der Interpolation entsprechende Darstellung der Eliminations-Resultante. , 1860 .

[11]  William H. Beyer,et al.  CRC standard mathematical tables , 1976 .

[12]  G. Pólya.  Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen , 1937 .

[13]  L. A. Segel,et al.  The Quasi-Steady-State Assumption: A Case Study in Perturbation , 1989, SIAM Rev..

[14]  J. Karamata,et al.  Sur Quelques Problemes Poses Par Ramanujan , 1960 .

[15]  Edward L. Reiss,et al.  A new asymptotic method for jump phenomena , 1980 .

[16]  Tony C. Scott,et al.  Resolution of a paradox in the calculation of exchange forces for H+2 , 1993 .

[17]  Robert M. Corless,et al.  What good are numerical simulations of chaotic dynamical systems , 1994 .

[18]  Edward M. Wright,et al.  The number of connected sparsely edged graphs , 1977, J. Graph Theory.

[19]  K. B. Ranger.  A complex variable integration technique for the two-dimensional Navier-Stokes equations , 1991 .

[20]  J. L. Coolidge,et al.  A Manual of Greek Mathematics , 1931 .

[21]  P. J. Rippon,et al.  A Note on Complex Iteration , 1985 .

[22]  Svante Janson,et al.  The Birth of the Giant Component , 1993, Random Struct. Algorithms.

[23]  Gaston H. Gonnet,et al.  Handbook Of Algorithms And Data Structures , 1984 .

[24]  G. Pólya,et al.  Problems and theorems in analysis , 1983 .

[25]  Luc Devroye,et al.  A note on the height of binary search trees , 1986, JACM.

[26]  H. P. Hudson,et al.  Squaring the circle , 1988, Nature.

[27]  F. D. Parker.  Integrals of Inverse Functions , 1955 .

[28]  E. M. Wright.  A non-linear difference-differential equation. , 1946 .

[29]  John Michael Robson,et al.  The Height of Binary Search Trees , 1979, Aust. Comput. J..

[30]  Tony C. Scott,et al.  The calculation of exchange forces: General results and specific models , 1993 .

[31]  D. Knuth,et al.  A recurrence related to trees , 1989 .

[32]  Robert M. Corless,et al.  Well … it isn't quite that simple , 1992, SIGS.

[33]  C. H. Anderson.  The linear differential-difference equation with constant coefficients , 1972 .

[34]  D. Sorensen,et al.  On the orthogonality of eigenvectors computed by divide-and-conquer techniques , 1991 .

[35]  Gaston H. Gonnet,et al.  Expected Length of the Longest Probe Sequence in Hash Code Searching , 1981, JACM.

[36]  B. Bollobás.  The evolution of random graphs , 1984 .

[37]  John David Anderson,et al.  Introduction to Flight , 1985 .

[38]  A. Iserles,et al.  The State of the art in numerical analysis : proceedings of the Joint IMA/SIAM Conference on the State of the Art in Numerical Analysis held at the University of Birmingham, 14-18 April 1986 , 1987 .

[39]  T. E. Hull,et al.  Comparing Numerical Methods for Ordinary Differential Equations , 1972 .

[40]  Robert M Corless,et al.  Sur l'inversion de yαey au moyen des nombres de Stirling associés , 1995 .

[41]  Richard Bellman,et al.  Differential-Difference Equations , 1967 .

[42]  N. D. Hayes.  Roots of the Transcendental Equation Associated with a Certain Difference‐Differential Equation , 1950 .

[43]  A. Rapoport,et al.  Connectivity of random nets , 1951 .

[44]  O. Skovgaard,et al.  Computation of Wave Heights Due to Refraction and Friction , 1975 .

[45]  F. Fritsch,et al.  Solution of the transcendental equation wew = x , 1973, CACM.

[46]  N. D. Bruijn.  Asymptotic methods in analysis , 1958 .

[47]  Samuel D. Conte,et al.  Elementary Numerical Analysis , 1980 .

[48]  Bruce W. Char,et al.  Maple V Library Reference Manual , 1992, Springer New York.

[49]  Charles Coulston Gillispie,et al.  Dictionary of scientific biography , 1970 .

[50]  Götz Alefeld,et al.  On the Convergence of Halley's Method , 1981 .

[51]  N. D. Hayes.  THE ROOTS OF THE EQUATION x=(cexp) n x AND THE CYCLES OF THE SUBSTITUTION (x/ce x ) , 1952 .

[52]  P. J. Rippon,et al.  Convergence of infinite exponentials , 1983 .

[53]  E.-M. Lémeray.  Racines de quelques équations transcendantes. Intégration d'une équation aux différences mêlées , 1897 .

[54]  H. Landau.  On some problems of random nets , 1952 .

[55]  T. Stieltjes.  Sur les racines de l'équationXn=0 , 1887 .