Contents

  1. Preface

  2. Historical prelude

  3. About the author

  4. Symbols

  5. I The classical Lambert W function

    1. 1 Basic properties of W

      1. 1.1 Equations solvable in terms of W

        1. 1.1.1 Defining equation of W

        2. 1.1.2 Equations of the form xbx=a. The base b Lambert function

        3. 1.1.3 Equations of the form x+bx=a

        4. 1.1.4 Equations of the form x+logbx=a, and x·logbx=a

        5. 1.1.5 Equations of the form xx=a, and xy=yx

      2. 1.2 Equations satisfied by W

        1. 1.2.1 The exponential and logarithm of W

        2. 1.2.2 W at some functional arguments

        3. 1.2.3 Linear combination of two Lambert function values

      3. 1.3 Differential and integral formulas

        1. 1.3.1 The derivative of W

        2. 1.3.2 Higher order derivatives

        3. 1.3.3 The generating function and the coefficients of the pn(x) polynomials

        4. 1.3.4 Derivatives of W(ex)

        5. 1.3.5 The differential equation of W

        6. 1.3.6 Some integrals of W

        7. 1.3.7 The Mellin transform of W

        8. 1.3.8 The Laplace transform of W(et)

      4. 1.4 The Taylor series of W0

      5. 1.5 The number of real solutions of xex. The two real branches of W

      6. 1.6 Convexity and concavity of W0 and W−1

      7. 1.7 The Ω constant

        1. 1.7.1 Ω is transcendental

        2. 1.7.2 The Adamchik integral (in a generalized form)

        3. 1.7.3 A log-exponential integral

        4. 1.7.4 A log-trigonometric integral

      8. Further notes

    2. 2 The branch structure of the Lambert W function

      1. 2.1 The exponential function

        1. 2.1.1 The mapping properties of exp

      2. 2.2 The definition of the logarithm

      3. 2.3 Branch cuts, covering maps, and Riemann surfaces

        1. 2.3.1 The z -plane and the W -plane

        2. 2.3.2 Branch cut and branch point

      4. 2.4 Riemann surfaces

        1. 2.4.1 Motivation

        2. 2.4.2 Covering maps

        3. 2.4.3 The notion of a surface

        4. 2.4.4 The Riemann surface of the logarithm -- heuristic way

        5. 2.4.5 The Riemann surface of the logarithm -- exact way

        6. 2.4.6 The monodromy group of the Riemann surface of the logarithm

      5. 2.5 The branches of the Lambert function

        1. 2.5.1 The partition of the z -plane

        2. 2.5.2 The branch separating curves

        3. 2.5.3 The trisectrix of Hippias

        4. 2.5.4 The closure of the branches

        5. 2.5.5 The Riemann surface of the Lambert function

        6. 2.5.6 Monodromies

        7. 2.5.7 The fixed points of the Lambert function

      6. Further notes

    3. 3 Unwinding number and branch differences

      1. 3.1 Why does the complex version of many identities fail?

        1. 3.1.1 Non-identities regarding the logarithm

        2. 3.1.2 Non-identities regarding roots

      2. 3.2 The unwinding number

        1. 3.2.1 The definition of the unwinding number

        2. 3.2.2 The use of K(z)

      3. 3.3 Corrected identities for the Lambert function

        1. 3.3.1 Correcting (1.21)

        2. 3.3.2 Correcting (1.23)

      4. 3.4 The Wright ω function

        1. 3.4.1 Looking at equation x+logx=a more carefully

        2. 3.4.2 The definition and basic properties of ω

        3. 3.4.3 The mapping properties of ω

      5. 3.5 The branch difference function Mm,n

        1. 3.5.1 An equation for Mm,n(z)

        2. 3.5.2 An equation for M0,−1(z)

      6. Further notes

    4. 4 Numerical approximations

      1. 4.1 A concise series representation of Wk

        1. 4.1.1 A first approximation of W

        2. 4.1.2 The integral representation of v(z)

        3. 4.1.3 The ck,m constants

        4. 4.1.4 The asymptotic series representation for Wk

        5. 4.1.5 The analysis of the remainder term in the asymptotic series representation for Wk

      2. 4.2 Series expansions and Padé approximants around −1/e

        1. 4.2.1 The behavior of W−1 and W1 around z=−1/e

        2. 4.2.2 The notion of Padé approximant

        3. 4.2.3 The Padé approximant of W0 around zero

      3. 4.3 Continued fraction representations for W0

        1. 4.3.1 The notion of C-fractions

        2. 4.3.2 Correspondence between C-fractions and Taylor series

        3. 4.3.3 The C-fraction expansion of W0(x) -- via Hankel determinants

        4. 4.3.4 The C-fraction expansion of W0(x) -- via the qd -algorithm

        5. 4.3.5 Associated continued fractions

        6. 4.3.6 The analysis of the relative errors of C- and associated fractions

        7. 4.3.7 The asymptotics of the elements in the C-fractions and associated fractions

        8. 4.3.8 The limit of the elements in the C-fraction expansion of W0 and Van Vleck's theorem

      4. Further notes

  6. II Generalized Lambert functions

    1. 5 Generalizations of the Lambert function

      1. 5.1 The generalized Lambert function

        1. 5.1.1 The polynomial-exponential and rational-exponential type equations

        2. 5.1.2 The definition of the generalized Lambert function

        3. 5.1.3 Simple cases

      2. 5.2 Transformations of rational-exponential type equations

      3. 5.3 The (1,1) -type function and the r -Lambert function

      4. 5.4 A series representation for the (1,1) -type function

      5. Further notes

    2. 6 The r -Lambert function

      1. 6.1 Equations solvable with the aid of Wr

        1. 6.1.1 Some special values of Wr

        2. 6.1.2 The base change formula

        3. 6.1.3 An equation for the classical W function

      2. 6.2 The derivative and integral of Wr

        1. 6.2.1 The derivative of Wr

        2. 6.2.2 Higher order derivatives

        3. 6.2.3 Derivatives of Wr(ez)

        4. 6.2.4 The derivative of Wr with respect to r

        5. 6.2.5 The approximation of the r -Lambert function with the Lambert function

        6. 6.2.6 The integral of Wr and of its powers

      3. 6.3 The branch structure

        1. 6.3.1 The real values

        2. 6.3.2 Setting up the scene

        3. 6.3.3 The branch separating curves in the z -plane

          1. 6.3.3.1 The case when r>1/e2

          2. 6.3.3.2 The case when r=1/e2

          3. 6.3.3.3 The case when 0<r<1/e2

          4. 6.3.3.4 The case when r<0

        4. 6.3.4 More on the points zk*

      4. Further notes

  7. III Applications

    1. 7 Physical applications

      1. 7.1 Falling body in resistant media

        1. 7.1.1 Free fall in vacuum

        2. 7.1.2 Taking the resistance of the media into account

      2. 7.2 Paramagnets and magnetization

        1. 7.2.1 The Brillouin function

        2. 7.2.2 The inverse Langevin function

      3. 7.3 Kepler's equation and Laplace's limit constant

        1. 7.3.1 The Kepler equation and λ

        2. 7.3.2 λ in terms of the (1,1) -Lambert function

      4. 7.4 Thermodynamic properties of an ensemble in a quantum well

        1. 7.4.1 The quantum well

        2. 7.4.2 Thermodynamics and ensembles in a quantum well

      5. 7.5 Einstein-Johnson sidewall correction in channel flow models

      6. 7.6 Radial speed and mass loss rate of solar wind

        1. 7.6.1 The governing equation of the solar wind

        2. 7.6.2 The mass loss rate due to solar wind

      7. 7.7 Wien's displacement law

      8. 7.8 Electric field around semi-infinite parallel plates

      9. 7.9 The electric potential around a collision presheath in plasma

        1. 7.9.1 Sheath and presheath

        2. 7.9.2 The potential χ in terms of W

      10. 7.10 Delta-potential model for hydrogen molecule ion

      11. Further notes

    2. 8 Biology, Ecology, and Probability

      1. 8.1 Cell proliferation

        1. 8.1.1 The need for delay differential equations

        2. 8.1.2 The Baker-Bocharov-Paul-Rihan model

      2. 8.2 Random parasite and prey models

        1. 8.2.1 The Random Parasite Equation

        2. 8.2.2 The Random Predator Equation

      3. 8.3 Blooming of phytoplanktons

        1. 8.3.1 Sverdrup's approach

        2. 8.3.2 Biomass growth and critical depth

      4. 8.4 Enzyme kinetics

        1. 8.4.1 The Michaelis-Menten equation

        2. 8.4.2 Exact solution of the Michaelis-Menten equation

        3. 8.4.3 Dark adaptation of the human eye

      5. 8.5 The Lindley distribution

        1. 8.5.1 The definition of the Lindley distribution

        2. 8.5.2 Lindley and Lambert W

      6. 8.6 The maximum of standard normal random variables

        1. 8.6.1 Norming constants

        2. 8.6.2 The norming constants and W

      7. 8.7 Superstars and retweet graphs

        1. 8.7.1 Tweets, retweets, and superstars

        2. 8.7.2 The superstar model

      8. Further notes

    3. 9 Mathematical applications

      1. 9.1 The infinite power tower

        1. 9.1.1 Tetration

        2. 9.1.2 The power tower in terms of W

      2. 9.2 Counting trees

        1. 9.2.1 Networks, minimal connectedness, and labelings

        2. 9.2.2 Cayley's theorem, generating functions, and W

      3. 9.3 The asymptotics of the Bell numbers

        1. 9.3.1 Number of groupings -- The Bell numbers

        2. 9.3.2 The asymptotics of the Bell numbers with the aid of W

      4. 9.4 The associate space of an Orlicz space

        1. 9.4.1 Basics on Orlicz spaces

        2. 9.4.2 The Orlicz space dual of LΦ

      5. 9.5 Kapteyn series

        1. 9.5.1 The notion of Kapteyn series

        2. 9.5.2 The geometry of the Kapteyn domain and the r -Lambert function

        3. 9.5.3 The proofs

        4. 9.5.4 The visualization of pa, qa and the domain Da

      6. 9.6 The best constants for the estimate of Weierstrass canonical products

      7. 9.7 Delay Differential Equations

        1. 9.7.1 A simple example

        2. 9.7.2 Some other examples

      8. Further notes

  8. Bibliography

  9. Index