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Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Values at fixed points

Values at infinities

General characteristics

Domain and analyticity

W(z) is an analytical function of z which is defined over the whole complex z-plane.

Symmetries and periodicities

Mirror symmetry

Periodicity

No periodicity

Poles and essential singularities

The function W(z) does not have poles and essential singularities.

Branch points

Branch cuts

Series representations

Generalized power series

Expansions at z==0

Expansions at z==-(1/e)

Expansions at z==∞

Integral representations

On the real axis

Of the direct function

Dispersion relation

Limit representations

Differential equations

Ordinary nonlinear differential equations

Transformations

Transformations and argument simplifications

Products, sums, and powers of the direct function

Sums of the direct function

Related transformations

Identities

Functional identities

Complex characteristics

Real part

Imaginary part

Absolute value

Argument

Conjugate value

Differentiation

Low-order differentiation

Symbolic differentiation

Eric Weisstein and Oleg Marichev

Fractional integro-differentiation

Integration

Indefinite integration

Involving only one direct function

Involving one direct function and elementary functions

Definite integration

For the direct function itself

Operations

Limit operation

Representations through more general functions

Through other functions

Representations through equivalent functions

With inverse function

With related functions

Theorems

Periodic solutions of the two species Lotka-Volterra model

A closed form solution for periodic solutions of the two species Lotka-Volterra model x'(t)==α x(t) (1-y(t)),  y'(t)==β y(t) (1-x(t)) is given by  y(t)==-W(-(Subscript[c, 1]) ((x(t))^(β/α)) (e^(-(β/α)x(t)))) .

Solution of the bootstrap equation

A solution of the bootstrap equation z==2Ω(z)-(e^(Ω(z)))+1 , appearing for instance in the renormalization group equations, is given as Ω(z)== (z-2 W(- (e^((z-1)/2))/2)-1)/2.

The number of rooted trees

The number of rooted trees on n labeled points is [(z^n)]W(z).

History

– J. H. Lambert (1758)
– L. Euler (1764, 1779)

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