Lambert W function ( $W$ )

The Lambert W function is the multivalued inverse of

w \mapsto w e^{w}

, so

W (z) e^{W (z)} = z

. Its principal real branch satisfies

W (z) = lo g z - lo g lo g z + o (1)

z \to + \infty

.

Table of contents
- Real branches of Lambert W Lambert W function

Real branches of Lambert W ( $W_{0}, W_{- 1}$ )

On

- e^{- 1} < z < 0

, the Lambert W function has two real branches:

- 1 < W_{0} (z) < 0

and

W_{- 1} (z) < - 1

. The derivative of

w e^{w}

is

e^{w} (1 + w)

, so this function decreases on

(- \infty, - 1)

and increases on

(- 1, \infty)

, proving the two-branch structure. As

u \to \infty

, series reversion gives

W_{0} (- e^{- u}) = - e^{- u} - e^{- 2 u} + O (e^{- 3 u}), W_{- 1} (- e^{- u}) = - u - lo g u + O (lo g u / u) .

Consequently the two positive roots of

x e^{1/ x} = e^{u}

are

- 1/ W_{0} (- e^{- u})

and

- 1/ W_{- 1} (- e^{- u})

.

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Lambert W function by

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The Lambert W function, often denoted as \( W(x) \), is a special function that is defined as the inverse of the function \( f(W) = W e^W \). In other words, if \( W = W(x) \), then: \[ x = W e^W \] This means that the Lambert W function gives solutions \( W \) for equation \( x = W e^W \) for various values of \( x \).

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