# Definition:Lambert W Function

## Definition

Let $e^z: \C \to \C$ be the complex exponential function.

Then $\map W z$ is defined to be the multifunction that satisfies:

$\map W z e^{\map W z} = z$

## Principal Branch:Real Valued

The principal branch of the Lambert W function is the real function $W_0: \hointr {-\dfrac 1 e} \to \to \hointr {-1} \to$ such that:

$x = \map {W_0} x e^{\map {W_0} x}$

## Lower Branch

The lower branch of the Lambert W function is the real function $W_{-1}: \left[{-\dfrac 1 e \,.\,.\, 0}\right) \to \left({\gets \,.\,.\, -1}\right]$ such that:

$x = W_{-1} \left({x}\right) e^{W_{-1} \left({x}\right)}$

## Graph

### Real Valued Branches

There are two real-valued branches of the Lambert $W$ Function:

The real valued principal branch, $W_0$

The lower branch, $W_{-1}$

## Also known as

The Lambert W function is also called:

• Lambert-W function
• Lambert W-function
• Lambert's W function
• The $\Omega$ (omega) function
• The Product Log function

## Also see

• Results about Lambert W Function can be found here.

## Source of Name

This entry was named for Johann Heinrich Lambert.