# Definition:Lambert W Function

Jump to navigation
Jump to search

## Contents

## 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.

## Sources

- 1996: R.M. Corless, G.H. Gonnet, D.E.G Hare, D.J. Jeffrey and D.E. Knuth:
*On the Lambert $W$ Function*(*Adv. Comput. Math.***Vol. 5**: 329 – 359)