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:

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

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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 $W_{-1}: \hointr {-\dfrac 1 e} 0 \to \hointl \gets {-1}$ such that:

$x = \map {W_{-1} } x e^{\map {W_{-1} } x}$

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: pp. 329 – 359)

• Weisstein, Eric W. "Lambert W-Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LambertW-Function.html