Definition:Lambert W Function
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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