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:

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

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, $W_{-1}$

WRealBranches.png


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