# Lambert W function

Lambert’s $W$ function^{} is the inverse of the function $f:\u2102\to \u2102$ given by $f(x):=x{e}^{x}$. That is, $W(x)$ is the complex valued function that satisfies

$$W(x){e}^{W(x)}=x,$$ |

for all $x\in \u2102$. In practice the definition of $W(x)$ requires a branch cut, which is usually taken along the negative real axis. Lambert’s W function is sometimes also called product log function.

This function allow us to solve the functional equation

$$g{(x)}^{g(x)}=x$$ |

since

$$g(x)={e}^{W(\mathrm{ln}(x))}.$$ |

## 1 References

A site with good information on Lambert’s W function is Corless’ page http://kong.apmaths.uwo.ca/ rcorless/frames/PAPERS/LambertW/“On the Lambert W Function”

Title | Lambert W function |
---|---|

Canonical name | LambertWFunction |

Date of creation | 2013-03-22 12:40:48 |

Last modified on | 2013-03-22 12:40:48 |

Owner | drini (3) |

Last modified by | drini (3) |

Numerical id | 8 |

Author | drini (3) |

Entry type | Definition |

Classification | msc 33B30 |

Synonym | product log |