Lambert W-Function

Question

Is there a standard name for the inverse of the Lambert W-Function, in the manner that the name "exponential function" is the name for the inverse function of the logarithmic function.

Answer 1

To my knowledge, there is no specific name given to the inverses of any of the special functions, like the error function, the beta and $\Gamma$ functions, hypergeometric functions, etc., and Lambert's W function is no exception to the general rule.

Answer 2

Also, be careful when you take an inverse function of $W$ since $W$ has two branches. Probably, you meant the principal branch $W_0$ of it that assumes values from the range $[-1/e,\infty]$. Then, the corresponding inverse function of $W_0$ will be $f(x) = xe^x$ on a domain $x \in [-1, \infty]$.