# Newton's method vs. Lambert W

## Main Question or Discussion Point

I understand the rational for using the Lambert W. function for solving equations such as $$x^x = z$$, where no derivative in terms of elementary functions exists for the expression. However, on the Wikipedia page about the Lambert W. function, an example is given with the equation $$2^t = 5t$$. In this case (since numerical evaluation of the W. function boils down to Newton's method in the end) is there any advantage to using the W. function instead of Newton's method directly to solve the equation, other than just getting it in an explicit formula for t? Thanks!

## Answers and Replies

Mute
Homework Helper
It depends. I'm not sure if Newton's method can get you the complex solutions, for example. There are always uses for the formal expressions. I recently used the Lambert-W function and its properties in the derivation of a distribution function. In particular, the principal branch W_0 has a series expansion about z = 0 which I used in an integral to get a closed form expression for something.

$$x^x = z$$

$$xlnx = lnz$$

$$\frac{x}{x} + lnx = 0$$

$$1 + lnx = 0$$

$$lnx = -1$$

$$e^{-1} = x$$
I'm pretty sure the derivative can be done in elementary functions. Solving for an equation like $$2^t = 5t$$ can't be done in elementary functions though.