Lambert W Function

The lambert W function, denoted as $W(x)$, is a special function that is defined as the inverse of the function $f(x)=x{e}^x$. In other words, if $y=W(x)$, then it satisfies the equation:

$$W(x)=\sum _{n=1}^{\infty }\frac{(-n{)}^{n-1}}{n!}{x}^n$$

This is only one way to define the Lambert W function, and there are other equivalent definitions as well.

However, the problem is that the Lambert W function is not defined for all real numbers. It is only defined for certain values of $x$, specifically for $x\ge -\frac{1}{e}$, where $e$ is the base of the natural logarithm.

Because of this, it cannot technically be called a function in the strictest sense, as it does not have a unique output for every input. Instead, it is a multi-valued function, meaning that for some inputs, there are multiple outputs.

Properties

A neat property of the Lambert W function is that it is the inverse of the function $f(x)=x{e}^x$. This means that if $y=W(x)$, then $x=y{e}^y$

Example

Take a number $z$ and use it as the input to the function $W(z)$. This returns a constant $W(z)$(duh). If you take this number and multiply it by $e$ raised to the power of that number, you get back $z$:

$$\begin{array}{rl}z& \to W(z)\\ W(z)& \times e^{W(z)}=z\end{array}$$

This property is useful in many applications, such as solving equations involving exponentials and logarithms. Namely, it allows us to solve those equations of the form:

$$t^t=z$$