TheLambertWFunction - crowlogic/arb4j GitHub Wiki

The Lambert W Function

The Lambert W function, also known as the product logarithm, is a special function denoted by $$W(x)$$ that solves the equation:

$$z = x e^x$$

for $x$ in terms of $z$. In other words, if $W(z)$ is the Lambert W function, then:

$$W(z)e^{W(z)} = z$$

Some important properties of the Lambert W function include:

Branches: The Lambert W function has multiple branches. The principal branch, denoted as $W_0(z)$, is real-valued for all real $z$. The secondary branch, denoted as $W_{-1}(z)$, is real-valued for $z \in [-\frac{1}{e}, 0)$. There are also infinitely many complex branches.
Derivative: The derivative of the Lambert W function can be computed as:

$$W'(z) = \frac{W(z)}{z(1 + W(z))}$$
Real-valued domain: For the principal branch $W_0(z)$, the function is real-valued for all real $z$, and the range is $-\frac{1}{e} \leq W_0(z) \leq \infty$.
Asymptotic behavior: As $z \to \infty$, $W_0(z) \sim \log(z) - \log(\log(z))$, and as $z \to 0$, $W_0(z) \sim z$.
Inverse relationship: The Lambert W function is related to the exponential function through their inverse relationship:

$$x = W(z)e^{W(z)}$$

The Lambert W function has many applications including: