The Product Logarithm

The Lambert W function, often called the product logarithm, solves equations of the form $w{e}^w=x$. In other words, $W\left(x\right)$ is the number $w$ such that $w{e}^w$ equals $x$. This special function appears in combinatorics, delay differential equations, and anywhere an unknown variable occurs both inside and outside an exponential.

Branches and Domain

The equation $w{e}^w=x$ generally has two real solutions when $x$ lies between $-\frac{1}{e}$ and $0$. These are denoted $W_0$ and $W_{-}$, representing the principal and lower branches. For $x$ greater than or equal to $0$, only the principal branch exists. This calculator focuses on that branch, ensuring a real solution for all $x$ above $-\frac{1}{e}$.

Newton Iteration

The Lambert W function lacks a simple closed form, so numerical methods are typically used. One approach is Newton's method, which refines guesses by iterating

$w_{\mathrm{n+1}}=w_n-\frac{w_n{e}^{w_n}-x}{e^{w_n}\left(w_n+1\right)}$

We start with an initial guess like $w=\mathrm{\backslash ln}\left(x\right)$ when $x$ is positive. Repeated iterations quickly converge, granting a solution that solves $w{e}^w=x$ to high precision.

Why It Matters

The Lambert W function unravels problems that defy elementary algebra. For instance, the solution to $ye$^y=a can be written as $y=W\left(a\right)$. This makes $W$ valuable in combinatorics for analyzing the growth of trees or networks, and in delay differential equations describing feedback systems.

Worked Example

If $x=1$, we seek $w$ satisfying $w{e}^w=1$. Newton's method with an initial guess of $0.5$ converges to $0.567143$. You can verify that $0.567143e^{0.567143}$ is indeed very close to $1$.

Using This Tool

Type a value for $x$ above and press "Compute". The script performs Newton iterations until the change falls below a small tolerance. Results are rounded to six decimal places for readability. Try values like $x=0.1$, $x=1$, or $x=-\frac{1}{e}$ to explore the behavior near the branch point.